pcalg (version 1.1-4)

condIndFisherZ: Conditional Independence by Fisher's Z-Transformation

Description

Using Fisher's z-transformation of the partial correlation, test for zero partial correlation for sets of normally distributed random variables.

Usage

condIndFisherZ(x, y, S, C, n, cutoff,
               verbose= isTRUE(getOption("verbose.pcalg.condIFz")))
zStat(x, y, S, C, n)

Arguments

x,y,S

it is tested, whether x and y are conditionally independent given the subset S of the remaining nodes. x, y, S all are integers, corresponding to variable or node numbers.

C

correlation matrix of nodes

n

integer specifying the number of observations (samples) used to estimate the correlation matrix C.

cutoff

numeric cutoff for significance level of individual partial correlation tests. Must be set to qnorm(1 - alpha/2) for a test significance level of alpha.

verbose

logical indicating whether some intermediate output should be shown (WARNING: This decreases the performance dramatically!)

Value

• zStat() gives a number $$Z = \sqrt{n - \left|S\right| - 3} \cdot \log((1+r)/(1-r))/2$$ which is asymptotically normally distributed under the null hypothesis of correlation 0.

  condIndFisherZ() returns a logical $L$ indicating whether the partial correlation of x and y given S is zero could not be rejected on the given significance level. More intuitively and for multivariate normal data, this means: If TRUE then it seems plausible, that x and y are conditionally independent given S. If FALSE then there was strong evidence found against this conditional independence statement.

Details

For gaussian random variables and after performing Fisher's z-transformation of the partial correlation, the test statistic zStat() is (asymptotically for large enough n) standard normally distributed. Partial correlation is tested in a two-sided hypothesis test, i.e., basically, condIndFisherZ(*) == abs(zStat(*)) > qnorm(1 - alpha/2). In a multivariate normal distribution, zero partial correlation is equivalent to conditional independence.

References

Markus Kalisch and Peter B"uhlmann (2005) Estimating high-dimensional directed acyclic graphs with the PC-algorithm; Research Report Nr.~130, ETH Zurich; http://stat.ethz.ch/research/research_reports/2005

See Also

pcorOrder for computing a partial correlation given the correlation matrix in a recursive way.

Examples

set.seed(42)
## Generate four independent normal random variables
n <- 20
data <- matrix(rnorm(n*4),n,4)
## Compute corresponding correlation matrix
corMatrix <- cor(data)
## Test, whether variable 1 (col 1) and variable 2 (col 2) are
## independent given variable 3 (col 3) and variable 4 (col 4) on 0.05
## significance level
x <- 1
y <- 2
S <- c(3,4)
n <- 20
alpha <- 0.05
cutoff <- 1-qnorm(alpha/2)
(b1 <- condIndFisherZ(x,y,S,corMatrix,n,cutoff))
   # -> 1 and 2 seem to be conditionally independent given 3,4

## Now an example with conditional dependence
data <- matrix(rnorm(n*3),n,3)
data[,3] <- 2*data[,1]
corMatrix <- cor(data)
(b2 <- condIndFisherZ(1,3,2,corMatrix,n,cutoff))
   # -> 1 and 3 seem to be conditionally dependent given 2