Last updated: May, 11 2020

Tetrad is a suite of software for the discovery, estimation, and simulation of causal models. Some of the functions that you can perform with Tetrad include, but are not limited to:

- Loading an existing data set, restricting potential models using your a priori causal knowledge, and searching for a model that explains it using one of Tetrad’s causal search algorithms
- Loading an existing causal graph and existing data set, and estimating a parameterized model from them
- Creating a new causal graph, parameterizing a model from it, and simulating data from that model

Tetrad allows for numerous types of data, graph, and model to be input and output, and some functions may be restricted based on what types of data or graph the user inputs. Other functions may simply not perform as well on certain types of data.

All analysis in Tetrad is performed graphically using a box paradigm, found in a sidebar to the left of the workspace. A box either houses an object such as a graph or a dataset, or performs an operation such as a search or an estimation. Some boxes require input from other boxes in order to work. Complex operations are performed by stringing chains of boxes together in the workspace. For instance, to simulate data, you would input a graph box into a parametric model box, the PM box into an instantiated model box, and finally the IM box into a simulation box.

In order to use a box, click on it in the sidebar, then click inside the workspace. This creates an empty box, which you can instantiated by double-clicking. Most boxes have multiple options available on instantiation, which will be explained in further detail in this manual.

In order to use one box as input to another, draw an arrow between them by clicking on the arrow tool in the sidebar, and clicking and dragging from the first box to the second in the workspace.

The graph box can be used to create a new graph, or to copy or edit a graph from another box.

- Another graph box
- A parametric model box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- A search box
- An updater box
- A regression box

- Another graph box
- A compare box
- A parametric model box
- A data box
- A simulation box
- A search box
- A knowledge box

When you first open a graph box with no parent, you will be presented with several options for which kind of graph you would like to create: a general graph, a directed acyclic graph (DAG), a structural equation model (SEM)graph, or a time lag graph. Once you have selected the type of graph you want to create, an empty graph box will open.

You can add variables to your graph by clicking on the variable button on the left, then clicking inside the graph area. Add edges by clicking on an edge type, then clicking and dragging from one variable to another. Variables may be measured (represented by rectangular icons) or latent (represented by elliptical icons). Edges may be directed, undirected, bidirected, or uncertain (represented by circles at the ends of an edge). Depending on the type of graph you choose to create, your choice of edges may be limited.

*DAGs* allow only directed edges. If an edge would create a cycle, it will not be accepted. A graph box
containing a DAG can be used as
input for any parametric model box, and is the only kind of graph box that can be used as input for a Bayes
parametric model.

*SEM graphs* allow only directed and bidirected edges. A graph box containing a SEM graph can be used as
input to a SEM parametric model or generalized SEM parametric model, where a bidirected edge between two
variables X and Y will be interpreted as X and Y having correlated error terms.

*Time lag graphs* allow only directed edges. New variables that you add will be initialized with a single
lag. (The number of lags in the graph may be changed under “Edit—Configuration…”) Edges from later lags to
earlier lags will not be accepted. Edges added within one lag will automatically be replicated in later lags.

The general *graph* option allows all edge types and configurations.

Instead of manually creating a new graph, you can randomly create one. To do so, open up a new empty graph box and click on “Graph—Random Graph.” This will open up a dialog box from which you can choose the type of random graph you would like to create by clicking through the tabs at the top of the window. Tetrad will randomly generate a DAG, a multiple indicator model (MIM) graph, or a scale-free graph. Each type of graph is associated with a number of parameters (including but not limited to the number of nodes and the maximum degree) which you can set.

Once a graph has been randomly generated, you can directly edit it within the same graph box by adding or removing any variables or edges that that type of graph box allows. So, for instance, although you cannot randomly generate a graph with bidirected edges, you can manually add bidirected edges to a randomly generated DAG in a SEM graph box.

Random graph generation is not available for time lag graphs.

If you have previously saved a graph from Tetrad, you can load it into a new graph box by clicking “File—Load…,” and then clicking on the file type of the saved graph. Tetrad can load graphs from XML, from text, and from JSON files.

To save a graph to file, click “File—Save…,” then click on the file type you would like to save your graph as. Tetrad can save graphs to XML, text, JSON, R and dot files. (If you save your graph to R or dot, you will not be able to load that file back into Tetrad.)

You can also save an image of your graph by clicking “File—Save Graph Image…” Tetrad cannot load graphs from saved image files.

There are two ways to copy a graph.

The first method allows you to copy a graph from any box which contains one. First, create a new graph box in the workspace, and draw an arrow from the box whose graph you want to copy to the new graph box. When opened, the new graph box will automatically contain a direct copy of the graph its parent box contains.

The second method allows you to copy a graph directly from most types of graph box. First, highlight the graph in the old graph box and click “Edit—Copy Selected Graph.” Then open up your new graph box and click “Edit—Paste Selected Graph.” (Some types of graph box do not have this functionality; see “Manipulating a Graph.”)

If you create a graph box as a child of another box, you can also choose to perform a graph manipulation on the parent graph. Your graph box will then contain the manipulated version of the parent graph.

The available graph manipulations are:

This option allows you to isolate a subgraph from the parent graph. Add variables to the subgraph by highlighting the variable name in the “Unselected” pane and clicking on the right arrow. The highlighted variable will then show up in the “Selected” pane. (You may also define which variables go in the “Selected” pane by clicking on the “Text Input…” button and typing the variable names directly into the window.) Choose the type of subgraph you want to display from the drop-down panel below. Then click “Graph It!” and the resulting subgraph of the selected variables will appear in the pane on the right. (Some types of subgraph, such as “Markov Blanket,” will include unselected variables if they are part of the subgraph as defined on the selected variables. So, for instance, an unselected variable that is in the Markov blanket of a selected variable will appear in the Markov Blanket subgraph. Edges between unselected variables will not be shown.) For large or very dense graphs, it may take a long time to isolate and display subgraphs.

The types of subgraphs that can be displayed are:

- Subgraph (displays the selected nodes and all edges between them)
- Adjacents (displays the selected nodes and all edges between them, as well as nodes adjacent to the selected nodes)
- Adjacents of adjacents (displays the selected nodes and all edges between them, as well as nodes adjacent to the selected nodes and nodes adjacent to adjacencies of the selected nodes)
- Adjacents of adjacents of adjacents (displays the selected nodes and all edges between them, as well as nodes adjacent to the selected nodes, nodes adjacent to adjacencies of the selected nodes, and nodes adjacent to adjacencies of adjacencies of the selected nodes)
- Markov Blankets (displays the selected nodes and all edges between them, as well as the Markov blankets of each selected node)
- Treks (displays the selected nodes, with an edge between each pair if and only if a trek exists between them in the full graph)
- Trek Edges (displays the selected nodes, and any treks between them, including nodes not in the selected set if they are part of a trek)
- Paths (displays the selected nodes, with an edge between each pair if and only if a path exists between them in the full graph)
- Path Edges (displays the selected nodes, and any paths between them, including nodes not in the selected set if they are part of a path)
- Directed Paths (displays the selected nodes, with a directed edge between each pair if and only if a directed path exists between them in the full graph)
- Directed Path Edges (displays the selected nodes, and any directed paths between them, including nodes not in the selected set if they are part of a path)
- Y Structures (displays any Y structures involving at least two of the selected nodes)
- Pag_Y Structures (displays any Y PAGs involving at least two of the selected nodes)
- Indegree (displays the selected nodes and their parents)
- Outdegree (displays the selected nodes and their children)
- Degree (displays the selected nodes and their parents and children)

If given a pattern as input, this chooses a random DAG from the Markov equivalence class of the pattern to display. The resulting DAG functions as a normal graph box.

If given a partial ancestral graph (PAG) as input, this chooses a random mixed ancestral graph (MAG) from the equivalence class of the PAG to display. The resulting MAG functions as a normal graph box.

If given a pattern as input, this displays all DAGs in the pattern’s Markov equivalence class. Each DAG is displayed in its own tab. Most graph box functionality is not available in this type of graph box, but the DAG currently on display can be copied by clicking “Copy Selected Graph.”

If given a DAG as input, this displays the pattern of the Markov equivalence class to which the parent graph belongs. The resulting pattern functions as a normal graph box.

Converts an input graph from partial ancestral to directed acyclic format. The resulting DAG functions as a normal graph box.

Converts an input graph from partial ancestral to time series DAG format. The resulting DAG functions as a normal graph box.

Replaces all bidirected edges in the input graph with undirected edges.

Replaces all undirected edges in the input graph with bidirected edges.

Replaces all edges in the input graph with undirected edges.

Creates a completely connected, undirected graph from the variables in the input graph.

Isolates the subgraph of the input graph involving all and only latent variables.

At the bottom of the graph box, the Edges and Edge Type Probabilities section provides an accounting of every edge in the graph, and how certain Tetrad is of its type. The first three columns contain a list, in text form, of all of the edges in the graph. The columns to the right are all blank in manually constructed graphs, user-loaded graphs, and graphs output by searches with default settings. They are only filled in for graphs that are output by searches performed with bootstrapping. In those cases, the fourth column will contain the percentage of bootstrap outputs in which the edge type between these two variables matches the edge type in the final graph. All of the columns to the right contain the percentages of the bootstrap outputs that output each possible edge type.

For more information on bootstrap searches, see the Search Box section of the manual.

You can change the layout of your graph by clicking on the “Layout” tab and choosing between several common layouts. You can also rearrange the layout of one graph box to match the layout of another graph box (so long as the two graphs have identical variables) by clicking “Layout—Copy Layout” and “Layout—Paste Layout.” You do not need to a highlight the graph in order to copy the layout.

Clicking on “Graph—Graph Properties” will give you a text box containing the following properties of your graph:

- Number of nodes
- Number of latent nodes
- Number of edges
- Number of directed edges
- Number of bidirected edges
- Number of undirected edges
- Max degree
- Max indegree
- Max outdegree
- Cyclicity

Clicking on “Graph—Paths” opens a dialog box that allows you to see all the paths between any two variables. You can specify whether you want to see only adjacencies, only directed paths, only semidirected paths, or all treks between the two variables of interest, and the maximum length of the paths you are interested in using drop boxes at the top of the pane. To apply those settings, click “update.”

You can automatically correlate or uncorrelated exogenous variables under the Graph tab.

You can highlight bidirected edges, undirected edges, and latent nodes under the Graph tab.

The compare box compares two or more graphs.

- A graph box
- An instantiated model box
- An estimator box
- A simulation box
- A search box
- A regression box

- None

An edgewise comparison compares two graphs, and gives a textual list of the edges which must be added to or taken away from one to make it identical to the other.

Take, for example, the following two graphs. The first is the reference graph, the second is the graph to be compared to it.

When these two graphs are input into the graph compare box, a window appears which allows you to specify which of the two graphs is the reference graph. When the comparison is complete, the following window results

When the listed changes have been made to the second graph, it will be identical to the first graph.

If one of the parent boxes contains multiple graphs, each graph will be compared separately to the reference graph (or the estimated graph will be compared separately to each reference graph, depending on which parent box is selected as the reference), and each comparison will be housed in its own tab, located on the left side of the window.

A stats list graph comparison tallies up and presents statistics for the differences and similarities between a true graph and a reference graph. Consider the example used in the above section; once again, we’ll let graph one be the true graph. Just as above, when the graphs are input to the tabular graph compare box, we must specify which of the graphs is the reference graph, and whether it contains latent variables. When the comparison is complete, the following window results:

The first columns gives an abbreviation for the statistic; the second columns gives a definition of the statistic. The third columns gives the statistic value.

A misclassification procedure organizes a graph comparison by edge type. The edge types (undirected, directed, uncertain, partially uncertain, bidirected, and null) are listed as the rows and columns of a matrix, with the true graph edges as the row headers and the target graph edges as the column headers. If, for example, there are three pairs of variables that are connected by undirected edges in the reference graph, but are connected by directed edges in the estimated graph, then there will be a 3 in the (undirected, directed) cell of the matrix. An analogous method is used to represent endpoint errors. For example:

If one of the parent boxes contains multiple graphs, then each estimated graph will be individually compared to the reference graph (or vice versa), and the results housed in their own tab, found on the left.

A graph intersection compares two or more graphs in the same comparison. It does so by ranking adjacencies (edges without regard to direction) and orientations based on how many of the graphs they appear in. In an n-graph comparison, it first lists any adjacencies found in all n graphs. Then it lists all adjacencies found in n – 1 graphs, then adjacencies found in n – 2 graphs, and so on.

After it has listed all adjacencies, it lists any orientations that are not contradicted among the graphs, again in descending order of how many graphs the orientation appears in. An uncontradicted orientation is one on which all graphs either agree or have no opinion. So if the edge X Y appears in all n graphs, it will be listed first. If the edge X Z appears in n – 1 graphs, it will be listed next, but only if the nth graph doesn’t contradict it—that is, only if the edge Z X does not appear in the final graph. If the undirected edge Z – X appears in the final graph, the orientation X Z is still considered to be uncontradicted.

Finally, any contradicted orientations (orientations that the graphs disagree on) are listed.

Rather than comparing edges or orientation, this option directly compares the implied dependencies in two graphs. When you initially open the box, you will see the following window:

The drop-down menu allows you to choose which variables you want to check the dependence of. If you select more than two variables, any subsequent variables will be considered members of the conditioning set. So, if you select variables X1, X2, and X3, in that order, the box will determine whether X1 is independent of X2, conditional on X3, in each of the graphs being compared. When you click “List,” in the bottom right of the window, the results will be displayed in the center of the window:

Edge weight (linear coefficient) similarity comparisons compare two linear SEM instantiated models. The output is a score equal to the sum of the squares of the differences between each corresponding edge weight in each model. Therefore, the lower the score, the more similar the two graphs are. The score has peculiarities: it does not take account of the variances of the variables, and may therefore best be used with standardized models; the complete absence of an edge is scored as 0—so a negative coefficient compares less well with a positive coefficient than does no edge at all.

Consider, for example, an edge weight similarity comparison between the following two SEM IMs:

When they are input into an edge weight similarity comparison, the following window results:

This is, unsurprisingly, a high score; the input models have few adjacencies in common, let alone similar parameters.

A model fit comparison takes a simulation box and a search box (ideally, a search that has been run on the simulated data in the simulation box), and provides goodness-of-fit statistics, including a Student’s t statistic and p value for each edge, for the output graph and the data, as well as estimating the values of any parameters. It looks and functions identically to the estimator box, but unlike the estimator box, it takes the search box directly as a parent, without needing to isolate and parameterize the graph output by the search.

The parametric model box takes a nonparameterized input graph and creates a causal model.

- A graph box
- Another parametric model box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- A search box
- A regression box

- A graph box
- Another parametric model box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- A search box
- A knowledge box

A Bayes parametric model takes as input a DAG. Bayes PMs represent causal structures in which all of the variables are categorical.

Bayes PMs consist of three components: the graphical representation of the causal structure of the model; for each named variable, the number of categories which that variable can assume; and the names of the categories associated with each variable.

You may either manually assign categories to the variables or have Tetrad assign them at random. If you choose to manually create a Bayes PM, each variable will initially be assigned two categories, named numerically. If you choose to have Tetrad assign the categories, you can specify a minimum and maximum number of categories possible for any given variable. You can then manually edit the number of categories and category names.

Take, for example, the following DAG:

One possible random Bayes PM that Tetrad might generate from the above DAG, using the default settings, looks like this:

To view the number and names of the categories associated with each variable, you can click on that variable in the graph, or choose it from the drop-down menu on the right. In this graph, X1 and X2 each have three categories, and the rest of the variables have four categories. The categories are named numerically by default.

The number of categories associated with a particular variable can be changed by clicking up or down in the drop-down menu on the right. Names of categories can be changed by overwriting the text already present.

Additionally, several commonly-used preset variable names are provided under the “Presets” tab on the right. If you choose one of these configurations, the number of categories associated with the current variable will automatically be changed to agree with the configuration you have chosen. If you want all of the categories associated with a variable to have the same name with a number appended (e.g., x1, x2, x3), choose the “x1, x2, x3…” option under Presets.

You can also copy category names between variables in the same Bayes PM by clicking on “Transfer—Copy categories” and “Transfer—Paste categories.”

The parametric model of a structural equation model (SEM) will take any type of graph as input, as long as the graph contains only directed and bidirected edges. SEM PMs represent causal structures in which all variables are continuous.

A SEM PM has two components: the graphical causal structure of the model, and a list of parameters used in a set of linear equations that define the causal relationships in the model. Each variable in a SEM PM is a linear function of a subset of the other variables and of an error term drawn from a Normal distribution.

Here is an example of a SEM graph and the SEM PM that Tetrad creates from it:

You can see the error terms in the model by clicking “Parameters—Show Error Terms.” In a SEM model, a bidirected edge indicates that error terms are correlated, so when error terms are visible, the edge between X1 and X2 will instead run between their error terms.

To change a parameter’s name or starting value for estimation, double click on the parameter in the window.

A generalized SEM parametric model takes as input any type of graph, as long as the graph contains only directed edges. (The generalized SEM PM cannot currently interpret bidirected edges.) Like a SEM PM, it represents causal structures in which all variables are continuous. Also like a SEM PM, a generalized SEM PM contains two components: the graphical causal structure of the model, and a set of equations representing the causal structure of the model. Each variable in a generalized SEM PM is a function of a subset of the other variables and an error term. By default, the functions are linear and the error terms are drawn from a Normal distribution (as in a SEM PM), but the purpose of a generalized SEM PM is to allow editing of these features.

Here is an example of a general graph and the default generalized SEM PM Tetrad creates using it:

You can view the error terms by clicking “Tools: Show Error Terms.”

The Variables tab contains a list of the variables and the expressions that define them, and a list of the error terms and the distributions from which their values will be drawn. Values will be drawn independently for each case if the model is instantiated (see IM box) and used to simulate data (see data box).

The Parameters tab contains a list of the parameters and the distributions from which they are drawn. When the model in instantiated in the IM box, a fixed value of each parameter will be selected according to the specified distribution.

To edit an expression or parameter, double click on it (in any tab). This will open up a window allowing you to change the function that defines the variable or distribution of the parameter.

For instance, if you double click on the expression next to X1 (b1*X5+E_X1), the following window opens:

The drop-down menu at the top of the window lists valid operators and functions. You could, for example, change the expression from linear to quadratic by replacing b1*X5+E_X1 with b1*X5^2+E_X1. You can also form more complicated expressions, using, for instance, exponential or sine functions. If the expression you type is well-formed, it will appear in black text; if it is invalid, it will appear in red text. Tetrad will not accept any invalid changes.

Parameters are edited in the same way as expressions.

If you want several expressions or parameters to follow the same non-linear model, you may wish to use the Apply Templates tool. This allows you to edit the expressions or parameters associated with several variables at the same time. To use the Apply Templates tool, click “Tools: Apply Templates….” This will open the following window:

You can choose to edit variables, error terms, or parameters by clicking through the “apply to” radio buttons. If you type a letter or expression into the “starts with” box, the template you create will apply only to variables, error terms, or parameters which begin with that letter for expression. For example, in the given generalized PM, there are two types of parameters: the standard deviations s1-s6 and the edge weights b1-b7. If you click on the “Parameters” radio button and type “b” into the “Starts with” box, only parameters b1-b7 will be affected by the changes you make.

