Installation

To install, run the following code in terminal:

# install the devtools package first if it's not yet installed
devtools::install_github("annaguo-bios/flexCausal")

The source code for flexCausal package is available on GitHub at flexCausal.

A Brief Introduction to ADMGs

The ADMG is one of the major input to the flexCausal package. It is a projection of a Directed Acyclic Graph (DAG) with unmeasured variables into only observed variables. Let \(G(O \cup U)\) denote a DAG with observed variables \(O\) and unmeasured variables \(U\). The ADMG is a projection of \(G(O \cup U)\) onto \(O\) only, denoted as \(G(O)\). The projection is guided by the following rules:

1. \(O_i \ \textcolor{blue}{\rightarrow} \ O_j\) is included in the ADMG \(G(O)\) if \(O_i \ \textcolor{blue}{\rightarrow} \ O_j\) exists in \(G(O \cup U)\) or if a directed path from \(O_i\) to \(O_j\) exists that passes through only unmeasured variables in \(G(O \cup U)\).
2. \(V_i \ \textcolor{red}{\leftrightarrow} \ V_j\) is included in the ADMG \(G(O)\) if a collider-free path, like \(V_i \ \textcolor{blue}{\leftarrow} \cdots \textcolor{blue}{\rightarrow} \ V_j\), exists in \(G(O \cup U)\) where all the intermediate variables belong to \(U\).

See Examples (a) and (b), where the DAGs with unmeasured variables on the left are projected onto their corresponding ADMGs on the right:

In all the following discussions, we will use the ADMG in example (a) above as a running example. The packages comes with simulated datasets data_example_a and data_example_b that are generated from the ADMGs in examples (a) and (b), respectively. Let’s take a look at the first few rows of data_example_a:

library(flexCausal) # load the package
data(data_example_a) # load the simulated dataset for example (a)
head(data_example_a) # take a glance of the data, which is a simulated dataset under above figure (a).

##            X          U A       M.1        M.2          L         Y
## 1 0.98890930  3.4036578 0 0.5423635 -1.6361458  0.7632388  3.257711
## 2 0.39774545 -0.7055857 0 2.4330827 -0.6538274  3.9498004  5.338658
## 3 0.11569778  1.0320105 1 4.5009622  2.0672577 11.4239744 19.819490
## 4 0.06974868  1.6994103 1 2.8610542 -1.1488686  5.0942460 10.803918
## 5 0.24374939  2.9995114 1 2.6677580 -1.1273291  3.2955105  8.205794
## 6 0.79201043  2.9700555 1 3.3190450  3.5674934 10.1107529 21.442111

Note that the \(M\) variable in the dataset is a multivariate variable, consisting of two components, \(M.1\) and \(M.2\). To input the ADMG into the flexCausal package, users need to specify the vertices, directed edges, and bi-directed edges in the ADMG, along with the components of any multivariate variables.

For example, to input the ADMG in example (a) above that aligns with the simulated dataset, we would create a graph object with the make.graph() function:

# create a graph object for the ADMG in example (a)
graph_a <- make.graph(vertices=c('A','M','L','Y','X'), # specify the vertices
                bi_edges=list(c('A','Y')), # specify the bi-directed edges
                di_edges=list(c('X','A'), c('X','M'), c('X','L'),c('X','Y'), c('M','Y'), c('A','M'), c('A','L'), c('M','L'), c('L','Y')), # specify the directed edges, with each pair of variables indicating an directed edge from the first variable to the second. For example, c('X', 'A') represents a directed edge from X to A.
                multivariate.variables = list(M=c('M.1','M.2'))) # specify the components of the multivariate variable M

With this graph object in hand, we can conveniently explore various properties of the ADMG using functions provided in the package. These functions include topological ordering, genealogical relations, and causal effect identifiability, and etc. For a detailed discussion, see Functions for learning the properties of ADMG.

