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Specifying Augmented Directed Acyclic Graphs

augmented_dags.Rmd

In this vignette, we introduce a specification for augmented Directed Acyclic Graphs (aDAGs). We will overload regular DAGs as specified using the dagitty R-package. While dagitty provides standard facilities for declaring nodes, edges, exposures, and outcomes in causal frameworks, we augment the DAG with additional metadata fields to make it more conducive to theory specification. These metadata fields do not interfere with regular use of the DAG in dagitty.

The metadata fields in an aDAG include:

  • tags for identifying nodes of interest in causal inference. This field can take on values like exposure, outcome, and unobserved.
  • pos for nodes, which defines the layout position in the X and Y dimension, e.g., pos="0,1 is positioned at coordinates X = 0 and Y = 1. This metadata field is used by dagitty.
  • label for nodes or edges: A descriptive label used for visualization and reporting. This is a new metadata field.
  • distribution for nodes: The assumed distribution-generating function for the variable associated with a node. For exogenous nodes, this constitutes the distribution of the variable associated with the node itself; for endogenous nodes, this constitutes the residual distribution of the associated variable. This is a new metadata field.
  • form for edges: A function specification (in a form interpretable by as.formula()) that describes how the variable associated with a child node is calculated from its parents. This is a new metadata field.

Throughout the vignette, we will illustrate how to write an augmented DAG, how to parse and inspect it with dagitty, theorytools, and tidySEM (for plotting), and how to leverage these additional properties for further modeling or simulation tasks.

library(theorytools)
library(dagitty)
library(tidySEM)
#> 
#> Attaching package: 'tidySEM'
#> The following object is masked from 'package:dagitty':
#> 
#>     edges

A Basic DAG

The usual syntax for specifying a DAG in the dagitty R-package is something like:

library(dagitty)
dagitty("dag {
  X -> Y
  Z -> X
  Z -> Y
}")

There are several tags that can be used in dagitty. Note that quotation marks used in tags must be double quotes ", so it makes sense to wrap the whole DAG syntax in single quotes ':

library(dagitty)
dagitty('dag {
  X [exposure, pos="0,1"]
  Y [outcome, pos="1,1"]
  Z [unobserved, pos="1,0"]
  X -> Y
  Z -> X
  Z -> Y
}')

Augmented DAGs (aDAGs)

In our augmented specification, we add additional properties as metadata fields.

Below, we detail each new property:

label (Nodes/Edges)

  • Usage: Provide a descriptive name for the node or edge.
  • Example: X [label="Study hours"]

The label is used, for example, by tidySEM to label nodes and edges:

library(tidySEM)
g <- dagitty('dag {
  X [label="Predictor", pos="0,0"]
  Y [label="Outcome", pos="1,0"]
  X -> Y [label="effect"]
}')
graph_sem(g, text_size = 2)

2. distribution (Nodes)

  • Usage: References a function that generates data for exogenous variables, or that describes the residual distribution for endogenous variables. The function can reference the argument n to determine sample size. For example, to specify a node comprising five groups with total sample size n, one could use sample.int(n = 5, size = n, replace=TRUE). If the argument n is not explicitly provided, theorytools checks if n is a formal argument of the function, and assigns it.

  • Examples:

    • X [distribution="rnorm()"]: Node X is an exogenous variable drawn from a normal distribution with default arguments.
    • Y [distribution="rnorm()"]: Node Y has residuals assumed to be normally distributed with default arguments.
g <- dagitty('dag {
  X [distribution="rbinom(size = 2, prob = .5)"]
  Y [distribution="rnorm()"]
  X -> Y [form=".2*X"]
}')

3. form (Edges)

  • Usage: Provides a formula-like specification for how a child node depends on its parent node(s). It should be something that as.formula() can parse.

  • Examples:

  • X -> Y [form=".2*X"] indicates that Y is a linear function of .2 times X

  • X -> Y [form="X:Z"] indicates that Y depends on an interaction between X and Z

  • X -> Y [form="X^2"] indicates that Y depends on a quadratic function of X

g <- dagitty('dag {
  X [distribution="rbinom(size = 2, prob = .5)"]
  Y [distribution="rnorm()"]
  X -> Y [form=".2*X"]
}')

Example: Augmented DAG Specification

Below is a simple, hypothetical DAG showing how to combine these ideas. This DAG posits:

