pSCCI {SCCI}

Stochastic Complexity-based Conditional Independence Criterium (p-value)

Description

This is an adapted version of SCCI for which the output can be interpreted as a p-value. For this, we adapted SCCI such that if SCCI = 0 (X is independent of Y given Z) it gives a p-value greater than 0.01 and for SCCI > 0 (X is not independent of Y given Z) gives a p-value smaller or equal to 0.01. Note that we just transformed the output of SCCI and do not obtain a real p-value. In essence, we define the artificial p-value as follows. Let v the output of SCCI divided by the number of samples n. p = 2^{-(6.643855 - v)}, which is equal to 0.01000001 if v = 0. Further, p \le 0.01 for SCCI \ge 0.000001. We restrict the p-values to be between 0 and 1.

Unlike SCCI, pSCCI is currently only instantiated with fNML.

pSCCI can be used directly in the PC algorithm developed by Spirtes et al. (2000), which was implemented in the 'pcalg' R-package by Kalisch et al. (2012), as shown in the example.

Usage

pSCCI(x, y, S, suffStat)	

Arguments

x

Position of x variabe (integer).

y

Position of y variabe (integer).

S

Vector of the position of zero or more conditioning variables (integer).

suffStat

This format was adapted such that it can be used in the PC algorithm and other algorithms from the 'pcalg' package. SCCI only need the filed "dm" that contains the data matrix.

References

Markus Kalisch, Martin Mächler, Diego Colombo, Marloes H. Maathuis, Peter Bühlmann; Causal inference using graphical models with the R package pcalg, Journal of Statistical Software, 2012

Alexander Marx and Jilles Vreeken; Testing Conditional Independence on Discrete Data using Stochastic Complexity, Proceedings of the 22nd International Conference on Artificial Intelligence and Statistics (AISTATS), PMLR, 2019

Peter Spirtes, Clark N. Glymour, Richard Scheines, David Heckerman, Christopher Meek, Gregory Cooper and Thomas Richardson; Causation, Prediction, and Search, MIT press, 2000

Examples

set.seed(1)
x = round((runif(1000, min=0, max=5)))
y = round((runif(1000, min=0, max=5)))
Z = data.frame(round((runif(1000, min=0, max=5))), round((runif(1000, min=0, max=5))))
## create data matrix
data_matrix = as.matrix(data.frame(x,y,S1=Z[,1], S2=Z[,2]))
suffStat = list(dm=data_matrix)
pSCCI(x=1,y=2,S=c(3,4),suffStat=suffStat)	## 0.01000001

### Using SCI within the PC algorithm
if(require(pcalg)){
  ## Load data
  data(gmD)
  V <- colnames(gmD$x)
  ## define sufficient statistics
  suffStat <- list(dm = gmD$x, nlev = c(3,2,3,4,2), adaptDF = FALSE)
  ## estimate CPDAG
  pc.D <- pc(suffStat,
            ## independence test: SCCI using fNML
            indepTest = pSCCI, alpha = 0.01, labels = V, verbose = TRUE)
}
if (require(pcalg) & require(Rgraphviz)) {
  ## show estimated CPDAG
  par(mfrow = c(1,2))
  plot(pc.D, main = "Estimated CPDAG")
  plot(gmD$g, main = "True DAG")
}