pcalg (version 2.0-3)

backdoor: Find Set Satisfying the Generalized Backdoor Criterion

Description

This function first checks if the total causal effect of one variable (x) onto another variable (y) is identifiable via the generalized backdoor criterion, and if this is the case it explicitly gives a set of variables that satisfies the generalized backdoor criterion with respect to x and y in the given graph.

Usage

backdoor(amat, x, y, type = "pag", max.chordal = 10, verbose=FALSE)

Arguments

amat

adjacency matrix (see Details for coding) of the given graph specified in type.

x,y

(integer) position of variable x or y in the adjacency matrix.

type

string specifying the type of graph of the adjacency matrix amat. It can be a DAG (type="dag"), a CPDAG (type="cpdag"), a MAG (type="mag"), or a PAG (type="pag").

max.chordal

only if type = "mag", is used in pag2magAM to determine paths too large to be checked for chordality.

verbose

logical; if true, some output is produced during computation.

Value

• Either NA if the total causal effect is not identifiable via the generalized backdoor criterion, or a set if the effect is identifiable via the generalized backdoor criterion. Note that if the set W is equal to the empty set, the output is NULL.

Details

This function is a generalization of Pearl's backdoor criterion, see Pearl (1993), defined for directed acyclic graphs (DAGs), for single interventions and single outcome variable to more general types of graphs (CPDAGs, MAGs, and PAGs) that describe Markov equivalence classes of DAGs with and without latent variables but without selection variables. For more details see Maathuis and Colombo (2013).

The motivation to find a set W that satisfies the generalized backdoor criterion with respect to x and y in the given graph relies on the result of the generalized backdoor adjustment:

If a set of variables W satisfies the generalized backdoor criterion relative to x and y in the given graph, then the causal effect of x on y is identifiable and is given by $$P(Y|do(X = x)) = \sum_W P(Y|X,W) \cdot P(W).$$

This result allows to write post-intervention densities (the one written using Pearl's do-calculus) using only observational densities estimated from the data.

If the input graph is a DAG (type="dag"), this function reduces to Pearl's backdoor criterion for single interventions and single outcome variable, and the parents of x in the DAG satisfy the backdoor criterion unless y is a parent of x.

If the input graph is a CPDAG C (type="cpdag"), a MAG M (type="mag"), or a PAG P (type="pag") (with both M and P not allowing selection variables), this function first checks if the total causal effect of x on y is identifiable via the generalized backdoor criterion (see Maathuis and Colombo, 2013). If the effect is not identifiable in this way, the output is NA. Otherwise, an explicit set W that satisfies the generalized backdoor criterion with respect to x and y in the given graph is found.

At this moment this function is not able to work with an RFCI-PAG.

It is important to note that there can be pair of nodes x and y for which there is no set W that satisfies the generalized backdoor criterion, but the total causal effect might be identifiable via some other technique.

Coding of adjacency matrix: If type = dag or type = cpdag: coding 0/1 for no edge or tail / arrowhead; e.g. amat[a,b] = 0 and amat[b,a] = 1 implies a -> b. Else: coding 0,1,2,3 for no edge, circle, arrowhead, tail; e.g., amat[a,b] = 2 and amat[b,a] = 3 implies a -> b.

References

M.H. Maathuis and D. Colombo (2013). A generalized backdoor criterion. arXiv preprint arXiv:1307.5636.

J. Pearl (1993). Comment: Graphical models, causality and intervention. Statistical Science 8, 266--269.

See Also

pc for estimating a CPDAG, dag2pag and fci for estimating a PAG, and pag2magAM for estimating a MAG.