The “Type Template” box itself works in the same way that the “Type Expression” box works in the “Edit Expression” window, with a few additions. If you scroll through the drop-down menu at the top of the window, you will see the options NEW, TSUM, and TPROD. Adding NEW to a template creates a new parameter for every variable the template is applied to. TSUM means “sum of the values of this variable’s parents,” and TPROD means “product of the values of this variable’s parents.” The contents of the parentheses following TSUM and TPROD indicate any operations which should be performed upon each variable in the sum or product, with the dollar sign ($) functioning as a wild card. For example, in the image above, TSUM(NEW(b)*$) means that, for each parent variable of the variable in question, a new “b” will be created and multiplied by the parent variable’s value, and then all of the products will be added together.

The instantiated model (IM) box takes a parametric model and assigns values to the parameters.

- A parametric model box
- Another instantiated model box
- An estimator box
- A simulation box
- An updater box

- A graph box
- A compare box
- A parametric model box
- Another instantiated model box
- An estimator box
- A simulation box
- A search box
- An updater box
- A classify box
- A knowledge box

A Bayes IM consists of a Bayes parametric model with defined probability values for all variables. This means that, conditional on the values of each of its parent variables, there is a defined probability that a variable will take on each of its possible values. For each assignment of a value to each of the parents of a variable X, the probabilities of the several values of X must sum to 1.

You can manually set the probability values for each variable, or have Tetrad assign them randomly. If you choose to have Tetrad assign probability values, you can manually edit them later.

Here is an example of a Bayes PM and its randomly created instantiated model:

In the model above, when X4 and X5 are both 0, the probability that X5 is 0 is 0.0346, that X5 is 1 is 0.4425, and that X5 is 2 is 0.5229. Since X5 must be 0, 1, or 2, those three values must add up to one, as must the values in every row.

To view the probability values of a variable, either double click on the variable in the graph or choose it from the drop-down menu on the right. You can manually set a given probability value by overwriting the text box. Be warned that changing the value in one cell will delete the values in all of the other cells in the row. Since the values in any row must sum to one, if all of the cells in a row but one are set, Tetrad will automatically change the value in the last cell to make the sum correct. For instance, in the above model, if you change the first row such that the probability that X5 = 0 is 0.5000 and the probability that X5 = 1 is 0.4000, the probability that X5 = 2 will automatically be set to 0.1000.

If you right click on a cell in the table (or two-finger click on Macs), you can choose to randomize the probabilities in the row containing that cell, randomize the values in all incomplete rows in the table, randomize the entire table, or randomize the table of every variable in the model. You can also choose to clear the row or table.

A Dirichlet instantiated model is a specialized form of a Bayes instantiated model. Like a Bayes IM, a Dirichlet IM consists of a Bayes parametric model with defined probability values. Unlike a Bayes IM, these probability values are not manually set or assigned randomly. Instead, the pseudocount is manually set or assigned uniformly, and the probability values are derived from it. The pseudocount of a given value of a variable is the number of data points for which the variable takes on that value, conditional on the values of the variable’s parents, where these numbers are permitted to take on non-negative real values. Since we are creating models without data, we can set the pseudocount to be any number we want. If you choose to create a Dirichlet IM, a window will open allowing you to either manually set the pseudocounts, or have Tetrad set all the pseudocounts in the model to one number, which you specify.

Here is an example of a Bayes PM and the Dirichlet IM which Tetrad creates from it when all pseudocounts are set to one:

In the above model, when X2=0 and X6=0, there is one (pseudo) data point at which X4=0, one at which X4=1, and one at which X4=2. There are three total (pseudo) data points in which X2=0 and X6=0. You can view the pseudocounts of any variable by clicking on it in the graph or choosing it from the drop-down menu at the top of the window. To edit the value of a pseudocount, double click on it and overwrite it. The total count of a row cannot be directly edited.

From the pseudocounts, Tetrad determines the conditional probability of a category. This estimation is done by taking the pseudocount of a category and dividing it by the total count for its row. For instance, the total count of X4 when X2=0 and X6=0 is 3. So the conditional probability of X4=0 given that X2=0 and X6=0 is 1/3. The reasoning behind this is clear: in a third of the data points in which X2 and X6 are both 0, X4 is also 0, so the probability that X4=0 given that X2 and X6 also equal 0 is probably one third. This also guarantees that the conditional probabilities for any configuration of parent variables add up to one, which is necessary.

To view the table of conditional probabilities for a variable, click the Probabilities tab. In the above model, the Probabilities tab looks like this:

A SEM instantiated model is a SEM parametric model in which the parameters and error terms have defined values. It assumes that relationships between variables are linear, and that error terms have Gaussian distributions. If you choose to create a SEM IM, the following window will open:

Using this box, you can specify the ranges of values from which you want coefficients, covariances, and variances to be drawn for the parameters in the model. In the above box, for example, all linear coefficients will be between -1.5 and -0.5 or 0.5 and 1.5. If you uncheck “symmetric about zero,” they will only be between 0.5 and 1.5.

Here is an example of a SEM PM and a SEM IM generated from it using the default settings:

You can now manually edit the values of parameters in one of two ways. Double clicking on the parameter in the graph will open up a small text box for you to overwrite. Or you can click on the Tabular Editor tab, which will show all of the parameters in a table which you can edit. The Tabular Editor tab of our SEM IM looks like this:

In the Tabular Editor tab of a SEM estimator box (which functions similarly to the SEM IM box), the SE, T, and P columns provide statistics showing how robust the estimation of each parameter is. Our SEM IM, however, is in an instantiated model box, so these columns are empty.

The Implied Matrices tab shows matrices of relationships between variables in the model. In the Implied Matrices tab, you can view the covariance or correlation matrix for all variables (including latents) or just measured variables. In our SEM IM, the Implied Matrices tab looks like this:

You can choose the matrix you wish to view from the drop-down menu at the top of the window. Only half of any matrix is shown, because in a well-formed acyclic model, the matrices should be symmetric. The cells in the Implied Matrices tab cannot be edited.

In an estimator box, the Model Statistics tab provides goodness of fit statistics for the SEM IM which has been estimated. Our SEM IM, however, is in an instantiated model box, so no estimation has occurred, and the Model Statistics tab is empty.

A standardized SEM instantiated model consists of a SEM parametric model with defined values for its parameters. In a standardized SEM IM, each variable (not error terms) has a Normal distribution with 0 mean and unit variance. The input PM to a standardized SEM IM must be acyclic.

Here is an example of an acyclic SEM PM and the standardized SEM IM which Tetrad creates from it

To edit a parameter, double click on it. A slider will open at the bottom of the window (shown above for the edge parameter between X1 and X2). Click and drag the slider to change the value of the parameter, or enter the specific value you wish into the box. The value must stay within a certain range in order for the Normal distribution to stay standardized, so if you attempt to overwrite the text box on the bottom right with a value outside the listed range, Tetrad will not allow it. In a standardized SEM IM, error terms are not considered parameters and cannot be edited, but you can view them by clicking Parameters: Show Error Terms.

The Implied Matrices tab works in the same way that it does in a normal SEM IM.

A generalized SEM instantiated model consists of a generalized SEM parametric model with defined values for its parameters. Since the distributions of the parameters were specified in the SEM PM, Tetrad does not give you the option of specifying these before it creates the instantiated model.

Here is an example of a generalized SEM PM and its generalized SEM IM:

Note that the expressions for X6 and X2 are not shown, having been replaced with the words “long formula.” Formulae over a certain length—the default setting is 25 characters—are hidden to improve visibility. Long formulae can be viewed in the Variables tab, which lists all variables and their formulae. You can change the cutoff point for long formulae by clicking Tools: Formula Cutoff.

If you double click on a formula in either the graph or the Variables tab, you can change the value of the parameters in that formula.

The data box stores or manipulates data sets.

- A graph box
- An estimator box
- Another data box
- A simulation box
- A regression box

- A graph box
- A parametric model box
- Another data box
- An estimator box
- A simulation box
- A search box
- A classify box
- A regression box
- A knowledge box

The data box stores the actual data sets from which causal structures are determined. Data can be loaded into the data box from a preexisting source, manually filled in Tetrad, or simulated from an instantiated model.

Data sets loaded into Tetrad may be categorical, continuous, mixed, or covariance data.

To load data, create a data box with no parent. When you double click it, an empty data window will appear:

Click "File -> Load Data" and select the text file or files that contain your data. The following window will appear:

The text of the source file appears in the Data Preview window. Above, there are options to describe your file, so that Tetrad can load it correctly. If you are loading categorical, continuous, or mixed data values, select the “Tabular Data” button. If you are loading a covariance matrix, select “Covariance Data.” Note that if you are loading a covariance matrix, your text file should contain only the lower half of the matrix, as Tetrad will not accept an entire matrix.

Below the file type, you can specify a number of other details about your file, including information about the type of data (categorical/continuous/mixed), metadata JSON file, delimiter between data values, variable names, and more. If your data is mixed (some variables categorical, and some continuous), you must specify the maximum number of categories discrete variables in your data can take on. All columns with more than that number of values will be treated as continuous; the others will be treated as categorical. If you do not list the variable names in the file, you should uncheck “First row variable names.” If you provide case IDs, check the box for the appropriate column in the “Case ID column to ignore” area. If the case ID column is labeled, provide the name of the label; otherwise, the case ID column should be the first column, and you should check “First column.”

Below this, you can specify your comment markers, quote characters, and the character which marks missing data values. Tetrad will use that information to distinguish continuous from discrete variables. You may also choose more files to load (or remove files that you do not wish to load) in the “Files” panel on the lower left.

Metadata is optional in general data handling. But it can be very helpful if you want to overwrite the data type of a given variable column. And the metadata MUST be a JSON file like the following example.

{ "domains": [ { "name": "raf", "discrete": false }, { "name": "mek", "discrete": true } ] }

You can specify the name and data type for each variable. Variables that are not in the metadata file will be treated as domain variables and their data type will be the default data type when reading in columns described previously.

When you are satisfied with your description of your data, click “Validate” at the bottom of the window. Tetrad will check that your file is correctly formatted. If it is, you will receive a screen telling you that validation has passed with no error. At this point, you can revisit the settings page, or click “Load” to load the data.

You can now save this data set to a text file by clicking File: Save Data.

In addition to loading data from a file, you can manually enter data values and variable names by overwriting cells in the data table.

Covariance matrices loaded into Tetrad should be ascii text files. The first row contains the sample size, the second row contains the names of the variables. The first two rows are followed by a lower triangular matrix. For example:

1000 X1 X2 X3 X4 X5 X6 1.0000 0.0312 1.0000 -0.5746 0.4168 1.0000 -0.5996 0.4261 0.9544 1.0000 0.8691 0.0414 -0.4372 -0.4487 1.0000 0.6188 0.0427 -0.1023 -0.0913 0.7172 1.0000

Categorical, continuous, or mixed data should also be an ascii text file, with columns representing variables and rows representing cases. Beyond that, there is a great deal of flexibility in the layout: delimiters may be commas, colons, tabs, spaces, semicolons, pipe symbols, or whitespace; comments and missing data may be marked by any symbol you like; there may be a row of variable names or not; and case IDs may be present or not. There should be no sample size row. For example:

X1 X2 X3 X4 X5 -3.0133 1.0361 0.2329 2.7829 -0.2878 0.5542 0.3661 0.2480 1.6881 0.0775 3.5579 -0.7431 -0.5960 -2.5502 1.5641 -0.0858 1.0400 -0.8255 0.3021 0.2654 -0.9666 -0.5873 -0.6350 -0.1248 1.1684 -1.7821 1.8063 -0.9814 1.8505 -0.7537 -0.8162 -0.6715 0.3339 2.6631 0.9014 -0.3150 -0.5103 -2.2830 -1.2462 -1.2765 -4.1204 2.9980 -0.3609 4.8079 0.6005 1.4658 -1.4069 1.7234 -1.7129 -3.8298

This is an advanced topic for datasets that contain interventional (i.e., experimental) variables. We model a single intervention using two variables: status variable and value variable. Below is a sample dataset, in which `raf`, `mek`, `pip2`, `erk`, `atk` are the 5 domain variables, and `cd3_s` and `cd3_v` are an interventional pair (status and value variable respectively). `icam` in another intervention variable, but it's a combined variable that doesn't have status.

raf mek pip2 erk akt cd3_s cd3_v icam 3.5946 3.1442 3.3429 2.81 3.2958 0 1.2223 * 3.8265 3.2771 3.2884 3.3534 3.7495 0 2.3344 * 4.2399 3.9908 3.0057 3.2149 3.7495 1 0 3.4423 4.4188 4.5304 3.157 2.7619 3.0819 1 3.4533 1.0067 3.7773 3.3945 2.9821 3.4372 4.0271 0 4.0976 *

And the sample metadata JSON file looks like this:

{ "interventions": [ { "status": { "name": "cd3_s", "discrete": true }, "value": { "name": "cd3_v", "discrete": false } }, { "status": null, "value": { "name": "icam", "discrete": false } } ], "domains": [ { "name": "raf", "discrete": false }, { "name": "mek", "discrete": false } ] }

Each intervention consists of a status variable and value variable. There are cases that you may have a combined interventional variable that doesn't have the status variable. In this case, just use `null`. The data type of each variable can either be discrete or continuous. We use a boolean flag to indicate the data type. From the above example, we only specified two domain variables in the metadata JSON, any variables not specifed in the metadata will be treated as domain variables.

The data box can also be used to manipulate data sets that have already been loaded or simulated. If you create a data box as the child of another box containing a data set, you will be presented with a list of operations that can be performed on the data. The available data manipulations are:

This operation allows you to make some or all variables in a data set discrete. If you choose it, a window will open.

When the window first opens, no variables are selected, and the right side of the window appears blank; in this case, we have already selected X1 ourselves. In order to discretize a variable, Tetrad assigns all data points within a certain range to a category. You can tell Tetrad to break the range of the dataset into approximately even sections (Evenly Distributed Intervals) or to break the data points themselves into approximately even chunks (Evenly Distributed Values). Use the scrolling menu to increase or decrease the number of categories to create. You can also rename categories by overwriting the text boxes on the left, or change the ranges of the categories by overwriting the text boxes on the right. To discretize another variable, simply select it from the left. If you want your new data set to include the variables you did not discretize, check the box at the bottom of the window.

You may discretize multiple variables at once by selecting multiple variables. In this case, the ranges are not shown, as they will be different from variable to variable.

If you choose this option, any discrete variables with numerical category values will be treated as continuous variables with real values. For example, “1” will be converted to “1.0.”

The Calculator option allows you to add and edit relationships between variables in your data set, and to add new variables to the data set.

In many ways, this tool works like the Edit Expression window in a generalized SEM parametric model. To edit the formula that defines a variable (which will change that variable’s values in the table) type that variable name into the text box to the left of the equals sign. To create a new variable, type a name for that variable into the text box to the left of the equals sign. Then, in the box on the right, write the formula by which you wish to define a new variable in place of, or in addition to, the old variable. You can select functions from the scrolling menu below. (For an explanation of the meaning of some the functions, see the section on generalized SEM models in the Parametric Model Box chapter.) To edit or create several formulae at once, click the “Add Expression” button, and another blank formula will appear. To delete a formula, check the box next to it and click the “Remove Selected Expressions” button.

When you click “Save” a table will appear listing the data. Values of variables whose formulae you changed will be changed, and any new variables you created will appear with defined values.

This option looks for pairs of interventional variables (currently only discrete variables) that are deterministic and merges them into one combined variable. For domain variables that are fully determinised, we'll add an attribute to them. Later in the knowledge box (Edges and Tiers), all the interventional variables (both status and value variables) and the fully-determinised domain variables will be automatically put to top tier. And all other domain variables will be placed in the second tier.

This operation takes two or more data boxes as parents and creates a data box containing all data sets in the parent boxes. Individual data sets will be contained in their own tabs in the resulting box.

This operation takes a tabular data set and outputs the lower half of the correlation matrix of that data set.

This operation takes a tabular data set and outputs the lower half of the covariance matrix of that data set.

This operation takes a covariance or correlation matrix and outputs its inverse. (Note: The output will not be acceptable in Tetrad as a covariance or correlation matrix, as it is not lower triangular.)

This operation takes a covariance matrix and outputs a tabular data set whose covariances comply with the matrix.

This operation takes two covariance matrices and outputs their difference. The resulting matrix will be a well-formatted Tetrad covariance matrix data set.

This operation takes two covariance matrices and outputs their sum. The resulting matrix will be a well-formatted Tetrad covariance matrix data set.

This operation takes two or more covariance matrices and outputs their average. The resulting matrix will be a well-formatted Tetrad covariance matrix data set.

This operation takes a tabular data set and outputs a time lag data set, in which each variable is recorded several times over the course of an experiment. You can specify the number of lags in the data. Each contains the same data, shifted by one “time unit.” For instance, if the original data set had 1000 cases, and you specify that the time lag data set should contain two lags, then the third stage variable values will be those of cases 1 to 998, the second stage variable values will be those of cases 2 to 999, and the first stage variable values will be those of cases 3 to 1000.

This operation takes a tabular data set and outputs a time lag data set in the same manner as “Convert to Time Lag Data,” then adds an index variable.

This operation is performed on a time lag data set. Tetrad performs a linear regression on each variable in each lag with respect to each of the variables in the previous lag, and derives the error terms. The output data set contains only the error terms.

Takes a continuous tabular data set and converts it to a data set whose covariance matrix is the identity matrix.

Takes a continuous tabular data set and increases its Gaussianity, using a nonparanormal transformation to smooth the variables. (Note: This operation increases only marginal Gaussanity, not the joint, and in linear systems may eliminate information about higher moments that can aid in non-Gaussian orientation procedures.)

The input for this operation is a directed acyclic graph (DAG) and a data set. Tetrad performs a linear regression on each variable in the data set with respect to all of the variables that the graph shows to be its parents, and derives the error terms. The output data set contains only the error terms.

This operation manipulates the data in your data set such that each variable has 0 mean and unit variance.

If you choose this operation, Tetrad will remove any row in which one or more of the values is missing.

If you choose this operation, Tetrad will replace any missing value markers with the most commonly used value in the column.

If you choose this operation, Tetrad will replace any missing value markers with the average of all of the values in the column. Replace Missing Values with Regression Predictions: If you choose this operation, Tetrad will perform a linear regression on the data in order to estimate the most likely value of any missing value.

This operation takes as input a discrete data set. For every variable which has missing values, Tetrad will create an extra category for that variable (named by default “Missing”) and replace any missing data markers with that category.