As a quick example, we can obtain the adjacency matrix of this ADMG with the f.adj_matrix() function:

f.adj_matrix(graph_a) # get the adjacency matrix of the ADMG in example (a)

##   A M L Y X
## A 0 0 0 0 1
## M 1 0 0 0 1
## L 1 1 0 0 1
## Y 0 1 1 0 1
## X 0 0 0 0 0

Quick Start on Estimation

The main function in this package is estADMG(), which estimates the Average Causal Effect (ACE) using both a one-step corrected plug-in estimator and a TMLE estimator. To get a sense of how to use this package, we provide a quick example below. We will use the ADMG in example (a) above to estimate the ACE of treatment \(A\) on outcome \(Y\) using a simulated dataset data_example_a. We can directly use the graph_a object created above as the input to the function.

est <- estADMG(a=c(1,0),
                data=data_example_a, 
                graph=graph_a,
                treatment='A', outcome='Y')

## The treatment is not fixable but is primal fixable. Estimation provided via extended front-door functional.

## Onestep estimated ACE: 1.96; 95% CI: (1.3, 2.62) 
## TMLE estimated ACE: 1.96; 95% CI: (1.29, 2.62)

## The graph is nonparametrically saturated. Results from the one-step estimator and TMLE are provided, which are in theory the most efficient estimators.

The code above estimates the ACE of treatment \(A\) on outcome \(Y\), defined as \(E(Y^1)-E(Y^0)\), using the data data_example_a generated based on Figure (a). The function estADMG() takes the following arguments:

• a: a vector of length 2, specifying the values of treatment \(A\) to compare. For example, a=c(1,0) compares causal effect under \(A=1\) verse \(A=0\). Alternatively, a can be input as a single integer if estimating the causal effect under only one treatment level is the goal. For example, to estimate \(E(Y^1)\), specify a = 1.
• data: a data frame containing the data.
• graph: a graph object specifying the ADMG.
• treatment: a string, specifying the name of the treatment variable.
• outcome: a string, specifying the name of the outcome variable.

Alternatively, instead of providing a graph object, users can directly specify the vertices, directed edges, and bi-directed edges within the estADMG() function, as shown below:

est <- estADMG(a=c(1,0),data=data_example_a, 
                vertices=c('A','M','L','Y','X'), # specify the vertices
                bi_edges=list(c('A','Y')), # specify the bi-directed edges
                di_edges=list(c('X','A'), c('X','M'), c('X','L'),c('X','Y'), c('M','Y'), c('A','M'), c('A','L'), c('M','L'), c('L','Y')), # specify the directed edges
                multivariate.variables = list(M=c('M.1','M.2')), # specify the components of the multivariate variable M
                treatment='A', outcome='Y')

## The treatment is not fixable but is primal fixable. Estimation provided via extended front-door functional.

## Onestep estimated ACE: 1.96; 95% CI: (1.3, 2.62) 
## TMLE estimated ACE: 1.96; 95% CI: (1.29, 2.62)

## The graph is nonparametrically saturated. Results from the one-step estimator and TMLE are provided, which are in theory the most efficient estimators.

Details on Estimation via TMLE estimator and onestep estimator

est <- estADMG(a=c(1,0),
                data=data_example_a, 
                graph = graph_a,
               treatment='A', outcome='Y',
                lib.seq = c("SL.glm", "SL.earth", "SL.ranger", "SL.mean"),
                lib.Y = c("SL.glm", "SL.earth", "SL.ranger", "SL.mean"),
                lib.A = c("SL.glm", "SL.earth", "SL.ranger", "SL.mean"),
                crossfit = TRUE,
                K=5)

est <- estADMG(a=c(1,0),
                data=data_example_a, 
                graph = graph_a,
                ratio.method.M = "dnorm")

Output

As an example, we estADMG() to estimate the average counterfactual outcome \(E(Y^1)\). The output is described as follows

est <- estADMG(a=1,
                data=data_example_a, 
                graph = graph_a,
                treatment='A', outcome='Y')

# TMLE and Onestep estimator
est$TMLE # a list contains the estimation result from TMLE estimator
est$Onestep # a list contains the estimation result from Onestep estimator

