pcalg (version 2.7-12)

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, maximal PDAG, 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")

Value

A list with three components:

gac

logical; TRUE if z satisfies the GAC relative to (x,y) in the graph represented by amat and type

res

logical vector of length three indicating if each of the three conditions (0), (1) and (2) are true

f

node positions of nodes in the forbidden set (see Perkovic et al. (2015, 2017b)

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 PDAG (type="pdag") or 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.

Author

Emilija Perkovic and Markus Kalisch ()

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, 2017a, 2017b).

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:

(0)

The graph is adjustment amenable relative to (x,y).

(1)

The intersection of z and the forbidden set (explained in Perkovic et al. (2015, 2017b) is empty.

(2)

All proper definite status non-causal paths in the graph from x to y are blocked by z.

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 details on the GAC for DAGs, CPDAGs, PAGs see Perkovic et. al (2015,2017a). For details on the GAC for MAGs see van der Zander et. al (2014) and for details on the GAC for maximal PDAGs see Perkovic et. al (2017b).

For the coding of the adjacency matrix see amatType. The input matrix can either be of class matrix or of class amat.

References

E. Perkovic, J. Textor, M. Kalisch and M.H. Maathuis (2015). A Complete Generalized Adjustment Criterion. In Proceedings of UAI 2015.

E. Perkovic, J. Textor, M. Kalisch and M.H. Maathuis (2017a). Complete graphical characterization and construction of adjustment sets in Markov equivalence classes of ancestral graphs. To appear in Journal of Machine Learning Research.

E. Perkovic, M. Kalisch and M.H. Maathuis (2017b). Interpreting and using CPDAGs with background knowledge. In Proceedings of UAI 2017.

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, 2017a)

##############################
## Example 4.1 in Perkovic et. al (2015), Example 2 in Perkovic et. al (2017a)
##############################
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 in Perkovic et. al (2015), Example 3 in Perkovic et. al (2017a)
##############################
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 in Perkovic et. al (2015), Example 4 in Perkovic et. al (2017a)
##############################
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 in Perkovic et. al (2015), Example 8 in Perkovic et. al (2017a)
##############################
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)

##############################
## Example 4.7 in Perkovic et. al (2017b)
##############################
mFig3a <- matrix(c(0,1,0,0, 1,0,1,1, 0,1,0,1, 0,1,1,0), 4,4)
mFig3b <- matrix(c(0,1,0,0, 0,0,1,1, 0,0,0,1, 0,0,1,0), 4,4)
mFig3c <- matrix(c(0,0,0,0, 1,0,1,0, 0,1,0,1, 0,1,1,0), 4,4)
type <- "pdag"
x <- 2; y <- 4
## Z does not satisfy GAC
gac(mFig3a,x,y, z=NULL, type) ## not amenable rel. to (X,Y)
gac(mFig3c,x,y, z=NULL, type) ## amenable rel. to (X,Y), but no set can block X <- Y
## Z satisfies GAC
gac(mFig3b,x,y, z=NULL, type) ## amenable rel. to (X,Y)