For discrete data, replaces missing values at random from the list of categories the variable takes in other cases. For continuous data, finds the minimum and maximum values of the column (ignoring the missing values) and picks a random number from U(min, max)

If you choose this operation, Tetrad will replace randomly selected data values with a missing data marker. You can set the probability with which any particular value will be replaced (that is, approximately the percentage of values for each variable which will be replaced with missing data markers).

This operation draws a random subset of the input data set (you specify the size of the subset) with replacement (that is, cases which appear once in the original data set can appear multiple times in the subset). The resulting data set can be used along with similar subsets to achieve more accurate estimates of parameters.

This operation allows you to split a data set into several smaller data sets. When you choose it, a window opens.

If you would like the subsets to retain the ordering they had in the original set, click “Original Order.” Otherwise, the ordering of the subsets will be assigned at random. You can also increase and decrease the number of subsets created, and specify the range of each subset.

This operation randomly reassigns the ordering of a data set’s cases.

This operation takes a tabular data set and outputs the first differences of the data (i.e., if X is a variable in the original data set and X’ is its equivalent in the first differences data set, X’1 = X2 – X1). The resulting data set will have one fewer row than the original.

This operation takes two or more datasets and concatenates. The parent datasets must have the same number of variables.

This operation takes as input a data set and creates a new data set containing only the continuous variables present in the original.

This operation takes as input a data set and creates a new data set containing only the discrete variables present in the original.

As explained above, you can select an entire column in a data set by clicking on the C1, C2, C3, etc… cell above the column. To select multiple columns, press and hold the “control” key while clicking on the cells. Once you have done so, you can use the Copy Selected Variables tool to create a data set in which only those columns appear.

This operation takes a data set as input, and creates a data set which contains all columns in the original data set except for those with constant values (such as, for example, a column containing nothing but 2’s).

This operation randomly reassigns the ordering of a data set’s variables.

Under the Edit tab, there are several options to manipulate data. If you select a number of cells and click “Clear Cells,” Tetrad will replace the data values in the selected cells with a missing data marker. If you select an entire row or column and click “Delete selected rows or columns,” Tetrad will delete all data values in the row or column, and the name of the row or column. (To select an entire column, click on the category number above it, labeled C1, C2, C3, and so on. To select an entire row, click on the row number to the left of it, labeled 1, 2, 3, and so on.) You can also copy, cut, and paste data values to and from selected cells. You can choose to show or hide category names, and if you click on “Set Constants Col to Missing,” then in any column in which the variable takes on only one value (for example, a column in which every cell contains the number 2) Tetrad will set every cell to the missing data marker.

Under the Tools tab, the Calculator tool allows you add and edit relationships between variables in the graph. For more information on how the Calculator tool works, see “Manipulating Data” section above.

Under the Tools tab, there are options to view information about your data in several different formats.

The Histograms tool shows histograms of the variables in the data set.

These show the distribution of data for each variable, with the width of each bar representing a range of values, and height of each bar representing how many data points fall into that range. Using histograms, you can determine whether each variable has a distribution that is approximately Normal. To select a variable to view, choose it from the drop-down menu on the right. You can increase or decrease the number of bars in the histogram (and therefore decrease or increase the range of each bar, and increase or decrease the accuracy of the histogram) using the menu on the right. You can also view only ranges with a certain amount of the data using the “cull bins” menu.

The Scatter Plots tool allows you to view scatter plots of two variables plotted against each other.

To view a variable as the x- or y-axis of the graph, select it from one the drop-down menus to the right. To view the regression line of the graph, check the box on the right.

You can see the correlation of two variables conditional on a third variable by using the Add New Conditional Variable button at the bottom of the window. This will open up a slider and a box in which you can set the granularity of the slider. By moving the slider to the left or right, you can change the range of values of the conditional variable for which the scatter plot shows the correlation of the variables on the x- and y- axes. You can increase and decrease the width of the ranges by changing the granularity of the slider. A slider with granularity 1 will break the values of the conditional variable into sections one unit long, etc. The granularity cannot be set lower than one.

In a well-formed model, the scatter plot of a variable plotted against itself should appear as a straight line along the line y = x.

The Q-Q Plot tool is a test for normality of distribution.

If a variable has a distribution which is approximately Normal, its Q-Q plot should appear as a straight line with a positive slope. You can select the variable whose Q-Q plot you wish to view from the drop-down menu on the right.

The Normality Tests tool gives a text box with the results of the Kolmogorov and Anderson Darling Tests for normality for each variable. The Descriptive Statistics tool gives a text box with statistical information such as the mean, median, and variance of each variable.

The estimator box takes as input a data box (or simulation box) and a parametric model box and estimates, tests, and outputs an instantiated model for the data. With the exception of the EM Bayes estimator, Tetrad estimators do not accept missing values. If your data set contains missing values, the missing values can interpolated or removed using the data box. (Note that missing values are allowed in various Tetrad search procedures; see the section on the search box.)

- A parametric model box

- A graph box
- A simulation box
- An updater box

Bayes nets are acyclic graphical models parameterized by the conditional probability distribution of each variable on its parents' values, as in the instantiated model box. When the model contains no latent variables, the joint distribution of the variables equals the product of the distributions of the variables conditional on their respective parents. The maximum likelihood (ML) estimate of the joint probability distribution under a model is the product of the corresponding frequencies in the sample.

The ML Bayes estimator, because it estimates Bayes IMs, works only on models with discrete variables. The model estimated must not include latent variables, and the input data set must not include missing data values. A sample estimate looks like this:

The Model tab works exactly as it does in a Bayes instantiated model. The Model Statistics tab provides the p-value for a chi square test of the model, degrees of freedom, the chi square value, and the Bayes Information Criterion (BIC) score of the model.

A Dirichlet estimate estimates a Bayes instantiated model using a Dirichlet distribution for each category. In a Dirichlet estimate, the probability of each value of a variable (conditional on the values of the variable’s parents) is estimated by adding together a prior pseudo count (which is 1, by default, of cases and the number of cases in which the variable takes that value in the data, and then dividing by the total number of cases in the pseudocounts and in the data with that configuration of values of parent variables. The default prior pseudo-count can be changed inside the box. (For a full explanation of pseudocounts and Dirichlet estimate, see the section on Dirichlet instantiated models.)

The Dirichlet estimator in TETRAD does not work if the input data set contains missing data values.

The EM Bayes estimator takes the same input and gives the same output as the ML Bayes estimator, but is designed to handle data sets with missing data values, and input models with latent variables.

A SEM estimator estimates the values of parameters for a SEM parametric model. SEM estimates do not work if the input data set contains missing data values. A sample output looks like this:

Tetrad provides five parameter optimizers: RICF,( Drton, M., & Richardson, T. S. (2004, July). Iterative
conditional fitting for Gaussian ancestral graph models. *In Proceedings of the 20th conference on Uncertainty
in artificial intelligence* (pp. 130-137). AUAI Press). expectation-maximization (EM), regression,
Powell Journal of Econometrics 25 (1984) 303-325) and random search. Accurate regression estimates assume that
the input parametric model is a DAG, and that its associated statistics are based on a linear, Gaussian model.
The EM optimizer has the same input constraints as regression, but can handle latent variables.

Tetrad also provides two scores that can be used in estimation: feasible generalized least squares (FGLS) and Full Information Maximum Likelihood (FML).

If the graph for the SEM is a DAG, and we may assume that the SEM is linear with Gaussian error terms, we use multilinear regression to estimate coefficients and residual variances. Otherwise, we use a standard maximum likelihoood fitting function (see Bollen, Structural Equations with Latent Variables, Wiley, 1989, pg. 107) to minimize the distance between (a) the covariance over the variables as implied by the coefficient and error covariance parameter values of the model and (b) the sample covariance matrix. Following Bollen, we denote this function Fml; it maps points in parameter values space to real numbers, and, when minimized, yields the maximum likelihood estimation point in parameter space.

In either case, an Fml value may be obtained for the maximum likelihood point in parameter space, either by regression or by direct minimization of the Fml function itself. The value of Fml at this minimum (maximum likelihood) point, multiplied by N - 1 (where N is the sample size), yields a chi square statistics (ch^2) for the model, which when referred to the chi square table with appropriate degrees of freedom, yields a model p value. The degrees of freedom (dof) in this case is equal to the m(m-1)/2 - f, where m is the number of measured variables, and f is the number of free parameters, equal to the number of coefficient parameters plus the number of covariance parameters. (Note that the degrees of freedom many be negative, in which case estimation should not be done.) The BIC score is calculated as ch^2 - dof * log(N).

You can change which score optimizer Tetrad uses by choosing them from the drop-down menus at the bottom of the window and clicking “Estimate Again.”

The Tabular Editor and Implied Matrices tabs function exactly as they do in the instantiated model box, but in the estimator box, the last three columns of the table in the Tabular Editor tab are filled in. The SE, T, and P columns provide the standard errors, t statistics, and p values of the estimation.

The Model Statistics tab provides the degrees of freedom, chi square, p value, comparative fit index (CFI), root mean square error of approximation (RMSEA) and BIC score of a test of the model. It should be noted that while these test statistics are standard, they are not in general correct. See Mathias Drton, 2009, Likelihood ratio tests and singularities. Annals of Statistics 37(2):979-1012. arXiv:math.ST/0703360.

When the EM algorithm is used with latent variable models, we recommend multiple random restarts. The number of restarts can be set in the lower right hand corner of the Estimator Box.

A generalized graphical model may have non-linear relations and non-Gaussian distributions. These models are automatically estimated by the Powell method, which seeks a maximum likelihood solution.

The updater box takes an instantiated model as input, and, given information about the values of parameters in that model, updates the information about the values and relationships of other parameters.

The Updater allows the user to specify values of variables as “Evidence.” The default is that the conditional probabilities (Bayes net models; categorical variables) or conditional means (SEM models; continuous variables) are computed. For any variable for which evidence is specified, the user can click on “Manipulated,” in which case the Updater will calculate the conditional probabilities or conditional means for other variables when the evidence variables are forced to have their specified values. In manipulated calculations, all connections into a measured variable are discarded, the manipulated variables are treated as independent of their causes in the graph, and probabilities for variables that are causes of the manipulated variables are unchanged.

There are four available updater algorithms in Tetrad: the approximate updater, the row summing exact updater, the CPT invariant updater, and the SEM updater. All except for the SEM updater function only when given Bayes instantiated models as input; the SEM updater functions when given a SEM instantiated model as input. None of the updaters work on cyclic models.

- An instantiated model box
- An estimator box

- An instantiated model box (Note that the instantiated model will have the updated parameters)

The approximated updater is a fast but inexact algorithm. It randomly draws a sample data set from the instantiated model and calculates the conditional frequency of the variable to be estimated.

Take, for example, the following instantiated model:

When it is input into the approximate updater, the following window results:

If we click “Do Update Now” now, without giving the updater any evidence, the right side of the screen changes to show us the marginal probabilities of the variables.

The blue lines, and the values listed across from them, indicate the probability that the variable takes on the given value in the input instantiated model. The red lines indicate the probability that the variable takes on the given value, given the evidence we’ve added to the updater.

Since we have added no evidence to the updater, the red and blue lines are very similar in length. To view the marginal probabilities for a variable, either click on the variable in the graph to the left, or choose it from the scrolling menu at the top of the window. At the moment, they should all be very close to the marginal probabilities taken from the instantiated model.

Now, we’ll return to the original window. We can do so by clicking “Edit Evidence” under the Evidence tab. Suppose we know that X1 takes on the value 1 in our model, or suppose we merely want to see how X1 taking that value affects the values of the other variables. We can click on the box that says “1” next to X1. When we click “Do Update Now,” we again get a list of the marginal probabilities for X1.

Now that we have added evidence, the “red line” marginal probabilities have changed; for X1, the probability that X1=1 is 1, because we’ve told Tetrad that that is the case. Likewise, the probabilities that X1=0 and X1=2 are both 0.

Now, let’s look at the updated marginal probabilities for X2, a parent of X1.

The first image is the marginal probabilities before we added the evidence that X1=1. The second image is the updated marginal probabilities. They have changed; in particular, it has become much more likely that X2=0.

Under the Mode tab, we can change the type of information that the updater box gives us. The mode we have been using so far is “Marginals Only (Multiple Variables).” We can switch the mode to “In-Depth Information (Single Variable).” Under this mode, when we perform the update, we receive more information (such as log odds and joints, when supported; joint probabilities are not supported by the approximate updater), but only about the variable which was selected in the graph when we performed the update. To view information about a different variable, we must re-edit the evidence with that variable selected.

If the variable can take one of several values, or if we know the values of more than one variable, we can select multiple values by pressing and holding the Shift key and then making our selections. For instance, in the model above, suppose that we know that X1 can be 1 or 2, but not 0. We can hold the Shift key and select the boxes for 1 and 2, and when we click “Do Update Now,” the marginal probabilities for X2 look like this:

Since X1 must be 1 or 2, the updated probability that it is 0 is now 0. The marginal probabilities of X2 also change:

The updated marginal probabilities are much closer to their original values than they were when we knew that X1 was 1.

Finally, if we are arbitrarily setting the value of a variable—that is, the values of its parents have no effect on its value—we can check the “Manipulated” box next to it while we are we editing evidence, and the update will reflect this information.

Note that multiple values cannot be selected for evidence for SEM models.

The row summing exact updater is a slower but more accurate updater than the approximate updater. The complexity of the algorithm depends on the number of variables and the number of categories each variable has. It creates a full exact conditional probability table and updates from that. Its window functions exactly as the approximate updater does, with two exceptions: in “Multiple Variables” mode, you can see conditional as well as marginal probabilities, and in “Single Variable” mode, you can see joint values.

The CPT invariant exact updater is more accurate than the approximate updater, but slightly faster than the row summing exact updater. Ifs window functions exactly as the approximate updater down, with one exception: in “Multiple Variables” mode, you can see conditional as well as marginal probabilities.

The SEM updater does not deal with marginal probabilities; instead, it estimates means.

When it is input to the SEM updater, the following window results:

Suppose we know that the mean of X1 is .5. When we enter that value into the text box on the left and click “Do Update Now,” the model on the right updates to reflect that mean, changing the means of both X1 and several other variables. In the new model, the means of X2, X4, and X5 will all have changed. If we click the “Manipulated” check box as well, it means that we have arbitrarily set the mean of X1 to .5, and that the value of its parent variable, X4, has no effect on it. The graph, as well as the updated means, changes to reflect this.

The rest of the window has the same functionality as a SEM instantiated model window, except as noted above.

The knowledge box takes as input a graph or a data set and imposes additional constraints onto it, to aid with search.

- A graph box
- A parametric model box
- An instantiated model box
- A data box
- A simulation box
- A search box
- Another knowledge box

- A search box
- Another knowledge box

The tiers and edges option allows you to sort variables into groupings that can or cannot affect each other. It also allows you to manually add forbidden and required edges one at a time.

The tiers tab for a graph with ten variables looks like this:

Tiers separate your variables into a time line. Variables in higher-numbered tiers occur later than variables in lower-numbered tiers, which gives Tetrad information about causation. For example, a variable in Tier 3 could not possibly be a cause of a variable in Tier 1.

To place a variable in a tier, click on the variable in the “Not in tier” box, and then click on the box of the tier. If you check the “Forbid Within Tier” box for a tier, variables in that tier will not be allowed to be causes of each other. To increase or decrease the number of tiers, use the scrolling box in the upper right corner of the window.

You can quickly search, select and place variables in a tier using the Find button associated with each tier. Enter a search string into the Find dialogue box using asterisks as wildcard indicators. E.g., "X1*" would find and select variables X1 and X10.

You can also limit the search such that edges from one tier only are added to the next immediate tier e.g,. if Tier 1 "Can cause only next tier" is checked then edges from variables in Tier 1 to variables in Tier 3 are forbidden.

If you have annotated your variables with interventional status and interventional value tags using a metadata JSON file (see Data Box section) the Tiers and Edges panel will automatically place these variables in Tier 1. If you have information about the effects of the intervention variables you can use the groups tab to indicate this.

The groups tab for a graph with four variables looks like this:

In the groups tab, you can specify certain groups of variables which are forbidden or required to cause other groups of variables. To add a variable to the “cause” section of a group, click on the variable in the box at the top, and then click on the box to the left of the group’s arrow. To add a variable to the “effect” section of a group, click on the variable in the box at the top, and then click on the box to the right of the group’s arrow. You can add a group by clicking on one of the buttons at the top of the window, and remove one by clicking the “remove” button above the group’s boxes.

The edges tab for a graph with four variables looks like this:

In the edges tab, you can require or forbid individual causal edges between variables. To add an edge, click the type of edge you’d like to create, and then click and drag from the “cause” variable to the “effect” variable.

You can also use this tab to see the effects of the knowledge you created in the other tabs by checking and unchecking the boxes at the bottom of the window. You can adjust the layout to mimic the layout of the source (by clicking “source layout”) or to see the variables in their timeline tiers (by clicking “knowledge layout”).

If you use a graph as input to a knowledge box with the “Forbidden Graph” operation, the box will immediately add all edges in the parent graph as forbidden edges. It will otherwise work like a Tiers and Edges box.

If you use a graph as input to a knowledge box with the “Required Graph” operation, the box will immediately add all edges in the parent graph as required edges. It will otherwise work like a Tiers and Edges box.

This option allows you to build clusters for a measurement model. When first opened, the window looks like this:

You can change the number of clusters using the text box in the upper right hand corner. To place a variable in a cluster, click and drag the box with its name into the cluster pane. To move multiple variables at once, shift- or command-click on the variables, and (without releasing the shift/command button or the mouse after the final click) drag. In the search boxes, these variables will be assumed to be children of a common latent cause.

The simulation box takes a graph, parametric model, or instantiated model and uses it to simulate a data set.

- A graph box
- A parametric model box
- An instantiated model box
- An estimator box
- A data box
- Another simulation box
- A search box
- An updater box
- A regression box

- A graph box
- A compare box
- A parametric model box
- An instantiated model box
- An estimator box
- A data box
- Another simulation box
- A search box
- A classify box
- A regression box
- A knowledge box

When you first open the simulation box, you will see some variation on this window:

The “True Graph” tab contains the graph from which data is simulated.

Because it has no input box to create constraints, a parentless simulation box offers the greatest freedom for setting the graph type, model type, and parameters of your simulated data. In particular, it is the only way that the simulation box will allow you to create a random graph or graphs within the box. (If you are simulating multiple data sets, and want to use a different random graph for each one, you can select “Yes” under “Yes if a different graph should be used for each run.”) You can choose the type of graph you want Tetrad to create from the “Type of Graph” drop-down list.

This option creates a DAG by randomly adding forward edges (edges that do not point to a variable’s ancestors) one at a time. You can specify graph parameters such as number of variables, maximum and minimum degrees, and connectedness.

This option creates a DAG whose variable’s degrees obey a power law. You can specify graph parameters such as number of variables, alpha, beta, and delta values.

This option creates a cyclic graph. You can specify graph parameters such as number of variables, maximum and average degrees, and the probability of the graph containing at least one cycle.