Examples

#####################################################################
##DAG
#####################################################################
## Simulate the true DAG
set.seed(123)
p <- 7
myDAG <- randomDAG(p, prob = 0.2) ## true DAG

## Extract the adjacency matrix of the true DAG
true.amat <- (amat <- as(myDAG, "matrix")) != 0 # TRUE/FALSE <==> 1/0
print.table(1*true.amat, zero.=".") # "visualization"

## Compute the effect using the generalized backdoor criterion
backdoor(true.amat, 5, 7, type="dag")
stopifnot(backdoor(true.amat, 5, 7, type="dag") == 3)
#####################################################################
##CPDAG
#####################################################################
##################################################
## Example not identifiable
## Maathuis and Colombo (2013), Fig. 3, p.14
##################################################

## create the graph
p <- 5
. <- 0
amat <- rbind(c(.,.,1,1,1),
              c(.,.,1,1,1),
              c(.,.,.,1,.),
              c(.,.,.,.,1),
              c(.,.,.,.,.))
colnames(amat) <- rownames(amat) <- as.character(1:5)
V <- as.character(1:5)
edL <- vector("list",length=5)
names(edL) <- V
edL[[1]] <- list(edges=c(3,4,5),weights=c(1,1,1))
edL[[2]] <- list(edges=c(3,4,5),weights=c(1,1,1))
edL[[3]] <- list(edges=4,weights=c(1))
edL[[4]] <- list(edges=5,weights=c(1))
g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed")

## estimate the true CPDAG
myCPDAG <- dag2cpdag(g)
## Extract the adjacency matrix of the true CPDAG
true.amat <- (as(myCPDAG, "matrix") != 0) # 1/0 <==> TRUE/FALSE

## The effect is not identifiable, in fact:
backdoor(true.amat, 3, 5, type="cpdag")
stopifnot(identical(NA, backdoor(true.amat, 3, 5, type="cpdag")))

##################################################
## Example identifiable
## Maathuis and Colombo (2013), Fig. 4, p.15
##################################################

## create the graph
p <- 6
amat <- rbind(c(0,0,1,1,0,1), c(0,0,1,1,0,1), c(0,0,0,0,1,0),
              c(0,0,0,0,1,1), c(0,0,0,0,0,0), c(0,0,0,0,0,0))
colnames(amat) <- rownames(amat) <- as.character(1:6)
V <- as.character(1:6)
edL <- vector("list",length=6)
names(edL) <- V
edL[[1]] <- list(edges=c(3,4,6),weights=c(1,1,1))
edL[[2]] <- list(edges=c(3,4,6),weights=c(1,1,1))
edL[[3]] <- list(edges=5,weights=c(1))
edL[[4]] <- list(edges=c(5,6),weights=c(1,1))
g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed")

## estimate the true CPDAG
myCPDAG <- dag2cpdag(g)
## Extract the adjacency matrix of the true CPDAG
true.amat <- as(myCPDAG, "matrix") != 0 # 1/0

## The effect is identifiable and
backdoor(true.amat, 6, 3, type="cpdag")
stopifnot(backdoor(true.amat, 6, 3, type="cpdag") == 1:2)

##################################################################
##PAG
##################################################################
##################################################
## Example identifiable
## Maathuis and Colombo (2013), Fig. 7, p.17
##################################################

## create the graph
p <- 7
amat <- t(matrix(c(0,0,1,1,0,0,0, 0,0,1,1,0,0,0, 0,0,0,1,0,1,0,
                   0,0,0,0,0,0,1, 0,0,0,0,0,1,1, 0,0,0,0,0,0,0,
                   0,0,0,0,0,0,0),  7, 7))
colnames(amat) <- rownames(amat) <- as.character(1:7)
V <- as.character(1:7)
edL <- vector("list",length=7)
names(edL) <- V
edL[[1]] <- list(edges=c(3,4),weights=c(1,1))
edL[[2]] <- list(edges=c(3,4),weights=c(1,1))
edL[[3]] <- list(edges=c(4,6),weights=c(1,1))
edL[[4]] <- list(edges=7,weights=c(1))
edL[[5]] <- list(edges=c(6,7),weights=c(1,1))
g <- new("graphNEL", nodes=V, edgeL=edL, edgemode="directed")
L <- 5

## compute the true covariance matrix of g
cov.mat <- trueCov(g)

## transform covariance matrix into a correlation matrix
true.corr <- cov2cor(cov.mat)
suffStat <- list(C=true.corr, n=10^9)
indepTest <- gaussCItest

## estimate the true PAG
true.pag <- dag2pag(suffStat, indepTest, g, L, alpha = 0.9999)

## The effect is identifiable  and the backdoor is  {1,2}:
stopifnot(backdoor(true.amat, 6, 3, type="cpdag") == 1:2)