# For either the TMLE or Onestep estimator, the output is a list that contains the following elements:
est$TMLE$EYa # the estimated average counterfactual outcome
est$TMLE$lower.ci # the lower bound of the 95% confidence interval
est$TMLE$upper.ci # the upper bound of the 95% confidence interval
est$TMLE$EIF # the estimated efficient influence function
est$TMLE.Ya$EIF.Y # the estimated efficient influence function at the tangent space associated with the outcome
est$TMLE.Ya$EIF.A # the estimated efficient influence function at the tangent space associated with the treatment
est$TMLE.Ya$EIF.v # the estimated efficient influence function at the tangent space associated with the rest of the variables
est$TMLE.Ya$p.a1.mpA # the estimated propensity score for treatment
est$TMLE.Ya$mu.next.A # the estimated sequential regression associated with variable that comes right after the treatment according to the topological ordering of the ADMG
est$TMLE.Ya$EDstar # mean of the estimated efficient influence function
est$TMLE.Ya$iter # iterations take for TMLE estimator to converge
est$TMLE.Ya$EDstar.record # the mean of the estimated efficient influence function at each iteration

Functions for learning the properties of ADMG

Apart from the estADMG() for causal effec estimation, we also provide functions for learning the properties of ADMG. The functions are described as follows:

• make.graph: create the graph object. For example, to create the graph object for the ADMG in Figure (a), we can use the following code:

graph_a <- make.graph(vertices=c('A','M','L','Y','X'), # specify the vertices
                bi_edges=list(c('A','Y')), # specify the bi-directed edges
                di_edges=list(c('X','A'), c('X','M'), c('X','L'),c('X','Y'), c('M','Y'), c('A','M'), c('A','L'), c('M','L'), c('L','Y')), # specify the directed edges, with each pair of variables indicating an directed edge from the first variable to the second. For example, c('X', 'A') represents a directed edge from X to A.
                multivariate.variables = list(M=c('M.1','M.2'))) # specify the components of the multivariate variable M

• f.adj_matrix: return the adjacency matrix of the graph. For example, to get the adjacency matrix of the graph object for the ADMG in Figure (a), we can use the following code:

f.adj_matrix(graph_a)

• f.top_order: return the topological ordering of the graph.

f.top_order(graph_a)

• f.parents: return the parents of a given vertex or vertices in the graph. For example, to get the parents of vertex Y in the graph object for the ADMG in Figure (a), we can use the following code:

f.parents(graph_a, 'Y')

• f.children: return the children of a given vertex or vertices in the graph. For example, to get the children of vertex A in the graph object for the ADMG in Figure (a), we can use the following code:

f.children(graph_a, 'A')

• f.descendants: return the descendants of a given vertex or vertices in the graph. For example, to get the descendants of vertex A in the graph object for the ADMG in Figure (a), we can use the following code:

f.descendants(graph_a, 'A')

• f.district: return the district of a given vertex or vertices in the graph. For example, to get the district of vertex A in the graph object for the ADMG in Figure (a), we can use the following code:

f.district(graph_a, 'A')

• f.markov_blanket: return the Markov blanket of a given vertex or vertices in the graph. For example, to get the Markov blanket of vertex A in the graph object for the ADMG in Figure (a), we can use the following code:

f.markov_blanket(graph_a, 'A')

• f.markov_pillow: return the Markov pillow of a given vertex or vertices in the graph. For example, to get the Markov pillow of vertex A in the graph object for the ADMG in Figure (a), we can use the following code:

f.markov_pillow(graph_a, 'A')

• is.fix: return whether a treatment variable is fixable in a graph object. For example, to check whether the treatment variable A is fixable in the graph object for the ADMG in Figure (a), we can use the following code:

is.fix(graph_a, 'A')

• is.p.fix: return whether a treatment variable is primal fixable in a graph object. For example, to check whether the treatment variable A is primal fixable in the graph object for the ADMG in Figure (a), we can use the following code:

is.p.fix(graph_a, 'A')

If the treatment is primal fixable, then the average causal effect of the treatment on any choice of the outcome in the given graph is always identified.

• is.np.saturated: return whether a graph object is NP-saturated. For example, to check whether the graph object for the ADMG in Figure (a) is NP-saturated, we can use the following code:

is.np.saturated(graph_a)

A graph being nonparametrically saturated means that the graph implies NO equality constraints on the observed data distribution.

• is.mb.shielded: return whether a graph is mb-shielded. For example, to check whether ADMG in Figure (a) is shielded, we can use the following code:

is.mb.shielded(graph_a)

A graph being mb-shielded means that the graph only implies ordinary equality constraints on the observed data distribution.