This option creates a one-factor multiple indicator model. You can specify graph parameters such as number of latent nodes, number of measurements per latent, and number of impure edges.

This option creates a two-factor multiple indicator model. You can specify graph parameters such as number of latent nodes, number of measurements per latent, and number of impure edges.

In addition to the graph type, you can also specify the type of model you would like Tetrad to simulate.

Simulates a Bayes instantiated model. You can specify model parameters including maximum and minimum number of categories for each variable.

Simulates a SEM instantiated model. You can specify model parameters including coefficient, variance, and covariance ranges.

Simulates data using a linear Markov 1 DBN without concurrent edges. The Fisher model suggests that shocks should be applied at intervals and the time series be allowed to move to convergence between shocks. This simulation has many parameters that can be adjusted, as indicated in the interface. The ones that require some explanation are as follows.

- Low end of coefficient range, high end of coefficient range, low end of variance range, high end of variance range. Each variable is a linear function of the parents of the variable (in the previous time lag) plus Gaussian noise. The coefficients are drawn randomly from U(a, b) where a is the low end of the coefficient range and b is the high end of the coefficient range. Here, a < b. The Gaussian noise is drawn uniformly from U(c, d), where c is the low end of the variance range and d is the high end of the variance range. Here, c < d.
- Yes, if negative values should be considered. If no, only positive values will be recorded. This should not be used for large numbers of variables, since it is more difficult to find cases with all positive values when the number of variables is large.
- Percentage of discrete variables. The model generates continuous data, but some or all of the variables may be discretized at random. The user needs to indicate the percentage of variables (randomly chosen that one wishes to have discretized. The default is zero—i.e., all continuous variables.
- Number of categories of discrete variables. For the variables that are discretized, the number of categories to use to discretize each of these variables.
- Sample size. The number of records to be simulated.
- Interval between shocks. The number of time steps between shocks in the model.
- Interval between data recordings. The data are recorded every so many steps. If one wishes to allow to completely converge between steps (i.e., produce equilibrium data), set this interval to some large number like 20 and set the interval between shocks likewise to 20 Other values can be used, however.
- Epsilon for convergence. Even if you set the interval between data recordings to a large number, you can specify an epsilon such that if all values of variables differ from their values one time step back by less than epsilon, the series will be taken to have converged, and the remaining steps between data recordings will be skipped, the data point being recorded at convergence.

This is a model for simulating mixed data (data with both continuous and discrete variables. The model is given in Lee J, Hastie T. 2013, Structure Learning of Mixed Graphical Models, Journal of Machine Learning Research 31: 388-396. Here, mixtures of continuous and discrete variables are treated as log-linear.

- Percentage of discrete variables. The model generates continuous data, but some or all of the variables may be discretized at random. The user needs to indicate the percentage of variables (randomly chosen that one wishes to have discretized. The default is zero—i.e., all continuous variables.
- Number of categories of discrete variables. For the variables that are discretized, the number of categories to use to discretize each of these variables.
- Sample size. The number of records to be simulated.

This is a special simulation for representing time series. Concurrent edges are allowed. This can take a Time Series Graph as input, in which variables in the current lag are written as functions of the parents in the current and previous lags.

- Sample size. The number of records to be simulated.

The instantiated model used to simulate the data will be re-parameterized for each run of the simulation.

If you input a graph, you will be able to simulate any kind of model, with any parameters. But the model will be constrained by the graph you have input (or the subgraph you choose in the “True Graph” tab.) Because of this, if you create a simulation box with a graph as a parent, you will not see the “Type of Graph” option.

At the time of writing, a simulation box with a parametric model input acts as though the PM’s underlying graph had been input into the box.

If you input an instantiated model, your only options will be the sample size of your simulation and the number of data sets you want to simulate; Tetrad will simulate every one of them based on the parameters of the IM. The model will not be re-parameterized for each run of the simulation.

The search box takes as input a data set (in either a data or simulation box) and optionally a knowledge box, and searches for causal explanations represented by directed graphs. The result of a search is not necessarily—and not usually—a unique graph, but an object such as a pattern that represents a set of graphs, usually a Markov Equivalence class. More alternatives can be found by varying the parameters of search algorithms.

- A graph box
- A parametric model box
- An instantiated model box
- An estimator box
- A data box
- A simulation box
- Another search box
- A regression box
- A knowledge box

- A graph box
- A compare box
- A parametric model box
- A simulation box
- Another search box
- A knowledge box

Using the search box requires you to select an algorithm (optionally select a test/score), confirm/change search parameters and finally run the search.

The search box first asks what algorithm, statistical tests and/or scoring functions you would like to use in the search. The upper left panel allows you to filter for different types of search algorithms with the results of filtering appearing in the middle panel. Selecting a particular algorithm will update the algorithm description on the right panel.

Choosing the correct algorithm for your needs is an important consideration. Tetrad provides over 30 search algorithms (and more are added all of the time) each of which makes different assumptions about the input data, uses different parameters, and produces different kinds of output. For instance, some algorithms produce Markov blankets or patterns, and some produce full graphs; some algorithms work best with Gaussian or non-Gaussian data; some algorithms require an alpha value, some require a penalty discount, and some require both or neither. You can narrow down the list using the “Algorithm filter" panel, which allows you to limit the provided algorithms according to whichever factor is important to you.

Depending on the datatype used as input for the search (i.e., continuous, discrete, or mixed data) and algorithm selected, the lower left panel will display available statistical tests (i.e., tests of independence) and Bayesian scoring functions.

After selecting the algorithm and desired test/score, click on "Set parameters" which will allow you to confirm/change the parameters of the search.

After optionally changing any search parameters, click on "Run Search and Generate Graph" which will execute the search

PC algorithm (Spirtes and Glymour, Social Science Computer Review, 1991) is a pattern search which assumes that the underlying causal structure of the input data is acyclic, and that no two variables are caused by the same latent (unmeasured) variable. In addition, it is assumed that the input data set is either entirely continuous or entirely discrete; if the data set is continuous, it is assumed that the causal relation between any two variables is linear, and that the distribution of each variable is Normal. Finally, the sample should ideally be i.i.d.. Simulations show that PC and several of the other algorithms described here often succeed when these assumptions, needed to prove their correctness, do not strictly hold. The PC algorithm will sometimes output double headed edges. In the large sample limit, double headed edges in the output indicate that the adjacent variables have an unrecorded common cause, but PC tends to produce false positive double headed edges on small samples.

The PC algorithm is correct whenever decision procedures for independence and conditional independence are available. The procedure conducts a sequence of independence and conditional independence tests, and efficiently builds a pattern from the results of those tests. As implemented in TETRAD, PC is intended for multinomial and approximately Normal distributions with i.i.d. data. The tests have an alpha value for rejecting the null hypothesis, which is always a hypothesis of independence or conditional independence. For continuous variables, PC uses tests of zero correlation or zero partial correlation for independence or conditional independence respectively. For discrete or categorical variables, PC uses either a chi square or a g square test of independence or conditional independence (see Causation, Prediction, and Search for details on tests). In either case, the tests require an alpha value for rejecting the null hypothesis, which can be adjusted by the user. The procedures make no adjustment for multiple testing. (For PC, CPC, JPC, JCPC, FCI, all testing searches.)

The algorithm effectively takes conditional independence facts as input. Thus it will work for any type of data for which a conditional independence facts are known. In the interface, it will work for linear, Gaussian data (the Fisher Z test), discrete multinomial data the Chi Square test) and mixed multinomial/Gaussian data (the Conditional Gaussian test).

The graph outputs a pattern (or CP-DAG). This is an equivalence class of directed acyclic graphs (DAGs). Each DAG in the equivalence class has all of the adjacencies (and no more) of the pattern. Each oriented edge in the pattern is so oriented in each of the DAG in the equivalence class. Unoriented edges in the equivalence class cannot be oriented by conditional independence facts. For example, if the model is X->Y->Z, the output will be X—Y—Z. There are not collider in this model, so the algorithm will not detect one. Since there are not colliders, the Meek cannot orient additional edges. If the model were X<-Y<-Z, the output would also be X—Y—Z; this model is in the same equivalence class as X->Y->Z. The model X->Y<-Z would be its own equivalence class, since the collider in this model can be oriented. See Spirtes et al. (2000) for more details.

The CPC (Conservative PC) algorithm (Ramsey et al., ??) modifies the collider orientation step of PC to make it more conservative—that is, to increase the precision of collider orientations at the expense of recall. It does this as follows. Say you want to orient X—Y—Z as a collider or a noncollider; the PC algorithm looks at variables adjacent to X or variables adjacent to Z to find a subset S such that X is independent of Z conditional on S. The CPC algorithm considers all possible such sets and records the set on which X is conditionally independent of Z. If all of these sets contain Y, it orients X—Y—Z as a noncollider. If none of them contains Z, if orient X—Y—Z as a collider. If some contain Z but other don’t, it marks it as ambiguous, with an underline. Thus, the output is ambiguous between patterns; in order to get a specific pattern out of the output, one needs first to decide whether the underlined triples are colliders or noncolliders and then to apply the orientation rules in Meek (1997).

The PC algorithm is correct whenever decision procedures for independence and conditional independence are available. The procedure conducts a sequence of independence and conditional independence tests, and efficiently builds a pattern from the results of those tests. As implemented in TETRAD, PC is intended for multinomial and approximately Normal distributions with i.i.d. data. The tests have an alpha value for rejecting the null hypothesis, which is always a hypothesis of independence or conditional independence. For continuous variables, PC uses tests of zero correlation or zero partial correlation for independence or conditional independence respectively. For discrete or categorical variables, PC uses either a chi square or a g square test of independence or conditional independence (see Causation, Prediction, and Search for details on tests). In either case, the tests require an alpha value for rejecting the null hypothesis, which can be adjusted by the user. The procedures make no adjustment for multiple testing. (For PC, CPC, JPC, JCPC, FCI, all testing searches.)

Same as for PC.

An e-pattern (extended pattern), consistent of directed and undirected edges where some of the triple may have been marked with underlines to indicate ambiguity, as above. It may be that bidirected edges are oriented as X->Y<->X<-W if two adjacent colliders are oriented; this is not ruled out.

See Drton and Maathuis (2017). The idea is to modify the adjacency search of PC so that if the order of the variables is randomized, the adjacency output is not affected. This is done as follows. The order of operations for the step where unconditional independencies are calculated is not affected; these may be done in any order. However, for the step in which one conditions on one variable, the output of that step could be affected by the order in which the operations are done. So instead of removing edges in this step, one simply records which edges one would remove, and then at the end of the step removes them all. Similarly for subsequence steps. In this way, the adjacencies of variables in the output of the adjacency step are fixed no matter the order in which the operations are visited. One them does collider orientation and applies the orientation rules in Meek (1997); there may be orientation differences from one run to the next still, if the order of the variables in the dataset is modified.

Same as for PC.

Same as for PC.

CPC, with the PC-Stable adjacency step substituted for the PC adjacency search.

Same as for PC.

Same as for CPC (an e-pattern).

Similar in spirit to CPC but orients all unshielded triples using maximum likelihood conditioning sets. The idea is as follows. The adjacency search is the same as for PC, but colliders are oriented differently. Let X—Y—Z be an unshielded triple (X not adjacent to Z) and find all subsets S from among the adjacents of X or the adjacents of Z such that X is independent of Z conditional on S. However, instead of using the CPC rule to orient the triple, instead just list the p-values for each of these conditional independence judgments and pick the set S’ that yields the highest such p-value. Then orient X->Y<-Z if S does not contain Y and X—Y—Z otherwise. This orients all unshielded triples. It’s possible (though rare) that adjacent triples both be oriented as 2-cycles, X->Y<->Z<-W. If this happens, pick one of the other of these triples or orient as a collider, arbitrarily. This guarantees that the resulting graph will be a pattern.

Same as for PC.

Same as PC, a pattern.

alpha, depth, useMaxPOrientationHeuristic, maxPOrientationMaxPathLength

FGES is an optimized and parallelized version of an algorithm developed by Meek [Meek, 1997] called the Greedy Equivalence Search (GES). The algorithm was further developed and studied by Chickering [Chickering, 2002]. GES is a Bayesian algorithm that heuristically searches the space of CBNs and returns the model with highest Bayesian score it finds. In particular, GES starts its search with the empty graph. It then performs a forward stepping search in which edges are added between nodes in order to increase the Bayesian score. This process continues until no single edge addition increases the score. Finally, it performs a backward stepping search that removes edges until no single edge removal can increase the score. More information is available here and here. The reference is Ramsey et al., 2017.

The algorithms requires a decomposable score—that is, a score that for the entire DAG model is a sum of logged scores of each variables given its parents in the model. The algorithms can take all continuous data (using the SEM BIC score), all discrete data (using the BDeu score) or a mixture of continuous and discrete data (using the Conditional Gaussian score); these are all decomposable scores.

Data that’s all continuous, all discrete, or a mixture of continuous and discrete variables. Continuous variables will be assumed to be linearly associated; discrete variable will be assumed to be associated by multinomial conditional probability tables. Continuous variables for the mixed case will be assumed to be jointly Gaussian.

A pattern, same as PC.

samplePrior, structurePrior, penaltyDiscount, symmetricFirstStep, faithfulnessAssumed, maxDegree, parallelism, verbose meekVerbose

Adjusts the discrete BDeu variable score of FGES so allow for multiple datasets as input. The BDeu scores for each data set are averaged at each step of the algorithm, producing a model for all data sets that assumes they have the same graphical structure across dataset. Note that in order to use this algorithm in a nontrivial way, one needs to have loaded or simulated multiple dataset.

A set of discrete datasets with the same variables and sample sizes.

A pattern, interpreted as a common model for all datasets.

All of the parameters from FGES are available for IMaGES. Additionally:

Adjusts the continuous variable score (SEM BIC) of FGES so allow for multiple datasets as input. The linear, Gaussian BIC scores for each data set are averaged at each step of the algorithm, producing a model for all data sets that assumes they have the same graphical structure across dataset.

A set of continuous datasets with the same variables and sample sizes.

A pattern, interpreted as a common model for all datasets.

All of the parameters from FGES are available for IMaGES. Additionally:

The FCI algorithm is a constraint-based algorithm that takes as input sample data and optional background knowledge and in the large sample limit outputs an equivalence class of CBNs that (including those with hidden confounders) that entail the set of conditional independence relations judged to hold in the population. It is limited to several thousand variables, and on realistic sample sizes it is inaccurate in both adjacencies and orientations. FCI has two phases: an adjacency phase and an orientation phase. The adjacency phase of the algorithm starts with a complete undirected graph and then performs a sequence of conditional independence tests that lead to the removal of an edge between any two adjacent variables that are judged to be independent, conditional on some subset of the observed variables; any conditioning set that leads to the removal of an adjacency is stored. After the adjacency phase, the resulting undirected graph has the correct set of adjacencies, but all of the edges are unoriented. FCI then enters an orientation phase that uses the stored conditioning sets that led to the removal of adjacencies to orient as many of the edges as possible. See [Spirtes, 1993].

The data are continuous, discrete, or mixed.

A partial ancestral graph (see Spirtes et al., 2000).

All of the parameters from FCI are below.

depth, maxPathLength, completeRuleSetUsed

A modification of the FCI algorithm in which some expensive steps are finessed and the output is somewhat differently interpreted. In most cases this runs faster than FCI (which can be slow in some steps) and is almost as informative. See Colombo et al., 2012.

Data for which a conditional independence test is available.

A partial ancestral graph (PAG). See Spirtes et al., 2000.

All of the parameters from FCI are available for RFCI. Additionally:

depth, maxPathLength, completeRuleSetUsed

RFCI-BSC is a combination of the RFCI [Colombo, 2012] algorithm and the Bayesian Scoring of Constraints (BSC) method [Jabbari, 2017] that can generate and probabilistically score multiple models, outputting the most probable one. This search algorithm is a hybrid method that derives a Bayesian probability that the set of independence tests associated with a given causal model are jointly correct. Using this constraint-based scoring method, we are able to score multiple causal models, which possibly contain latent variables, and output the most probable one. See [Jabbari, 2017].

The data are discrete only.

A partial ancestral graph (PAG). See Spirtes et al., 2000.

All of the parameters from RFCI are available for RFCI-BSC. Additionally:

numRandomizedSearchModels, thresholdNoRandomDataSearch, cutoffDataSearch, thresholdNoRandomConstrainSearch, cutoffConstrainSearch, numBscBootstrapSamples, lowerBound, upperBound, outputRBD

GFCI is a combination of the FGES [CCD-FGES, 2016] algorithm and the FCI algorithm [Spirtes, 1993] that improves upon the accuracy and efficiency of FCI. In order to understand the basic methodology of GFCI, it is necessary to understand some basic facts about the FGES and FCI algorithms. The FGES algorithm is used to improve the accuracy of both the adjacency phase and the orientation phase of FCI by providing a more accurate initial graph that contains a subset of both the non-adjacencies and orientations of the final output of FCI. The initial set of nonadjacencies given by FGES is augmented by FCI performing a set of conditional independence tests that lead to the removal of some further adjacencies whenever a conditioning set is found that makes two adjacent variables independent. After the adjacency phase of FCI, some of the orientations of FGES are then used to provide an initial orientation of the undirected graph that is then augmented by the orientation phase of FCI to provide additional orientations. A verbose description of GFCI can be found here (discrete variables) and here (continuous variables).

Same as for FCI.

Same as for FCI.

Uses all of the parameters of FCI (see Spirtes et al., 1993) and FGES (see CCD-FGES et al., 2016).

The tsFCI algorithm is a version of FCI for time series data. See the FCI documentation for a description of the FCI algorithm, which allows for unmeasured (hidden, latent) variables in the data-generating process and produces a PAG (partial ancestral graph). tsFCI takes as input a “time lag data set,” i.e., a data set which includes time series observations of variables X1, X2, X3, ..., and their lags X1:1, X2:1, X3:1, ..., X1:2, X2:2,X3:2, ... and so on. X1:n is the nth-lag of the variable X1. To create a time lag data set from a standard tabular data set (i.e., a matrix of observations of X1, X2, X3, ...), use the “create time lag data” function in the data manipulation toolbox. The user will be prompted to specify the number of lags (n), and a new data set will be created with the above naming convention. The new sample size will be the old sample size minus n.

The (continuous) data has been generated by a time series.

A PAG over the input variables with stated number of lags.

tsGFCI uses a BIC score to search for a skeleton. Thus, the only user-specified parameter is an optional “penalty score” to bias the search in favor of more sparse models. See the description of the GES algorithm for discussion of the penalty score. For the traditional definition of the BIC score, set the penalty to 1.0. The orientation rules are the same as for FCI. As is the case with tsFCI, tsGFCI will automatically respect the time order of the variables and impose a repeating structure. Firstly, it puts lagged variables in appropriate tiers so, e.g., X3:2 can cause X3:1 and X3 but X3:1 cannot cause X3:2 and X3 cannot cause either X3:1 or X3:2. Also, it will assume that the causal structure is the same across time, so that if the edge between X1 and X2 is removed because this increases the BIC score, then also the edge between X1:1 and X2:1 is removed, and so on for additional lags if they exist. When some edge is removed as the result of a score increase, all similar (or “homologous”) edges are also removed.

