pcalg (version 2.4-3)

gac: Test If Set Satisfies Generalized Adjustment Criterion (GAC)

Description

This function tests if z satisfies the Generalized Adjustment Criterion (GAC) relative to (x,y) in the graph represented by adjacency matrix amat and interpreted as type (DAG, CPDAG, MAG, PAG). If yes, z can be used in covariate adjustment for estimating causal effects of x on y.

Usage

gac(amat, x, y, z, type = "pag")

Arguments

amat

adjacency matrix of type amat.cpdag or amat.pag

x,y,z

(integer) positions of variables in x, y or z in the adjacency matrix. x, y and z can be vectors representing several nodes.

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.

Value

A list with three components:

Details

This work is a generalization of the work of Shpitser et al. (2012) (necessary and sufficient criterion in DAGs/ADMGs) and van der Zander et al. (2014) (necessary and sufficient criterion in MAGs). Moreover, it is a generalization of the Generalized Backdoor Criterion (GBC) of Maathuis and Colombo (2013): While GBC is sufficient but not necessary, GAC is both sufficient and necessary for DAGs, CPDAGs, MAGs and PAGs. For more details see Perkovic et al. (2015).

The motivation to find a set z that satisfies the GAC with respect to (x,y) is the following:

A set of variables z satisfies the GAC relative to (x,y) in the given graph, if and only if the causal effect of x on y is identifiable by covariate adjustment and is given by $$% P(Y|do(X = x)) = \sum_Z P(Y|X,Z) \cdot P(Z),$$ (for any joint distribution “compatible” with the graph; the formula is for discrete variables with straightforward modifications for continuous variables). This result allows to write post-intervention densities (the one written using Pearl's do-calculus) using only observational densities estimated from the data.

For z to satisfy the GAC relative to (x,y) and the graph, the following three conditions must hold:

It is important to note that there can be x and y for which there is no set Z that satisfies the GAC, but the total causal effect might be identifiable via some technique other than covariate adjustment.

For the coding of the adjacency matrix see amatType.

References

E. Perkovic, J. Textor, M. Kalisch and M.H. Maathuis (2015). A Complete Generalized Adjustment Criterion. In Proceedings of UAI 2015. http://arxiv.org/abs/1507.01524.

I. Shpitser, T. VanderWeele and J.M. Robins (2012). On the validity of covariate adjustment for estimating causal effects. In Proceedings of UAI 2010.

B. van der Zander, M. Liskiewicz and J. Textor (2014). Constructing separators and adjustment sets in ancestral graphs. In Proceedings of UAI 2014.

M.H. Maathuis and D. Colombo (2013). A generalized backdoor criterion. Annals of Statistics 43 1060-1088.

See Also

backdoor for the Generalized Backdoor Criterion, pc for estimating a CPDAG and fci and fciPlus for estimating a PAG.

Examples

## We reproduce the four examples in Perkovic et.al. (2015)

##############################
## Example 4.1
##############################
mFig1 <- matrix(c(0,1,1,0,0,0, 1,0,1,1,1,0, 0,0,0,0,0,1,
                  0,1,1,0,1,1, 0,1,0,1,0,1, 0,0,0,0,0,0), 6,6)
type <- "cpdag"
x <- 3; y <- 6
## Z satisfies GAC :
gac(mFig1, x,y, z=c(2,4),    type)
gac(mFig1, x,y, z=c(4,5),    type)
gac(mFig1, x,y, z=c(4,2,1),  type)
gac(mFig1, x,y, z=c(4,5,1),  type)
gac(mFig1, x,y, z=c(4,2,5),  type)
gac(mFig1, x,y, z=c(4,2,5,1),type)
## Z does not satisfy GAC :
gac(mFig1,x,y, z=2,    type)
gac(mFig1,x,y, z=NULL, type)

##############################
## Example 4.2
##############################
mFig3a <- matrix(c(0,1,0,0, 1,0,1,1, 0,1,0,1, 0,1,1,0), 4,4)
mFig3b <- matrix(c(0,2,0,0, 3,0,3,3, 0,2,0,3, 0,2,2,0), 4,4)
mFig3c <- matrix(c(0,3,0,0, 2,0,3,3, 0,2,0,3, 0,2,2,0), 4,4)
type <- "pag"
x <- 2; y <- 4
## Z does not satisfy GAC
gac(mFig3a,x,y, z=NULL, type) ## not amenable rel. to (X,Y)
gac(mFig3b,x,y, z=NULL, type) ## not amenable rel. to (X,Y)
## Z satisfies GAC
gac(mFig3c,x,y, z=NULL, type) ## amenable rel. to (X,Y)

##############################
## Example 4.3
##############################
mFig4a <- matrix(c(0,0,1,0,0,0, 0,0,1,0,0,0, 2,2,0,3,3,2,
                   0,0,2,0,2,2, 0,0,2,1,0,2, 0,0,1,3,3,0), 6,6)
mFig4b <- matrix(c(0,0,1,0,0,0, 0,0,1,0,0,0, 2,2,0,0,3,2,
                   0,0,0,0,2,2, 0,0,2,3,0,2, 0,0,2,3,2,0), 6,6)
type <- "pag"
x <- 3; y <- 4
## both PAGs are amenable rel. to (X,Y)
## Z satisfies GAC in Fig. 4a
gac(mFig4a,x,y, z=6, type)
gac(mFig4a,x,y, z=c(1,6), type)
gac(mFig4a,x,y, z=c(2,6), type)
gac(mFig4a,x,y, z=c(1,2,6), type)
## no Z satisfies GAC in Fig. 4b
gac(mFig4b,x,y, z=NULL, type)
gac(mFig4b,x,y, z=6, type)
gac(mFig4b,x,y, z=c(5,6), type)

##############################
## Example 4.4
##############################
mFig5a <- matrix(c(0,1,0,0,0, 1,0,1,0,0, 0,0,0,0,1, 0,0,1,0,0, 0,0,0,0,0), 5,5)
type <- "cpdag"
x <- c(1,5); y <- 4
## Z satisfies GAC
gac(mFig5a,x,y, z=c(2,3), type)
## Z does not satisfy GAC
gac(mFig5a,x,y, z=2, type)

mFig5b <- matrix(c(0,1,0,0,0,0,0, 2,0,2,3,0,3,0, 0,1,0,0,0,0,0,
0,2,0,0,3,0,0, 0,0,0,2,0,2,3, 0,2,0,0,2,0,0, 0,0,0,0,2,0,0), 7,7)
type <- "pag"
x<-c(2,7); y<-6
## Z satisfies GAC
gac(mFig5b,x,y, z=c(4,5), type)
gac(mFig5b,x,y, z=c(4,5,1), type)
gac(mFig5b,x,y, z=c(4,5,3), type)
gac(mFig5b,x,y, z=c(1,3,4,5), type)
## Z does not satisfy GAC
gac(mFig5b,x,y, z=NULL, type)