pcalg (version 2.7-12)

backdoor: Find Set Satisfying the Generalized Backdoor Criterion (GBC)

Description

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

Usage

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

Value

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

Arguments

amat

adjacency matrix of type amat.cpdag or amat.pag.

x,y

(integer) position of variable \(X\) and \(Y\), respectively, 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"); then the type of the adjacency matrix is assumed to be amat.cpdag. It can also be a MAG (type="mag"), or a PAG (type="pag"); then the type of the adjacency matrix is assumed to be amat.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.

Author

Diego Colombo and Markus Kalisch ()

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 (2015).

The motivation to find a set W that satisfies the GBC 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 GBC 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 GBC (see Maathuis and Colombo, 2015). If the effect is not identifiable in this way, the output is NA. Otherwise, an explicit set W that satisfies the GBC 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 GBC, but the total causal effect might be identifiable via some other technique.

For the coding of the adjacency matrix see amatType.

References

M.H. Maathuis and D. Colombo (2015). A generalized backdoor criterion. Annals of Statistics 43 1060-1088.

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

See Also

gac for the Generalized Adjustment Criterion (GAC), which is a generalization of GBC; pc for estimating a CPDAG, dag2pag and fci for estimating a PAG, and pag2magAM for estimating a MAG.

Examples

#####################################################################
##DAG
#####################################################################
## Simulate the true DAG
suppressWarnings(RNGversion("3.5.0"))
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 set satisfying the GBC:
backdoor(true.amat, 5, 7, type="dag")
stopifnot(backdoor(true.amat, 5, 7, type="dag") == 3)
#####################################################################
##CPDAG
#####################################################################
##################################################
## Example not identifiable
## Maathuis and Colombo (2015), Fig. 3a, p.1072
##################################################
## 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 (2015), Fig. 3b, p.1072
##################################################

## 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

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

##################################################################
##PAG
##################################################################
##################################################
## Example identifiable
## Maathuis and Colombo (2015), Fig. 5a, p.1075
##################################################

## 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)@amat

## The effect is identifiable  and the backdoor set is:
backdoor(true.pag, 3, 5, type="pag")
stopifnot(backdoor(true.pag, 3, 5, type="pag") == 1:2)