The (continuous) data has been generated by a time series.

A PAG over the input variables with stated number of lags.

Uses all of the parameters of FCI (see Spirtes et al., 1993) and FGES (see CCD-FGES et al., 2016).

tsIMAGES is a version of tsGFCI which averages BIC scores across multiple data sets. Thus, it is used to search for a PAG (partial ancestral graph) from time series data from multiple units (subjects, countries, etc). tsIMAGES allows both for unmeasured (hidden, latent) variables and the possibility that different subjects have different causal parameters, though they share the same qualitative causal structure. As with IMAGES, the user can specify a “penalty score” to produce more sparse models. For the traditional definition of the BIC score, set the penalty to 1.0. See the documentation for IMAGES and tsGFCI.

The (continuous) data has been generated by a time series.

A PAG over the input variables with stated number of lags.

Uses the parameters of IMaGES.

This is a restriction of the FGES algorithm to union of edges over the combined Markov blankets of a set of targets, including the targets. In the interface, just one target may be specified. See Ramsey et al., 2017 for details. In the general case, finding the graph over the Markov blanket variables of a target (including the target) is far faster than finding the pattern for all of the variables.

The same as FGES

A graph over a selected group of nodes that includes the target and each node in the Markov blanket of the target. This will be the same as if FGES were run and the result restricted to just these variables, so some edges may be oriented in the returned graph that may not have been oriented in a pattern over the selected nodes.

Uses the parameters of FGES (see CCD-FGES et al., 2016).

Markov blanket fan search. Similar to FGES-MB (see CCD-FGES, 2016) but using PC as the basic search instead of FGES. The rules of the PC search are restricted to just the variables in the Markov blanket of a target T, including T; the result is a graph that is a pattern over these variables.

Same as for PC

A pattern over a selected group of nodes that includes the target and each node in the Markov blanket of the target.

Uses the parameters of PC.

This is just the adjacency search of the PC algorithm, included here for times when just the adjacency search is needed, as when one is subsequently just going to orient variables pairwise.

Same as for PC

An undirected graph over the variables of the input dataset. In particular, parents of a variables are not married by FAS, so the resulting graph is not a Markov random field. For example, if X->Y<-Z, the output will be X—Y—Z with X—Z. The parents of Y will be joined by an undirected edge, morally, only if they are joined by a trek in the true graph.

Need reference. Finds a Markov random field (with parents married) for a dataset in which continuous and discrete variables are mixed together. For example, if X->Y<-Z, the output will be X—Y—Z with X—Z. The parents of Y will be joined by an undirected edge, morally, even though this edge does not occur in the true model.

Data are mixed.

A Markov random field for the data.

mgmParam1, mgmParam2, mgmParam3

A translation of the Fortran code for GLASSO (Graphical LASSO—see Friedman, Tibshirani anad Hastie, 2007) Like MGM, this produces an undirected graph in which parents are always married.

The data are continuous.

A Markov random field.

Searches for causal structure over latent variables, where the true models are Multiple Indicator Models (MIM’s) as described in the Graphs section. The idea is this. There is a set of latent (unmeasured) variables over which a directed acyclic model has been defined, Then for each of these latent L there are 3 (preferably 4) or more measures of that variable—that is, measured variables that are all children of L. Under these conditions, one may define tetrad constraints (see Spirtes et al., 2000). There is a theorem to the effect that if certain patterns of these tetrad constraints hold, there must be a latent common cause of all of them (the Tetrad Representation Theorem). The FOFC (Find One Factor Clusters) takes advantage of this fact. The basic idea is to build up clusters one at a time by adding variables that keep them pure in the sense that all relevant tetrad constraints still hold. There are different ways of going about this. One could try to build one cluster up as far as possible, then remove all of those variables from the set, and try to make a another cluster using the remaining variables (SAG, Seed and Grow). Or one can try in parallel to grow all possible clusters and then choose among the grown clusters using some criterion such as cluster size (GAP, Grow and Pick). In general, GAP is more accurate. The result is a clustering of variables. Once one has such a “measurement model, one can estimate (using the ESTIMATOR box) a covariance matrix over the latent variables that are parents of the measures and use some algorithm such as PC or GES to estimate a pattern over the latent variables. The algorithm to run PC or GES on this covariance matrix is called MimBuild (“MIM” is the graph, Multiple Indicator Model; “Build” means build). MimBUILD is an optional choice inside FOFC In this way, one may recover causal structure over the latents. The more measures one has for each latent the better the result is, generally. At least 3 measured indicator variables are needed for each latent variable. The larger the sample size the better. One important issue is that the algorithm is sensitive to so-called “impurities”—that is,causal edges among the measured variables, or between measured variables and multiple latent variables. The algorithm will in effect remove one measure in each impure pair from consideration.

FTFC (Find Two Factor Clusters) is similar to FOFC, but instead of each cluster having one latent that is the parent of all of the measure in the cluster, it instead has two such latents. So each measure has two latent parents; these are two “factors.” Similarly to FOFC, constraints are checked for, but in this case, the constraints must be sextad constraints, and more of them must be satisfied for each pure cluster (see Kummerfelt et al., 2014). Thus, the number of measures in each cluster, once impure edges have been taken into account, must be at least six, preferably more.

Continuous data over the measures with at least six variable variables in each cluster once variables involve in impure edges have been removed.

A clustering of measures. It may be assumed that each cluster has at least two factors and that the clusters are pure.

LiNGAM (Shimizu et al., 2006) was one of the first of the algorithms that assumed linearity among the variables and non-Gaussianity of error term, and still one of the best for smaller models, for the basic algorithm, implemented here. The idea is to use the Independent Components Analysis (ICA) algorithm to check all permutations of the variables to find one that is a causal order—that is, one in which earlier variables can cause later variables but not vice-versa. The method is clever. First, since we assume the model is a directed acyclic graph (DAG), there must be some permutation of the variables for which the main diagonal of the inverse of the weight matrix contains no zeros. This gives us a permuted estimate of the weight matrix. Then we look for a permutation of this weight matrix that is lower triangular. There must be one, since the model is assumed to be a DAG. But a lower triangular weight matrix just gives a causal order, so we’re done.

In the referenced paper, we implement Algorithm A, which is described above. Once one has a causal order the only thing one needs to do is to eliminate the extra edges. For this, we use the causal order to define knowledge of tiers and run FGES.

Our implementation of LiNGAM has one parameter, penalty discount, used for the FGES adjacency search. The method as implemented does not scale much beyond 10 variables, because it is checking every permutation of all of the variables (twice). The implementation of ICA we use is FastIca (Hyvärinen et al., 2004).

Shimizu, S., Hoyer, P. O., Hyvärinen, A., & Kerminen, A. (2006). A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7(Oct), 2003-2030.

Hyvärinen, A., Karhunen, J., & Oja, E. (2004). Independent component analysis (Vol. 46). John Wiley & Sons.

FASK learns a linear model in which all of the variables are skewed.

The idea is as follows. First, FAS-stable is run on the data, producing an undirected graph. We use the BIC score as a conditional independence test with a specified penalty discount c. This yields undirected graph G0 . The reason FAS-stable works for sparse cyclic models where the linear coefficients are all less than 1 is that correlations induced by long cyclic paths are statistically judged as zero, since they are products of multiple coefficients less than 1. Then, each of the X − Y adjacencies in G0 is oriented as a 2-cycle X += Y , or X → Y , or X ← Y . Taking up each adjacency in turn, one tests to see whether the adjacency is a 2-cycle by testing if the difference between corr(X, Y ) and corr(X, Y |X > 0), and corr(X, Y ) and corr(X, Y |Y > 0), are both significantly not zero. If so, the edges X → Y and X ← Y are added to the output graph G1 . If not, the Left-Right orientation is rule is applied: Orient X → Y in G1, if (E(X Y |X > 0)/ E(X 2|X > 0)E(Y 2 |X > 0) − E(X Y |Y > 0)/ E(X 2 |Y > 0)E(Y 2|Y > 0)) > 0; otherwise orient X ← Y . G1 will be a fully oriented graph. For some models, where the true coefficients of a 2-cycle between X and Y are more or less equal in magnitude but opposite in sign, FAS-stable may fail to detect an edge between X and Y when in fact a 2-cycle exists. In this case, we check explicitly whether corr(X, Y |X > 0) and corr(X, Y |Y > 0) differ by more than a set amount of 0.3. If so, the adjacency is added to the graph and oriented using the aforementioned rules.

We include pairwise orientation rule RSkew, Skew, and Tanh from Hyvärinen, A., & Smith, S. M. (2013). Pairwise likelihood ratios for estimation of non-Gaussian structural equation models. Journal of Machine Learning Research, 14(Jan), 111-152, so in some configurations FASK can be made to implement an algorithm that has been called in the literature "Pairwise LiNGAM"--this is intentional; we do this for ease of comparison. You'll get this configuration if you choose one of these pairwise orientation rules, together with the FAS with orientation alpha and two-cycle threshold set to zero and skewness threshold set to 1, for instance.

See Sanchez-Romero R, Ramsey JD, Zhang K, Glymour MR, Huang B, Glymour C. Causal discovery of feedback
networks with functional magnetic resonance imaging. *Network Neuroscience* 2018.

Continuous, linear data in which all of the variables are skewed.

A fully directed, potentially cyclic, causal graph.

Multi-FASK is a metascript that learns a model from a list of datasets in a method similar to IMaGES (see). For adjacencies, it uses FAS-Stable with the voting-based score from IMaGES used as a test (using all of the datasets, standardized), producing a single undirected graph G. It then orients each edge X--Y in G for each dataset using the FASK (see) left-right rule and orient X->Y if that rule orients X--Y as such in at least half of the datasets. The final graph is returned.

For FASK, See Sanchez-Romero R, Ramsey JD, Zhang K, Glymour MR, Huang B, Glymour C. Causal discovery of feedback networks with functional magnetic resonance imaging. Network Neuroscience 2018.

Same as FASK.

Same as FASK.

This is an algorithm that orients an edge X--Y for continuous variables based on non-Gaussian information. This rule in particular uses an entropy calculation to make the orientation. Note that if the variables X and Y are both Gaussian, and the model is linear, it is not possible to orient the edge X--Y pairwise; any attempt to do so would result in random orientation. But if X and Y are non-Gaussian, the orientation is fairly easy. This rule is similar to Hyvarinen and Smith's (2013) EB rule, but using Anderson Darling for the measure of non-Gaussianity, to somewhat better effect. See Ramsey et al. (2012).

This is an algorithm that orients an edge X--Y for continuous variables based on non-Gaussian information. This rule in particular uses a skewness to make the orientation. Note that if the variables X and Y are both Gaussian, and the model is linear, it is not possible to orient the edge X--Y pairwise; any attempt to do so would result in random orientation. But if X and Y are non-Gaussian, in particular in this case, if X and Y are skewed, the orientation is relatively straightforward. See Hyvarinen and Smith (2013) for details.

The Skew rule is differently motivated from the RSkew rule (see), though they both appeal to the skewness of the variables.

This is an algorithm that orients an edge X--Y for continuous variables based on non-Gaussian information. This rule in particular uses a skewness to make the orientation. Note that if the variables X and Y are both Gaussian, and the model is linear, it is not possible to orient the edge X--Y pairwise; any attempt to do so would result in random orientation. But if X and Y are non-Gaussian, in particular in this case, if X and Y are skewed, the orientation is relatively straightforward. See Hyvarinen and Smith (2013) for details.

The RSkew rule is differently motivated from the Skew rule (see), though they both appeal to the skewness of the variables.

Continuous data in which the variables are non-Gaussian. Non-Gaussianity can be assessed using the Anderson-Darling score, which is available in the Data box.

Orients all of the edges in the input graph using the selected score.

All of the below tests do testwise deletion as a default way of dealing with missing values. For testwise deletion, if a test, say, I(X, Y | Z), is done, columns for X, Y, and Z are scanned for missing values. If any row occurs in which X, Y, or Z is missing, that row is deleted from the data for those three variables. So if a different test, I(R, W | Q, T) is done, different rows may be stricken from the data. That is, the deletion is done testwise. For a useful discussion of the testwise deletion condition, see for instance Tu, R., Zhang, C., Ackermann, P., Mohan, K., Kjellström, H., & Zhang, K. (2019, April). Causal discovery in the presence of missing data. In The 22nd International Conference on Artificial Intelligence and Statistics (pp. 1762-1770). PMLR. For all of these tests, if no data are missing, the behavior will be as if testwise deletion were not being done.

This is a test based on the BDeu score given in Heckerman, D., Geiger, D., & Chickering, D. M. (1995).
Learning Bayesian networks: The combination of knowledge and statistical data. *Machine learning, 20*(3),
197-243, used as a test. This gives a score for any two variables conditioned on any list of others which is
more positive for distributions which are more strongly dependent. The test for X _||_ Y | Z compares two
different models, X conditional on Y, and X conditional on Y and Z; the scores for the two models are
subtracted, in that order. If the difference is negative, independence is inferred.

equivalentSamplelSize, structurePrior

Fisher Z judges independence if the conditional correlation is cannot statistically be distinguished from zero. Primarily for the linear, Gaussian case.

This uses the SEM BIC Score to create a test for the linear, Gaussian case, where we include an additional
penalty term, which is commonly used. We call this the *penalty discount*. So our formulas has BIC = 2L
- ck log N,where L is the likelihood, c the penalty discount (usually greater than or equal to 1), and N the
sample size. Since the assumption is that the data are distributed as Gaussian, this reduces to BIC = -n log
sigma - ck ln N, where sigma is the standard deviation of the linear residual obtained by regressing a child
variable onto all of its parents in the model.

The Probabilistic Test applies a Bayesian method to derive the posterior probability of an independence constraint R = (X⊥Y|Z) given a dataset D. This is intended for use with datasets with discrete variables. It can be used with constraint-based algorithms (e.g., PC and FCI). Since this test provides a probability for each independence constraint, it can be used stochastically by sampling based on the probabilities of the queried independence constraints to obtain several output graphs. It can also be used deterministically by using a fixed decision threshold on the probabilities of the queried independence constraints to generate a single output graph.

noRandomlyDeterminedIndependence cutoffIndTest priorEquivalentSampleSize

CCI ("Conditional Correlation Independence") is a fairly general independence test—not completely general, but general for additive noise models—that is, model in which each variable is equal to a (possibly nonlinear) function of its parents, plus some additive noise, where the noise may be arbitrarily distributed. That is, X = f(parent(X)) + E, where f is any function and E is noise however distributed; the only requirement is that thre be the “+” in the formula separating the function from the noise. The noise can’t for instance, be multiplicative, e.g., X = f(parent(X)) x E. The goal of the method is to estimate whether X is independent of Y given variables Z, for some X, Y, and Z. It works by calculating the residual of X given Z and the residual of Y given Z and looking to see whether those two residuals are independent. This test may be used with any constraint-based algorithm (PC, FCI, etc.).

alpha, numBasisFunctions, kernelType, kernelMultiplier, basisType, kernelRegressionSampleSize

This is the usual Chi-Square test for discrete variables; consult an introductory statistics book for
details for the unconditional case, where you're just trying, e.g., to determine if X and Y are independent.
For the conditional case, the test proceeds as in Fienberg, S. E. (2007). *The analysis of
cross-classified categorical data*, Springer Science & Business Media, by identifying and removing from
consideration zero rows or columns in the conditional tables and judging dependence based on the remaining
rows and columns.

This is the usual test of d-separation, a property of graphs, not distributions. It's not really a test, but it can be used in place of a test of the true graph is known. This is a way to find out, for constraint-based algorithms, or even for some score-based algorithms like FGES, what answer the algorithm would give if all of the statistical decisions made are correct. Just draw an edge from the true graph to the algorithm--the d-separation option will appear, and you can then just run the search as usual.

This is a BIC score for the discrete case, used as a test. The likelihood is judged by the multinomial tables directly, and this is penalized as is usual for a BIC score. The only surprising thing perhaps is that we use the formula BIC = 2L - k ln N, where L is the likelihood, k the number of parameters, and N the sample size, instead of the usual L + k / 2 ln N. So higher BIC scores will correspond to greater dependence. In the case of independence, the BIC score will be negative, since the likelihood will be zero, and this will be penalized. The test yields a p-value; we simply use alpha - p as the score, where alpha is the cutoff for rejecting the null hypothesis of independence. This is a number that is positive for dependent cases and negative for independent cases.

penaltyDiscount, structurePrior

This is completely parallel to the Chi-Square statistic, using a slightly different method for estimating
the statistic. The alternative statistic is still distributed as chi-square in the limit. In practice, this
statistic is more or less indistinguishable in most cases from Chi-Square. For an explanation, see Spirtes,
P., Glymour, C. N., Scheines, R., Heckerman, D., Meek, C., Cooper, G., & Richardson, T. (2000). *Causation,
prediction, and search*. MIT press.

KCI ("Kernel Conditional Independence") is a general independence test for model in which X = f(parents(X), eY); here, eY does not need to be additive; it can stand in any functional relationships to the other variables. The variables may even be discrete. The goal of the method is to estimate whether X is independent of Y given Z, completely generally. It uses the kernel trick to estimate this. As a result of using the kernel trick, the method is complex in the direction of sample size, meaning that it may be very slow for large samples. Since it’s slow, individual independence results are always printed to the console so the user knows how far a procedure has gotten. This test may be used with any constraint-based algorithm (PC, FCI, etc.)

alpha, kciUseAppromation, kernelMultiplier, kciNumBootstraps, thresholdForNumEigenvalues, kciEpsilon

Conditional Gaussian Test is a likelihood ratio test based on the conditional Gaussian likelihood function. This is intended for use with datasets where there is a mixture of continuous and discrete variables. It is assumed that the continuous variables are Gaussian conditional on each combination of values for the discrete variables, though it will work fairly well even if that assumption does not hold strictly. This test may be used with any constraint-based algorithm (PC, FCI, etc.). See See Andrews, B., Ramsey, J., & Cooper, G. F. (2018). Scoring Bayesian networks of mixed variables. International journal of data science and analytics, 6(1), 3-18.

Degenerate Gaussian Likelihood Ratio Test may be used for the case where there is a mixture of discrete and Gaussian variables. Calculates a a likelihood ratio based on likelihood that is calculated using a conditional Gaussian assumption. See Andrews, B., Ramsey, J., & Cooper, G. F. (2019). Learning high-dimensional directed acyclic graphs with mixed data-types. Proceedings of machine learning research, 104, 4.

Most TETRAD searches can be performed with resampling. This option is available on the Set Parameters screen. When it is selected, the search will be performed multiple times on randomly selected subsets of the data, and the final output graph will be the result of a voting procedure among all of the graphs. These subsets may be selected with replacement (bootstrapping) or without. There are also options for the user to set the size of the subset, and the number of resampling runs. The default number of resampling runs is zero, in which case no resampling will be performed.