  • X: Number of study hours, an exposure. Values are randomly sampled from 1-20 hours.
  • Z: Stress level, an exogenous covariate, exponentially distributed (i.e., right-skewed, most people are not very stressed).
  • Y: Exam performance an outcome depending on X and Z, with normally distributed residuals.s
g <- dagitty('dag {
  X [exposure,
     pos="0,0",
     label="Study Hours",
     distribution="sample.int(n = 20, size = n, replace = TRUE)"]
  Z [label="Stress Level",
     pos=".5,1",
     distribution="rexp()"]
  Y [outcome,
     pos="1,.2",
     label="Exam Performance", distribution="rnorm()"]
  X -> Y [label="direct", form="0.5+X"]
  X -> Z
  Z -> Y [label="indirect", form="2*Z"]
}')
graph_sem(g, text_size = 3)

Parsing DAG Properties

Augmented DAGs are interoperable with dagitty, but the dagitty package is not natively aware of the additional metadata fields used in theorytools, like distribution or form. To access the augmented properties of aDAGs, the theorytools package uses tidySEM. The purpose of the tidySEM package is to plot graphs (structural equation models and DAGs) as ggplot objects, which can be further customized using regular ggplot2 code. It contains parsing functions to extract nodes and edges from a variety of objects, including dagitty graphs. The functions get_nodes() and get_edges() parse the nodes and edges of aDAGs, respectively:

get_nodes(g)
#>   name exposure   x   y            label
#> 1    X     TRUE 0.0 0.0      Study Hours
#> 2    Y       NA 1.0 0.2 Exam Performance
#> 3    Z       NA 0.5 1.0     Stress Level
#>                                   distribution outcome shape
#> 1 sample.int(n = 20, size = n, replace = TRUE)      NA  none
#> 2                                      rnorm()    TRUE  none
#> 3                                       rexp()      NA  none
get_edges(g)
#>   from to  e    label     form arrow  color
#> 1    X  Y ->   direct -X^2+4*X  last gray80
#> 2    X  Z ->     <NA>     <NA>  last gray80
#> 3    Z  Y -> indirect      2*Z  last gray80

Interpreting distribution and form in Simulation

A primary motivation for these augmented properties is simulation. For example, you might simulate data by:

  1. Generating X from sample.int(n).
  2. Generating Z from rexp(n).
  3. Generating Y using a formula that includes X and Z plus a residual from rnorm(n).

Code to simulate data in line with these metadata can be automatically generated:

set.seed(1)
cat(simulate_data(g, run = FALSE), sep = "\n")
#> # Set random seed
#> set.seed(1140350788)
#> # Set simulation parameters
#> n <- 500
#> # Simulate exogenous nodes
#> X <- sample.int(n = 20, size = n, replace = TRUE)
#> # Simulate endogenous nodes
#> Z <- 0.09 * X + rexp(n = n)
#> Y <- 2 * Z + X * (4 - 0.49 * X) + rnorm(n = n)
#> df <- data.frame(
#> X = X,
#> Y = Y,
#> Z = Z
#> )

To illustrate, we show a scatter plot of data simulated using this code:

df <- simulate_data(g, run = TRUE)
ggplot2::ggplot(df, aes(x=X,y=Y,color=Z))+geom_point()

You can use this script, for example, to generate synthetic data and build a reproducible analysis pipeline for a Preregistration-As-Code (Peikert, Van Lissa, and Brandmaier 2021; Van Lissa 2022).

Additional Notes

  1. Syntax Quoting: In R, there are rules about using quotes within quotes. The dagitty package only recognizes double quotes (" ") inside graph specifications. This means you must wrap the graph specification text in single quotes (' '). Alternatively, you can escape every double quote inside the graph specification, which is not recommended because it is a hassle.
  2. Multiple Parents: If a node has multiple parents, you can either specify multiple edges with their own form properties or a single edge with a combined formula. They are combined, and unique terms are retained.
  3. Order of Declaration: dagitty does not mind the order in which nodes are declared, but you’ll need a topological order (no cycles) for valid DAG generation and simulation.
  4. Integration with dagitty Functions: The standard dagitty functions (e.g., adjustmentSets()) only look for recognized tags like exposure and outcome. They ignore custom properties like distribution and form, but these do not interfere with normal usage.

References

Peikert, Aaron, Caspar J. Van Lissa, and Andreas M. Brandmaier. 2021. “Reproducible Research in R: A Tutorial on How to Do the Same Thing More Than Once.” Psych 3 (4): 836–67. https://doi.org/10.3390/psych3040053.
Van Lissa, Caspar J. 2022. “Complementing Preregistered Confirmatory Analyses with Rigorous, Reproducible Exploration Using Machine Learning.” Religion, Brain & Behavior 0 (0): 1–5. https://doi.org/10.1080/2153599X.2022.2070254.