For each potential edge in the final output graph, the individual sampled graphs may contain a directed edge in one direction, the other direction, a bidirected edge, an uncertain edge, or no edge at all. The voting procedure reconciles all of these possible answers into a single final graph, and the "ensemble method," which can be set by the user in the parameter settings screen, determines how it will do that.

The three available ensemble methods are Preserved, Highest, and Majority. Preserved tends to return the densest graphs, then Highest, and finally Majority returns the sparsest. The Preserved ensemble method ensures that an edge that has been found by some portion of the individual sample graphs is preserved in the final graph, even if the majority of sample graphs returned [no edge] as their answer for that edge. So the voting procedure for Preserved is to return the edge orientation that the highest percentage of sample graphs returned, other than [no edge]. The Highest ensemble method, on the other hand, simply returns the edge orientation which the highest proportion of sample graphs returned, even if that means returning [no edge]. And the Majority method requires that at least 50 percent of the sample graphs agree on an edge orientation in order to return any edge at all. If the highest proportion of sample graphs agree on, for instance, a bidirected edge, but only 40 percent of them do so, then the Majority ensemble method will return [no edge] for that edge.

Like the tests, above, all of the below tests do testwise deletion as a default way of dealing with missing values. For testwise deletion, if a score, say, score(X | Y, Z), is done, columns for X, Y, and Z are scanned for missing values. If any row occurs in which X, Y, or Z is missing, that row is deleted from the data for those three variables. So if a different test, score(R | W, Q, T) is done, different rows may be stricken from the data. That is, the deletion is done testwise. For a useful discussion of the testwise deletion condition, see for instance Tu, R., Zhang, C., Ackermann, P., Mohan, K., Kjellström, H., & Zhang, K. (2019, April). Causal discovery in the presence of missing data. In The 22nd International Conference on Artificial Intelligence and Statistics (pp. 1762-1770). PMLR. For all of these tests, if no data are missing, the behavior will be as if testwise deletion were not being done.

This is the BDeu score given in Heckerman, D., Geiger, D., & Chickering, D. M. (1995). Learning Bayesian
networks: The combination of knowledge and statistical data. *Machine learning, 20*(3), 197-243. This
gives a score for any two variables conditioned on any list of others which is more positive for
distributions which are more strongly dependent.

equivalentSampleSize, samplePrior

Conditional Gaussian BIC Score may be used for the case where there is a mixture of discrete and Gaussian variables. Calculates a BIC score based on likelihood that is calculated using a conditional Gaussian assumption. See Andrews, B., Ramsey, J., & Cooper, G. F. (2018). Scoring Bayesian networks of mixed variables. International journal of data science and analytics, 6(1), 3-18.

Degenerate Gaussian BIC Score may be used for the case where there is a mixture of discrete and Gaussian variables. Calculates a BIC score based on likelihood that is calculated using a conditional Gaussian assumption. See Andrews, B., Ramsey, J., & Cooper, G. F. (2019). Learning high-dimensional directed acyclic graphs with mixed data-types. Proceedings of machine learning research, 104, 4.

This uses d-separation to make something that acts as a score if you know the true graph. A score in Tetrad, for FGES, say, is a function that for X and Y conditional on Z, returns a negative number if X _||_ Y | Z and a positive number otherwise. So to get this behavior in no u certain terms, we simply return -1 for independent cases and +1 for dependent cases. Works like a charm. This can be used for FGES to check what the ideal behavior of the algorithm should be. Simply draw an edge from the true graph to the search box, select FGES, and search as usual.

This is a BIC score for the discrete case. The likelihood is judged by the multinomial tables directly, and this is penalized as is usual for a BIC score. The only surprising thing perhaps is that we use the formula BIC = 2L - k ln N, where L is the likelihood, k the number of parameters, and N the sample size, instead of the usual L + k / 2 ln N. So higher BIC scores will correspond to greater dependence. In the case of independence, the BIC score will be negative, since the likelihood will be zero, and this will be penalized.

This is specifically a BIC score for the linear, Gaussian case, where we include an additional penalty term,
which is commonly used. We call this the *penalty discount*. So our formulas has BIC = 2L - ck log N,
where L is the likelihood, c the penalty discount (usually greater than or equal to 1), and N the sample
size. Since the assumption is that the data are distributed as Gaussian, this reduces to BIC = -n log sigma
- ck ln N, where sigma is the standard deviation of the linear residual obtained by regressing a child
variable onto all of its parents in the model.

This is the Extended BIC (EBIC) score of Chen and Chen (Chen, J., & Chen, Z. (2008). Extended Bayesian information criteria for model selection with large model spaces. Biometrika, 95(3), 759-771.). This score is adapted to score-based search in high dimensions. There is one parameter, gamma, which takes a value between 0 and 1; if it's 0, the score is standard BIC. A value of 0.5 or 1 is recommended depending on how many variables there are per sample.

Note: You must specify the "Value Type" of each parameter, and the value type must be one of the following: Integer, Double, String, Boolean.

- Short Description: Yes, if adding an original dataset as another bootstrapping
- Long Description: It has been shown that adding in the algorithm result one would get using the original data to those found by the bootstrap method can improve accuracy of summary graphs. Select “Yes” here to include an extra run using the original dataset.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Cutoff for p values (alpha) (min = 0.0)
- Long Description: Statistical tests often compare a test statistic to a distribution and make a judgment that the null hypothesis has been rejected based on whether the area in the tails for the distribution for that test statistic is greater than some cutoff alpha. For tests of independence, for instance, a lower alpha level makes it easier to judge independence, and a higher alpha makes it harder to judge independence. Thus, a lower alpha for a search generally results in a sparser graph. The default for this is 0.01, though for discrete searches we recommend using a value of 0.05.
- Default Value: 0.01
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Yes if the orient away from arrow rule should be applied
- Long Description: The Orient Away from Arrow rule is usually applied for PC if there is a structure X->Y—Z to yield X->Y->Z, to avoid the creation of a collider known not to exist. Set this parameter to “No” if a chain of directed edges pointing in the same direction when only the first few such orientations are justified based on the data.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Average degree of graph (min = 1)
- Long Description: The average degree of a graph is equal to 2E / V, where E is the number of edges in the graph and V the number of variables (vertices) in the graph, since each edge has two endpoints. This allows one to have control over the density of randomly generated graphs. The default average degree is 2, which corresponds to a graph in which there are the same number of edges as nodes. For denser graphs, larger average degrees can be specified.
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Double

- Short Description: Basis type (1 = Polynomial, 2 = Cosine)
- Long Description: For CCI, this determines which basis type will be used (1 = Polynomial, 2 = Cosine)
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 2
- Value Type: Integer

- Short Description: Cutoff for p values (alpha) (min = 0.0)
- Long Description: Alpha level (0 to 1)
- Default Value: 0.01
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Yes if the exact algorithm should be used for continuous parents and discrete children
- Long Description: For the conditional Gaussian likelihood, if the exact algorithm is desired for discrete children and continuous parents, set this parameter to “Yes”.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: High end of coefficient range (min = 0.0)
- Long Description: When simulating data from linear models, one needs to specify the distribution of the coefficient parameters. Here, we draw coefficients from U(-m2, -m1) U U(m1, m2); m2 is what is being called the “high end of the coefficient range” and must be greater than m1.
- Default Value: 0.7
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Low end of coefficient range (min = 0.0)
- Long Description: When simulating data from linear models, one needs to specify the distribution of the coefficient parameters. Here, we draw coefficients from U(-m2, -m1) U U(m1, m2); m1 is what is being called the “low end of the coefficient range” and has a minimum value of 0.
- Default Value: 0.2
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Yes if negative coefficient values should be considered
- Long Description: Usually coefficient values for linear models are chosen from U(-b, -a) U U(a, b) for some a, b; this is called the “symmetric” model (symmetric about zero). If only positive values should be considered, this parameter should be set to false (“No” selected),
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Collider discovery: 1 = Lookup from adjacency sepsets, 2 = Conservative (CPC), 3 = Max-P
- Long Description: For variants of PC, one may choose from one of three different ways for orienting colliders. One may look them up from sepsets, as in the original PC, or estimate them conservatively, as from the Conservative PC algorithm, or by choosing the sepsets with the maximum p-value. In simulation, CPC (default = 2) works the best for sample sizes greater than 100; for very small sample sizes, PC-Max generally works better.
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 3
- Value Type: Integer

- Short Description: Yes if the complete FCI rule set should be used
- Long Description: For the FCI algorithm, to final orientation rules sets are available, one due to P. Spirtes, guaranteeing arrow completeness, and a second due to J. Zhang, guaranteeing additional tail completeness. If this parameter is set to “Yes,” the tail-complete rule set will be used.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if a concurrent FAS should be done
- Long Description: Various versions of the PC adjacency search lend themselves to concurrent processing—that is, doing different independence tests in parallel to speed up the processing. If this parameter is set to ‘Yes’, and this option is available, it will be used.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Collider conflicts: 1 = Overwrite, 2 = Orient bidirected, 3 = Prioritize existing colliders
- Long Description: It is not possible to avoid collider orientation conflicts in PC entirely. We offer three ways to deal with them. One may use the “overwrite” rule as introduced in the PCALG R package, or one may mark all collider conflicts using bidirected edges, or one may prioritize existing colliders, ignoring subsequent conflicting information.
- Default Value: 3
- Lower Bound: 1
- Upper Bound: 3
- Value Type: Integer

- Short Description: Yes if graph should be connected
- Long Description: It is possible to generate a random graph in which paths exists from every node to every other. This places some constraints on how the graph may be generated, but it is feasible in most cases. Setting this flag to “Yes” generates connected graphs.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: High end of covariance range (min = 0.0)
- Long Description: When simulating data from linear models, one needs to specify the distribution of the covariance parameters. Here, we draw coefficients from U(-c2, -c1) U U(c1, c2); c2 is what is being called the “high end of the covariance range” and must be greater than c1. The default value is 1.5.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Low end of covariance range (min = 0.0)
- Long Description: When simulating data from linear models, one needs to specify the distribution of the covariance parameters. Here, we draw coefficients from U(-c2, -c1) U U(c1, c2); c1 is what is being called the “low end of the covariance range” and has a minimum value of 0. The default value is 0.5.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Yes if negative covariance values should be considered
- Long Description: Usually covariance values are chosen from U(-b, -a) U U(a, b) for some a, b; this is called the “symmetric” model (symmetric about zero). If only positive values should be considered, this parameter should be set to false (“No” selected)
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Constraint-independence cutoff threshold
- Long Description: null
- Default Value: 0.5
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Independence cutoff threshold
- Long Description: null
- Default Value: 0.5
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Independence cutoff threshold
- Long Description: null
- Default Value: 0.5
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: "continuous" or "discrete"
- Long Description: For a mixed data type simulation, if this is set to “continuous” or “discrete”, all variables are taken to be of that sort. This is used as a double-check to make sure the percent discrete is set appropriately.
- Default Value: categorical
- Lower Bound:
- Upper Bound:
- Value Type: String

- Short Description: Maximum size of conditioning set (unlimited = -1)
- Long Description: This variable is usually called “depth” for algorithms such as PC in which conditioning sets are considered of increasing size from zero up to some limit, called “depth”. For example, if depth = 3, conditioning sets will be considered of sizes 0, 1, 2, and 3. In order to express that no limit should be imposed, use the value -1.
- Default Value: -1
- Lower Bound: -1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Threshold for judging a regression of a variable onto its parents to be deternimistic (min = 0.0)
- Long Description: When regressing a child variable onto a set of parent variables, one way to test for determinism is to see whether the relevant matrix is singular. We may instead ask how close to singular the matrix is; this gives a threshold for this. The default value is 0.1.
- Default Value: 0.1
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Yes if a different graph should be used for each run
- Long Description: When doing an analysis where, repeatedly, a random graph is chosen, with some further processing downstream, one may either keep using the same graph (with different simulated random datasets based on that graph) or pick a new graph every time. This parameter determines that behavior; if ‘Yes’ a new graph is chosen every time; if ‘No’, the same graph is always used.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if continuous variables should be discretized when child is discrete
- Long Description: For the conditional Gaussian likelihood, when scoring X->D, where X is continuous and D discrete, it is possible to write out the formula for that longhand, but a fast way to do it (and in fact more accurate usually) is to simply discretize X for just those cases. If this parameter is set to “Yes”, this discretization will be done.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the Euclidean norm squared should be calculated (slow), No if not
- Long Description: The generalized information criterion is defined with an information term that take a Euclidean norm squares; the can be calcuted directly.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes logs should be taken, No if not
- Long Description: The formula for the score allows a log to be taken optionally in the information term.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if unshielded collider orientation should be done
- Long Description: Please see the description of this algorithm in Thomas Richardson and Peter Spirtes in Chapter 7 of Computation, Causation, & Discovery by Glymour and Cooper eds.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if errors should be Normal; No if they should be abs(Normal) (i.e., non-Gaussian)
- Long Description: A “quick and dirty” way to generate linear, non-Gaussian data is to set this parameter to “No”; then the errors will be sampled from a Beta distribution.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Threshold for including edges detectable by skewness
- Long Description: For FASK, this includes an adjacency X—Y in the model if |corr(X, Y | X > 0) – corr(X, Y | Y > 0)| exceeds some threshold. The default for this threshold is 0.3. Sanchez-Romero, Ramsey et al., (2018) Network Neuroscience.
- Default Value: 0.3
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Upper bound for |left-right| to count as 2-cycle. (Set to zero to turn off pre-screening.)
- Long Description: 2-cycles are screened by looking to see if the left-right rule returns a difference smaller than this threshold. To turn off the screening, set this to zero.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Alpha threshold used for orientation (where necessary). (Set to zero to turn use of alpha.)
- Long Description: Used for orienting 2-cycles and testing for zero edges.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: For FASK v1 and v2, the bias for orienting with negative coefficients (Set to zero for no bias.)
- Long Description: The bias procedure for v1 is given in the published description.
- Default Value: 0.0
- Lower Bound: -Infinity
- Upper Bound: Infinity
- Value Type: Double

- Short Description: The left right rule: 1 = FASK v1, 2 = FASK v2, 3 = RSkew, 4 = Skew, 5 = Tanh
- Long Description: The FASK left right rule v2 is default, but two other (related) left-right rules are given for relation to the literature, and the v1 FASK rule is included for backward compatibility.
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 5
- Value Type: Integer

- Short Description: Linearity assumed
- Long Description: True if a linear, non-Gaussian, additive model is assume; false if a nonlinear, non-Gaussian, additive model is assumed.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Variables should be assumed to have positive skewness
- Long Description: If false (default), each variable is multiplied by the sign of its skewness in the left-right rule.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Acceptance Proportion
- Long Description: An edge occurring in this proportion of individual FASK graphs will appear in the final graph.
- Default Value: 0.5 ,.
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Adjacency Method: 1 = FAS Stable, 2 = FAS Stable Concurrent, 3 = FGES, 4 = External Graph
- Long Description: This is the method FASK will use to find adjacencies. For 4 = External graph, an external graph must be supplied.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 4
- Value Type: Integer

- Short Description: Yes if (one edge) faithfulness should be assumed
- Long Description: Assumes that if X _||_ Y, by an independence test, then X _||_ Y | Z for nonempty Z.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Test ordering: 1 = PC-1 1, 2 = PC-2, 3 = PC-3
- Long Description: See Causation, Prediction, and Search (2000).
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 3
- Value Type: Integer

- Short Description: Adjacency search: 1 = PC, 2 = PC-Stable, 3 =f Concurrent PC-Stable
- Long Description: For variants of PC, one may select either to use the usual PC adjacency search, or the procedure from the PC-Stable algorithm (Diego and Maathuis), or the latter using a concurrent algorithm (that is, one that runs in parallel on multiple processors).
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 3
- Value Type: Integer

- Short Description: Fast ICA 'a' parameter.
- Long Description: This is the 'a' parameter of Fast ICA. (See Hyvarinen, A. (2001). Independent Component Analysis/Hyvarinen A., Karhunen J., Oja E. NY: John Wiley & Sons Inc.) It ranges between 1 and 2; we use a default of 1.1. This scales values evaluated by the tanh function for the log cosh option.
- Default Value: 1.1
- Lower Bound: 1.0
- Upper Bound: 2.0
- Value Type: Double

- Short Description: The maximum number of optimization iterations.
- Long Description: This is the maximum number if iterations of the optimization procedure of ICA. (See Hyvarinen, A. (2001). Independent Component Analysis/Hyvarinen A., Karhunen J., Oja E. NY: John Wiley & Sons Inc.) It's an integer greater than 0; we use a default of 2000.
- Default Value: 2000
- Lower Bound: 1
- Upper Bound: 500000
- Value Type: Double

- Short Description: Fast ICA tolerance parameter.
- Long Description: This is the tolerance parameter of Fast ICA. (See Hyvarinen, A. (2001). Independent Component Analysis/Hyvarinen A., Karhunen J., Oja E. NY: John Wiley & Sons Inc.) It must be greater than or equal to 0.00; we use a default of 1e-6.
- Default Value: 1e-6
- Lower Bound: 0.0
- Upper Bound: 1000.0
- Value Type: Double

- Short Description: Epsilon where |xi.t - xi.t-1| < epsilon, criterion for convergence
- Long Description: This is a parameter for the linear Fisher option. The idea of Fisher model (for the linear case) is to shock the system every so often and let it converge by applying the rules of transformation (that is, the linear model) repeatedly until convergence. This sets the criterion for convergence—the process continues until the differences from one time step to the next fall below this epsilon.
- Default Value: 0.001
- Lower Bound: 4.9E-324
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: General function for error terms
- Long Description: This template specifies how distributions for error terms are to be generated. For help in constructing such templates, see the Generalized SEM PM model.
- Default Value: Beta(2, 5)
- Lower Bound:
- Upper Bound:
- Value Type: String

- Short Description: General function template for latent variables
- Long Description: This template specifies how equations for latent variables are to be generated. For help in constructing such templates, see the Generalized SEM PM model.
- Default Value: TSUM(NEW(B)*$)
- Lower Bound:
- Upper Bound:
- Value Type: String

- Short Description: General function template for measured variables
- Long Description: This template specifies how equations for measured variables are to be generated. For help in constructing such templates, see the Generalized SEM PM model.
- Default Value: TSUM(NEW(B)*$)
- Lower Bound:
- Upper Bound:
- Value Type: String

- Short Description: General function for parameters
- Long Description: This template specifies how distributions for parameter terms are to be generated. For help in constructing such templates, see the Generalized SEM PM model.
- Default Value: Split(-1.0, -0.5, 0.5, 1.0)
- Lower Bound:
- Upper Bound:
- Value Type: String

- Short Description: IA parameter (GLASSO)
- Long Description: The R Fortan implementation of GLASSO (https://CRAN.R-project.org/package=glasso) includes a number of parameters, of which this is one. This is the maximum number of iterations of the optimization loop.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if negative coefficients should be included in the model
- Long Description: One may include positive coefficients, negative coefficients, or both, in the model. To include negative coefficients, set this parameter to “Yes”.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if negative skew values should be included in the model, if Beta errors are chosen
- Long Description: Yes if negative skew values should be included in the model, if Beta errors are chosen.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if positive coefficients should be included in the model
- Long Description: e may include positive coefficients, negative coefficients, or both, in the model. To include positive coefficients, set this parameter to “Yes”.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if positive skew values should be included in the model, if Beta errors are chosen
- Long Description: Yes if positive skew values should be included in the model, if Beta errors are chosen.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Interval between data recordings for the linear Fisher model (min = 1)
- Long Description:
- Default Value: 10
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Interval beween shocks (R. A. Fisher simulation model) (min = 1)
- Long Description: This is a parameter for the linear Fisher option. The idea of Fisher model (for the linear case) is to shock the system every so often and let it converge by applying the rules of transformation (that is, the linear model) repeatedly until convergence. This sets the number of step between shocks.
- Default Value: 10
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: IPEN parameter (GLASSO)
- Long Description: The R Fortan implementation of GLASSO (https://CRAN.R-project.org/package=glasso) includes a number of parameters, of which this is one. This is the maximum number of iterations of the optimization loop.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: IS parameter (GLASSO)
- Long Description: The R Fortan implementation of GLASSO (https://CRAN.R-project.org/package=glasso) includes a number of parameters, of which this is one. This is the maximum number of iterations of the optimization loop.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: ITR parameter (GLASSO)
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Cutoff for p values (alpha) (min = 0.0)
- Long Description: Alpha level (0 to 1)
- Default Value: 0.05
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Cutoff
- Long Description: Cutoff for p-values.
- Default Value: 6
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Epsilon for Proposition 5, a small positive number
- Long Description: We have a number of parameters here for the Kernel Conditional Independence Test (KCI). In order to understand the parameters, it is necessary to read the paper on which this test is based, here: Zhang, K., Peters, J., Janzing, D., & Schölkopf, B. (2012). Kernel-based conditional independence test and application in causal discovery. arXiv preprint arXiv:1202.3775. This parameter is the epsilon for Proposition 5, a small positive number. The default value is 0.001; it must be a positive real number.
- Default Value: 0.001
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Number of bootstraps for Theorems 4 and Proposition 5 for KCI
- Long Description: We have a number of parameters here for the Kernel Conditional Independence Test (KCI). In order to understand the parameters, it is necessary to read the paper on which this test is based, here: Zhang, K., Peters, J., Janzing, D., & Schölkopf, B. (2012). Kernel-based conditional independence test and application in causal discovery. arXiv preprint arXiv:1202.3775. This parameter is the number of bootstraps for Theorems 4 and Proposition 5. The default is 5000; it must be positive integer.
- Default Value: 5000
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Use the approximate Gamma approximation algorithm
- Long Description: We have a number of parameters here for the Kernel Conditional Independence Test (KCI). In order to understand the parameters, it is necessary to read the paper on which this test is based, here: Zhang, K., Peters, J., Janzing, D., & Schölkopf, B. (2012). Kernel-based conditional independence test and application in causal discovery. arXiv preprint arXiv:1202.3775. If this parameter is set to ‘Yes’, the Gamma approximation algorithm is used, as described in this paper; otherwise, the non-approximate procedure is used.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Bowman and Azzalini (1997) default kernel bandwidhts should be multiplied by...
- Long Description: For the conditional correlation algorithm. Bowman, A. W., & Azzalini, A. (1997, Applied smoothing techniques for data analysis: the kernel approach with S-Plus illustrations (Vol. 18), OUP Oxford), give a formula for default optimal kernel widths. We allow these defaults to be multiplied by some factor, which we call the “kernel multiplier”, to capture more or less than this optimal signal. This multiplier must be a positive real number.
- Default Value: 1.0
- Lower Bound: 4.9E-324
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Minimum sample size to use per conditioning for kernel regression
- Long Description: Kernel regression for X _||_ Y | Z looks for dependencies between X and Y for Z values near to some particular Z values of interest. One can find the m nearest points to a given Z = z’ by expanding the search radius until you get that many points. This parameter specifies the smallest such set of nearest points on which to allow a judgment to be based.
- Default Value: 100
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Kernel type (1 = Gaussian, 2 = Epinechnikov)
- Long Description: For CCI, this determine which kernel type will be used (1 = Gaussian, 2 = Epinechnikov).
- Default Value: 2
- Lower Bound: 1
- Upper Bound: 2
- Value Type: Integer

- Short Description: Kernel width
- Long Description: A larger kernel width means that more information will be taken into account but possibly less focused information.
- Default Value: 1.0
- Lower Bound: 4.9E-324
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Number of Latent --> Measured impure edges
- Long Description: It is possible for structural nodes to have as children measured variables that are children of other structural nodes. These edges in the graph will be considered impure.
- Default Value: 0
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Lower bound cutoff threshold
- Long Description: null
- Default Value: 0.3
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Maximum number of categories (min = 2)
- Long Description: The maximum number of categories to be used for randomly generated discrete variables. The default is 2. This needs to be greater or equal to than the minimum number of categories.
- Default Value: 3
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Maximum absolute correlation considered
- Long Description: For the Nandy rule, the absolute max correlation r. For the standard BIC or high-dimensional rule, the maximum absolute residual correlation.
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: The maximum degree of the graph (min = -1)
- Long Description: It is possible for a random graph to have a single node with very high degree—i.e. number of adjacent edges. This parameter places an upper bound on the maximum such degree. If no limit is to be placed on the maximum degree, use the value -1.
- Default Value: 100
- Lower Bound: -1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The maximum number of distinct values in a column for discrete variables (min = 0)
- Long Description: Discrete variables will be simulated using any number of categories from 2 up to this maximum. If set to 0 or 1, discrete variables will not be generated.
- Default Value: 0
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Maximum indegree of graph (min = 1)
- Long Description: It is possible for a random graph to have a node in which there is a very large “indegree”—that is, number of parents, or number of edges into that node. This parameter places a bound on the maximum such indegree.
- Default Value: 100
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Maximum indegree of true graph (min = 0)
- Long Description: This is the maximum number of parents one expects any node to have in the true model.
- Default Value: 4
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The maximum number of iterations the algorithm should go through orienting edges
- Long Description: In orienting, this algorith may go through a number of iterations, conditioning on more and more variables until orientations are set. This sets that number.
- Default Value: 15
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Maximum outdegree of graph (min = 1)
- Long Description: It is possible for a random graph to have a node in which there is a very large “outdegree”—that is, number of children, or number of edges out of that node. This parameter places a bound on the maximum such outdegree. If no limit is to be placed on the max outdegree, use the value -1.
- Default Value: 100
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Maximum path length for the unshielded collider heuristic for max P (min = 0)
- Long Description: For the Max P “heuristic” to work, it must be the case that X and Z are only weakly associated—that is, that paths between them are not too short. This bounds the length of paths for this purpose.
- Default Value: 3
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The maximum length for any discriminating path. -1 if unlimited (min = -1)
- Long Description: See Spirtes, Glymour, and Scheines (2000), Causation, Prediction, and Search for the definition of discrimination path. Finding discriminating paths can be expensive. This sets the maximum length of such paths that the algorithm tries to find.
- Default Value: -1
- Lower Bound: -1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: MAXIT parameter (GLASSO) (min = 1)
- Default Value: 10000
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: High end of mean range (min = 0.0)
- Long Description: For a linear model, means of variables may be randomly shifted. The default is for there to be no shift, but shifts from a minimum value to a maximum value may be specified. The minimum must be less than or equal to the maximum.
- Default Value: 1.5
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Low end of mean range (min = 0.0)
- Long Description: For a linear model, means of variables may be randomly shifted. The default is for there to be no shift, but shifts from a minimum value to a maximum value may be specified. The minimum must be less than or equal to the maximum.
- Default Value: 0.5
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Number of Measured <-> Measured impure edges
- Long Description: It is possible for measures from two different structural nodes to be confounded. These confounding (bidirected) edges will be considered to be impure.
- Default Value: 0
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of Measured --> Measured impure edges
- Long Description: It is possible for measures from two different structural nodes to have directed edges between them. These edges will be considered to be impure.
- Default Value: 0
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of measurements per Latent
- Long Description: Each structural node in the MIM will be created to have this many measured children.
- Default Value: 5
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Additive measurement noise variance (min = 0.0)
- Long Description: One difficult problem one encounters for analyzing real data is measurement noise—that is, once the actual values V of a variable are determined, there is some additional noise M added on top of that, so that the effect value is V + M, not V. We will assume here what is often assumed, that this noise is additive and Gaussian, with some variance. If this parameter value is set to zero, no measurement noise is added to the dataset per variable. Otherwise, if the value is greater than zero, independent Gaussian noise will be added with mean zero and the given variance.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: MGM tuning parameter #1 (min = 0.0)
- Long Description: The MGM algorithm has three internal tuning parameters, of which this is one.
- Default Value: 0.1
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: MGM tuning parameter #2 (min = 0.0)
- Long Description: The MGM algorithm has three internal tuning parameters, of which this is one.
- Default Value: 0.1
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: MGM tuning parameter #3 (min = 0.0)
- Long Description: The MGM algorithm has three internal tuning parameters, of which this is one.
- Default Value: 0.1
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Minimum number of categories (min = 2)
- Long Description: The minimum number of categories to be used for randomly generated discrete variables. The default is 2.
- Default Value: 3
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Yes, if use the cutoff threshold for the independence test.
- Long Description: null
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Number of functions to use in (truncated) basis
- Long Description: The types of bases that we’re using here (Gaussian, Epinechnikov) have an infinite number of functions in them, but we’re only using a finite number of them, the most significant ones. This parameter specifies how many of the most significant basis functions to use. The default is 30.
- Default Value: 30
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The number of bootstrappings drawing from posterior dist. (min = 1)
- Long Description: null
- Default Value: 50
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of categories for discrete variables (min = 2)
- Long Description: The number of categories to be used for randomly generated discrete variables. The default is 4; the minimum is 2.
- Default Value: 4
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The number of categories used to discretize continuous variables, if necessary (min = 2)
- Long Description: If the exact algorithm is desired for discrete children and continuous parents is not used, the conditional Gaussian likelihood needs to keep a copy of all continuous variables on hand, discretized with a certain number of categories. This parameter gives the number of categories to use for this second backup copy of the continuous variables.
- Default Value: 3
- Lower Bound: 2
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The number of lags in the time lag model
- Long Description: A time lag model may take variables from previous time steps into account. This determines how many steps back these relevant variables might go.
- Default Value: 1
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of latent variables (min = 0)
- Long Description: A latent (or ‘unmeasured’) variable is one for which values are not recorded in the dataset being analyzed. These are variables that affect the measured variables but which are not included in the final dataset. This situation comes up frequently when analyzing real data, so it is important to be able to generate random datasets with this feature, in order to see how the algorithms react to the existence of latent variables. Some algorithms, like FCI, FFCI, FOFC, FTFC, and such, are correct when some variables not measured affect the data.
- Default Value: 0
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of measured variables (min = 1)
- Long Description: A measured variable is one for which values are recorded in the dataset being analyzed. This parameter sets the number of measured variables to be randomly simulated; this will be the number of columns in the dataset.
- Default Value: 10
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The number of search probabilistic model (min = 1)
- Long Description: null
- Default Value: 10
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of runs (min = 1)
- Long Description: An analysis(randomly pick graph, randomly simulate a dataset, run an algorithm on it, look at the result) may be run over and over again, repeatedly, and results summarized. This parameter indicates the number of repetitions that should be done for the analysis. The minimum is 1.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of structural edges
- Long Description: This is a parameter for generating random multiple indictor models (MIMs). A structural edge is an edge connecting two structural nodes.
- Default Value: 3
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Number of structural nodes
- Long Description: This is a parameter for generating random multiple indictor models (MIMs). A structural node is one of the latent variables in the model; each structural node has a number of child measured variables.
- Default Value: 3
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The number of bootstraps/resampling iterations (min = 0)
- Long Description: For bootstrapping, the number of bootstrap iterations that should be done by the algorithm, with results summarized.
- Default Value: 0
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Yes if Richardson's step C (orient toward d-connection) should be used
- Long Description: Please see the description of this algorithm in Thomas Richardson and Peter Spirtes in Chapter 7 of Computation, Causation, & Discovery by Glymour and Cooper eds.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if visible feedback loops should be oriented
- Long Description: Please see the description of this algorithm in Thomas Richardson and Peter Spirtes in Chapter 7 of Computation, Causation, & Discovery by Glymour and Cooper eds.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Constraint Scoring: Yes: Dependent Scoring, No: Independent Scoring.
- Long Description: null
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Penalty discount (min = 0.0)
- Long Description: A BIC score is of the form 2L – k ln N, where L is the likelihood, k the number of degrees of freedom, and N the sample size. Tests based on this statistic can often yield too dense a graph; to compensate for this, we add a factor c, as follows: 2L – c k ln N, where usually c >= 1. If c is chosen to be greater than 1, say 2, the output graph will be sparser. We call this c the “penalty discount”; similar mechanisms have been proposed elsewhere.
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: EBIC Gamma (0-1)
- Long Description: The gamma parameter for Extended BIC (Chen and Chen). In [0, 1].
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: True error variance
- Long Description: The true error variance of the model, assuming this is the same for all variables.
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Risk bound
- Long Description: This is the probability of getting the true model if a correct model is discovered. Could underfit.
- Default Value: 0.001
- Lower Bound: 0
- Upper Bound: 1
- Value Type: Double

- Short Description: Correlation Threshold
- Long Description: The algorithm will complain if correlations are found that are greater than this in absolute value.
- Default Value: 1
- Lower Bound: 0
- Upper Bound: 1
- Value Type: Double

- Short Description: Lambda (manually set)
- Long Description: The manually set lambda for GIC--the default is 10, though this should be set by the user to a good value.
- Default Value: 10.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Error Threshold
- Long Description: Adjusts the threshold for judging conditional dependence.
- Default Value: 0.5
- Lower Bound: 0.0
- Upper Bound: 1
- Value Type: Double

- Short Description: The number of threads to use in the search
- Long Description: If this parameter is offered, it will control the number of threads used in the search for the sections of the search that can be parallelized.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Percentage of discrete variables (0 - 100) for mixed data
- Long Description: For a mixed data type simulation, specifies the percentage of variables that should be simulated (randomly) as discrete. The rest will be taken to be continuous. The default is 0—i.e. no discrete variables.
- Default Value: 50.0
- Lower Bound: 0.0
- Upper Bound: 100.0
- Value Type: Double

- Short Description: The percentage of resample size (min = 0.1)
- Long Description: Each bootstrap iteration uses a certain portion of the data drawn randomly either with replacement or without replacement. This parameter specifies the percentage of records in the bootstrap (as a percentage of the total original sample size of the data being bootstrapped), in the range 1 to 100.
- Default Value: 90
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Yes if the possible dsep search should be done
- Long Description: This algorithm has a possible d-sep path search, which can be time-consuming. See Spirtes, Glymour, and Scheines, Causation, Prediction and Search for details.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: The probability of adding a cycle to the graph
- Long Description: One way to add cycles to a graph is to pick a group of 3, 4, or 5 nodes and create a cycle between those variables. A graph may be constructed in this way consisting entirely of cycles, and this graph may then be used to test algorithms that should be able to handle cycles. This parameter sets the probability that any particular such set of nodes will be used to form a cycle in the graph.
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: The probability of creating a 2-cycles in the graph (0 - 1)
- Long Description: The types of bases that we’re using here (Gaussian, Epinechnikov) have an infinite number of functions in them, but we’re only using a finite number of them, the most significant ones. This parameter specifies how many of the most significant basis functions to use. The default is 30.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: The number of datasets that should be taken in each random sample
- Long Description: This parameter is for algorithms that take multiple datasets as input, such as IMaGES. The idea is that maybe you have 100 dataset but want to take a random sample of 5 such datasets. This parameter, in this example, is ‘5’. It is the number of dataset that should be taken in each random sample of datasets.
- Default Value: 1
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Yes if the order of the columns in each datasets should be randomized
- Long Description: It is usually the case that for graphs that are faithful to the true model the order of the columns in the dataset should not matter; you should always end up with the same model. However, in the real world where unfaithfulness is an issue this may not be true. To test the resilience of methods to random reordering of the columns in the data, set this parameter to “Yes”.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: The number of random features to use
- Long Description:
- Default Value: 10
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Ensemble method: Preserved (0), Highest (1), Majority (2)
- Long Description: This parameter governs how summary graphs are generated based on graphs learned from individual bootstrap samples. If “Preserved”, an edge is kept and its orientation is chosen based on the highest probability. If “Highest”, an edge is kept the same way the preserved ensemble one does except when [no edge]'s probability is the highest one, then the edge is ignored. If “Majority”, the edge is kept only if its chosen orientations' probability is more than 0.5.
- Default Value: 0
- Lower Bound: 0
- Upper Bound: 2
- Value Type: Integer

- Short Description: Yes, if sampling with replacement (bootstrapping)
- Long Description: Resampling can be done with replacement or without replacement. If with replacement, it is possible to have more than one copy of some of the records in the original dataset being included in the bootstrap. This is what is usually meant by “bootstrap”. For this option, select “Yes” here. It is also possible to prevent repetitions and do so-called “random subsampling”; for this option, select “No” here.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Prior equivalent sample size (min = 1.0)
- Long Description: This sets the prior equivalent sample size. This number is added to the sample size for each conditional probability table in the model and is divided equally among the cells in the table. A prior of 15 (default) does well as optimizing the estimate of the output pattern; a prior of 5 will optimize the structural Hamming distance.
- Default Value: 10.0
- Lower Bound: 1.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Sample size (min = 1)
- Long Description: One of the main features of a dataset is the number of records in the data, or sample size, or N. When simulating random data, this parameter determines now many records should be generated for the data. The minimum number of records is 1; the default is set to 1000.
- Default Value: 1000
- Lower Bound: 1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Save latent variables.
- Long Description: When saving datasets, even though latent variables in simulation are supposed to be left out of the data, if one wishes to see what values those latent variables took on, one may opt to save the latent variables out with the rest of the data.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: For scale-free graphs, the parameter alpha (min = 0.0)
- Long Description: We use the algorithm for generating scale free graphs described in B. Bollobas,C. Borgs, J. Chayes, and O. Riordan, Directed scale-free graphs, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, 132--139, 2003. Please see this article for a description of the parameters.
- Default Value: 0.05
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: For scale-free graphs, the parameter beta (min = 0.0)
- Long Description: We use the algorithm for generating scale free graphs described in B. Bollobas,C. Borgs, J. Chayes, and O. Riordan, Directed scale-free graphs, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, 132--139, 2003. Please see this article for a description of the parameters.
- Default Value: 0.9
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: For scale-free graphs, the parameter delta_in (min = 0.0)
- Long Description: We use the algorithm for generating scale free graphs described in B. Bollobas,C. Borgs, J. Chayes, and O. Riordan, Directed scale-free graphs, Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms, 132--139, 2003. Please see this article for a description of the parameters.
- Default Value: 3
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: For scale-free graphs, the parameter delta_out (min = 0.0)
- Default Value: 3
- Lower Bound: -2147483648
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: The coefficient for the self-loop (default 0.0)
- Long Description: For simulation time series data, each variable depends on itself one time-step back with a linear edge that has this coefficient.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Lambda: 1 = Chickering, 2 = Nandy
- Long Description: The Chickering Rule is the local scoring consistency criterion in Chickering's formulation of GES, which is a difference of BIC scores, though we allow a multiplier on the penalty term called "penalty discount". The Nandy et al. rule is a reformulation of the Chickering rule using a single calculation of a partial correlation in place of the likelihood difference.
- Default Value: 1
- Lower Bound: 1
- Upper Bound: 2
- Value Type: Integer

- Short Description: Lambda: 1 = ln n, 2 = pn^1/3, 3 = 2 ln pn, 4 = 2(ln pn + ln ln pn), 5 = ln ln n ln pn, 6 = ln n ln pn, 7 = Manual
- Long Description: The rule used for calculating the lambda term of the score. a score. We follow Kim, Y., Kwon, S., & Choi, H. (2012). Consistent model selection criteria on high dimensions. The Journal of Machine Learning Research, 13(1), 1037-1057 and articles referenced therein. The selling point for these scores is that with large numbers of variables, if the density is not too high, adjacency precision will be kept in check.
- Default Value: 4
- Lower Bound: 1
- Upper Bound: 7
- Value Type: Integer

- Short Description: Structure Prior for SEM BIC (default 0)
- Long Description: Structure prior; default is 0 (turned off); may be any positive number otherwise
- Default Value: 0
- Lower Bound: 0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Number of records that should be skipped between recordings (min = 0)
- Long Description: This is a parameter for the linear Fisher option. The idea of Fisher model (for the linear case) is to shock the system every so often and let it converge by applying the rules of transformation (that is, the linear model) repeatedly until convergence. Data recordings are made every so many steps. This is an additional parameter indicating how many data recordings are skipped before actually inserting a record into the returned dataset. This is useful to test the reaction of a method to missing time steps.
- Default Value: 0
- Lower Bound: 0
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: Yes if the 'stable' FAS should be done
- Long Description: In Colombo, D., & Maathuis, M. H. (2014, Order-independent constraint-based causal structure learning, The Journal of Machine Learning Research, 15(1), 3741-3782), a modification of the adjacency search of PC was proposed that results in invariance under order permutations of the variables in the data. If this parameter is set to ‘Yes’, this version of the PC adjacency search is used.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the data should be standardized
- Long Description: An operation one often wants to perform on a dataset is to standardize each of its variables by subtracting the mean and dividing the standard deviation. This yields a dataset in which each variable has mean zero and unit variance. If this parameter is set to ‘Yes’, this operation will be performed on the simulated dataset.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Structure prior coefficient (min = 0.0)
- Long Description: For Tetrad, we use as a structure prior the default number of parents for any conditional probability table. Higher weight is accorded to tables with about that number of parents. The prior structure weights are distributed according to a binomial distribution.
- Default Value: 1.0
- Lower Bound: 0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Yes if the first step step for FGES should do scoring for both X->Y and Y->X
- Long Description: For discrete searches, and in some other situations, it may make a difference for an edge X—Y whether you score X->Y or X<-Y, even though theoretically they should have the same score. If this parameter is set to “Yes”, both scores will be calculated and the higher score used. (Recall we are calculating BIC as 2L – c k ln N, where c is the penalty discount.)
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Target variable name
- Long Description: This parameter is for searches, such as Markov blanket searches, that require a target variable. In the case of a Markov blanket search, one is searching the graph over the Markov blanket of some target variable named V—this parameter specifies the name ‘V’.
- Default Value:
- Lower Bound:
- Upper Bound:
- Value Type: String

- Short Description: "T-Depth", the maximum number of neighbors considered in power set calculations
- Long Description: For FGES, this is the maximum number of T-neighbors or H-complement-neights that are considered in any scoring step. Default is -1 (unlimited).
- Default Value: -1
- Lower Bound: -1
- Upper Bound: 2147483647
- Value Type: Integer

- Short Description: THR parameter (GLASSO) (min = 0.0)
- Default Value: 1.0E-4
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Threshold to determine how many eigenvalues to use--the lower the more (0 to 1)
- Long Description: We have a number of parameters here for the Kernel Conditional Independence Test (KCI). In order to understand the parameters, it is necessary to read the paper on which this test is based, here: Zhang, K., Peters, J., Janzing, D., & Schölkopf, B. (2012). Kernel-based conditional independence test and application in causal discovery. arXiv preprint arXiv:1202.3775. This parameter is the threshold to determine how many eigenvalues to use--the lower the more (0 to 1). The default value is 0.001; it must be a positive real number.
- Default Value: 0.001
- Lower Bound: 0.0
- Upper Bound: Infinity
- Value Type: Double

- Short Description: Yes, if use the cutoff threshold for the meta-constraints independence test (stage 2).
- Long Description: null
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes, if use the cutoff threshold for the constraints independence test (stage 1).
- Long Description: null
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Alpha orienting 2-cycles (min = 0.0)
- Long Description: The alpha level of a T-test used to determine where 2-cycles exist in the graph. A value of zero turns off 2-cycle detection.
- Default Value: 0.0
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Time limit
- Long Description: T-Separation requires a time limit. Default 1000.
- Default Value: 1000.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Yes, if the orientation adjustment step should be included
- Long Description: null
- Default Value: false
- Lower Bound: g
- Upper Bound:
- Value Type: Boolean

- Short Description: Upper bound cutoff threshold
- Long Description: null
- Default Value: 0.7
- Lower Bound: 0.0
- Upper Bound: 1.0
- Value Type: Double

- Short Description: Yes if adjacencies from conditional correlation differences should be used
- Long Description: FASK can use adjacencies X—Y where |corr(X,Y|X>0) – corr(X,Y|Y>0)| > threshold. This expression will be nonzero only if there is a path between X and Y; heuristically, if the difference is greater than, say, 0.3, we infer an adjacency. To see adjacencies included for this reason, set this parameter to “Yes”. Sanchez-Romero, Ramsey et al., (2018) Network Neuroscience.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if adjacencies from the FAS search (correlation) should be used
- Long Description: Determines whether adjacencies found by conditional correlation should be included in the final model.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if recursive simulation, No if reduced form simulation
- Long Description: Determines the type of simulation done. If recursive, the graph must be a DAG in causal order. "Reduced form" means X = (I - B)^-1 e, which requires a possibly large matrix inversion.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the equivalent sample size should be used in place of N
- Long Description: We calculate the equivalent sample size by assuming that all record are equally correlated
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the GAP algorithms should be used. No if the SAG algorithm should be used
- Long Description: This is a parameter for FOFC (Find One Factor Clusters). There are two procedures implemented for growing pure clusters of variables. In principle they give the same answer, but in practice they could give different answers. The first is GAP, “Grow and Pick”, where you specify all the possible initial sets, grown them all to their maximum sizes, and pick a set of non-overlapping such largest sets from these. The second is SAG, “Seed and Grow”, where you grow pure clusters one at a time, excluding variables found in earlier clusters from showing up in later ones. This parameter specifies which of these algorithms should be used, ‘Yes’ for GAP, ‘No’ for SAG.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the heuristic for orienting unshielded colliders for max P should be used
- Long Description: The “Max P” method for orienting an unshielded triple X—Y—Z records p-values for X _||_ Z | S for all S in adj(X) or adj(Z), finds the set S0 with the highest p-value, and orients X->Y<-Z just in case Y is not in S0. Another way to do the orientation if X and Z are only weakly dependent, is to simply see whether the p-value for X _||_ Z | Y is greater than the p-value for X _||_ Z. This is the “heuristic” referred to her; the purpose is to speed up the search.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if adjacencies based on skewness should be used
- Long Description: FASK can use adjacencies X—Y where |corr(X,Y|X>0) – corr(X,Y|Y>0)| > threshold. This expression will be nonzero only if there is a path between X and Y; heuristically, if the difference is greater than, say, 0.3, we infer an adjacency. To see adjacencies included for this reason, set this parameter to “Yes”. Sanchez-Romero, Ramsey et al., (2018) Network Neuroscience.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the Wishart test shoud be used. No if the Delta test should be used
- Long Description: This is a parameter for the FOFC (Find One Factor Clusters) algorithm. There are two tests implemented there for testing for tetrads being zero, Wishart and Delta. This parameter picks which of these tests should be use: ‘Yes’ for Wishart and ‘No’ for Delta.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: High end of variance range (min = 0.0)
- Long Description: When simulating data from linear models, one needs to specify the distribution of the variance parameters. Here, we draw coefficients from U(v1, v2); v2 is what is being called the “high end of the variance range” and must be greater than v1. The default value is 3.0.
- Default Value: 3.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Low end of variance range (min = 0.0)
- Long Description: When simulating data from linear models, one needs to specify the distribution of the variance parameters. Here, we draw coefficients from U(v1, v2); v1 is what is being called the “low end of the variance range” and has a minimum 0. The default value is 1.0.
- Default Value: 1.0
- Lower Bound: 0.0
- Upper Bound: 1.7976931348623157E308
- Value Type: Double

- Short Description: Yes if verbose output should be printed or logged
- Long Description: If this parameter is set to ‘Yes’, extra (“verbose”) output will be printed if available giving some details about the step-by-step operation of the algorithm.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if verbose output for Meek rule applications should be printed or logged
- Long Description: If this parameter is set to ‘Yes’, Meek rule appications will be printed out to the log.
- Default Value: false
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

- Short Description: Yes if the (MimBuild) stucture model should be included in the output graph
- Long Description: FOFC proper yields a measurement model--that is, a set of pure children for each of the discovered latents. One can estimate the structure over the latents (the structure model) using Mimbuild. This struture model is included in the output if this parameter is set to Yes.
- Default Value: true
- Lower Bound:
- Upper Bound:
- Value Type: Boolean

The regression box performs regression on variables in a data set, in an attempt to discover causal correlations between them. Both linear and regression are available.

- A data box
- A simulation box

- A graph box
- A compare box
- A parametric model box
- A data box
- A simulation box
- A search box

Linear regression is performed upon continuous data sets. If you have a categorical data set upon which you would like to perform linear regression, you can make it continuous using the data manipulation box.

Take, for example, a data set with the following underlying causal structure:

When used as input to the linear regression box, the following window results:

To select a variable as the response variable, click on it in the leftmost box, and then click on the top right-pointing arrow. If you change your mind about which variable should be the response variable, simply click on another variable and click on the arrow again.

To select a variable as a predictor variable, click on it in the leftmost box, and then click on the second right- pointing arrow. To remove a predictor variable, click on it in the predictor box and then click on the left-pointing arrow.

Clicking “Sort Variables” rearranges the variables in the predictor box so that they follow the same order they did in the leftmost box. The alpha value in the lower left corner is a threshold for independence; the higher it is set, the less discerning Tetrad is when determining the independence of two variables.

When we click “Execute,” the results of the regression appear in the box to the right. For each predictor variable, Tetrad lists the standard error, t value, and p value, and whether its correlation with the response variable is significant.

The Output Graph tab contains a graphical model of the information contained in the Model tab. For the case in which X4 is the response variable and X1, X2, and X3 are the predictors, Tetrad finds that only X1 is significant, and the output graph looks like this:

Comparison to the true causal model shows that this correlation does exist, but that it runs in the opposite direction.

Logistic regression may be run on discrete, continuous, or mixed data sets; however, the response variable must be binary. In all other ways, the logistic regression box functions like the linear regression box.

Peter Spirites

The output of the FCI algorithm [Spirtes, 2001] is a partial ancestral graph (PAG), which is a graphical object that represents a set of causal Bayesian networks (CBNs) that cannot be distinguished by the algorithm. Suppose we have a set of cases that were generated by random sampling from some CBN. Under the assumptions that FCI makes, in the large sample limit of the number of cases, the PAG returned by FCI is guaranteed to include the CBN that generated the data.

An example of a PAG is shown in Figure 2. This PAG represents the pair of CBNs in Figure 1a and 1b (where measured variables are in boxes and unmeasured variables are in ovals), as well as an infinite number of other CBNs that may have an arbitrarily large set of unmeasured confounders. Despite the fact that there are important differences between the CBNs in Figure 1a and 1b (e.g., there is an unmeasured confounder of X1 and X2 in Figure 1 b but not in Figure 1a), they share a number of important features in common (e.g., in both CBNs, X2 is a direct cause of X6, there is no unmeasured confounder of X2 and X6, and X6 is not a cause of X2). It can be shown that every CBN that a PAG represents shares certain features in common. The features that all CBNs represented by a PAG share in common can be read off of the output PAG according to the rules described next.

There are 4 kinds of edges that occur in a PAG: A -> B, A o-> B, A o–o B, and A <-> B. The edges indicate what the CBNs represented by the PAG have in common. A description of the meaning of each edge in a PAG is given in Table A1.

Parameter Description Cutoff for p-values (alpha) Conditional independence tests with p-values greater than this will be judged to be independent (H0). Default 0.01. Maximum size of conditioning set (depth) PC in the adjacency phase will consider conditioning sets for conditional independences of increasing size, up to this value. For instance, for depth 3, the maximum size of a conditioning set considered will be 3.Table A1: Types of edges in a PAG.

Edge type | Relationships that are present | Relationships that are absent |
---|---|---|

A --> B | A is a cause of B. It may be a direct or indirect cause that may include other measured variables. Also, there may be an unmeasured confounder of A and B. | B is not a cause of A. |

A <-> B | There is an unmeasured variable (call it L) that is a cause of A and B. There may be measured variables along the causal pathway from L to A or from L to B. | A is not a cause of B. B is not a cause of A. |

A o-> B | Either A is a cause of B, or there is an unmeasured variable that is a cause of A and B, or both. | B is not a cause of A. |

A o–o B | Exactly one of the following holds: (a) A is a cause of B, or (b) B is a cause of A, or (c) there is an unmeasured variable that is a cause of A and B, or (d) both a and c, or (e) both b and c. |

Table A1 is sufficient to understand the basic meaning of edge types in PAGs. Nonetheless, it can be helpful to know the following additional perspective on the information encoded by PAGs. Each edge has two endpoints, one on the A side, and one on the B side. For example A --> B has a tail at the A end, and an arrowhead at the B end. Altogether, there are three kinds of edge endpoints: a tail "–", an arrowhead ">", and a "o." Note that some kinds of combinations of endpoints never occur; for example, A o– B never occurs. As a mnemonic device, the basic meaning of each kind of edge can be derived from three simple rules that explain what the meaning of each kind of endpoint is. A tail "–" at the A end of an edge between A and B means "A is a cause of B"; an arrowhead ">" at the A end of an edge between A and B means "A is not a cause of B"; and a circle "o" at the A end of an edge between A and B means "can't tell whether or not A is a cause of B". For example A --> B means that A is a cause of B, and that B is not a cause of A in all of the CBNs represented by the PAG.

The PAG in Figure 2 shows examples of each type of edge, and the CBNs. Figure 1. show some examples of what kinds of CBNs can be represented by that PAG.

Figure 1. Two CBNs that FCI (as well as FCI+, GFCI, and RFCI) cannot distinguish.

Figure 2. The PAG that represents the CBN s in both Figures 1a and 1b.

This section describes two types of arc specializations that provide additional information about the nature of an arc in a PAG.

One arc specialization is colored green and is called definitely visible. In a PAG P without selection bias, a green (definitely visible) arc from A to B denotes that A and B do not have a latent confounder. If an arc is not definitely visible (represented as black) then A and B may have a latent confounder.

Another arc specialization is shown as bold and is called definitely direct. In a PAG P without selection bias, a bold (definitely direct) arc from A to B denotes that A is a direct cause of B, relative to the other measured variables. If an arc is not definitely direct (represented as not bolded) then A may not be a direct cause of B, in which case there may be one or more measured variables on every causal path from A to B.

In the following examples, the DAG representing a causal process is on the left, and the corresponding PAG is on the right. All variables are observed except for latent variable L.

Example of an edge C ➔ D that is definitely visible (green) and definitely direct (bold):

Example of an edge (C ➔ E) that is definitely visible (green) and not definitely direct (not bold):

Example of an edge (F ➔ E) that is not definitely visible (black) and not definitely direct (not bold):

It is conjectured that it is not possible for an edge to be definitely direct (bold) and not definitely visible (black).

By default Java will allocate the smaller of 1/4 system memory or 1GB to the Java virtual machine (JVM). If you run out of memory (heap memory space) running your analyses you should increase the memory allocated to the JVM with the following switch '-XmxXXG' where XX is the number of gigabytes of ram you allow the JVM to utilize. Do run Tetrad with more memory you need to start it from the command line or terminal. For example to allocate 8 gigabytes of ram you would add -Xmx8G immediately after the java command e.g., java -Xmx8G -jar tetrad-gui.jar.

Two vertices in a graph are adjacent if there is a directed, or undirected, or double headed edge between them.

The total number of edges directed both into and out of a vertex.

The number of edges directed into a vertex.

In a variable set V, with joint probability Pr, the Markov Blanket of a variable X in V is the smallest subset M of V \ {X} such that X II V \ M | M. In a DAG model, the Markov Blanket of X is the union of the set of direct causes (parents) of X, the set of direct effects (children) of X, and the set of direct causes of direct effects of X.

Two directed acyclic graphs (DAGS) are Markov Equivalent if they have the same adjacencies and for every triple X – Y – Z of adjacent vertices, if X and Z are not adjacent, X -> Y <- Z in both graphs or in neither graph.

Rules for finding all directions of edges implied by a pattern, consistent with any specified “knowledge” constraints on directions. See https://arxiv.org/pdf/1302.4972.pdf

An acyclic graph with directed and undirected edges. Directed edges have the same interpretation as in DAGs.
Undirected edges represent common causes. See Richardson, T. (2003). Markov properties for acyclic directed
mixed graphs. *Scandinavian Journal of Statistics*, 30(1), 145-157.

A graphical model in which unmeasured variables each have multiple measured effects. There may be directed edges between unmeasured variables, but no directed edges from measured variables to unmeasured variables are allowed.

The number of edges directed out of a vertex.

See PAG description in this manual.

A graphical representation of a Markov Equivalence Class or Classes, having both directed and undirected edges, with an undirected edge indicating that for each possible direction of the edge, there is a graph in the class or classes having that edge direction.

A network in which the frequency of nodes with degree k obeys a power law--the relation between log of degree and log of frequency is roughly linear. See https://cs.brynmawr.edu/Courses/cs380/spring2013/section02/slides/10_ScaleFreeNetworks.pdf.

A trek between X and Y is a directed path from X to Y or from Y to X, or two directed paths from a third variable Z into X and Y that do not intersect except at Z.