cond.fun.chisq.test {FunChisq}

Conditional Functional Chi-Squared Test

Description

Asymptotic chi-squared test to determine the model-free functional dependency of effect variable Y on a cause variable X, conditioned on a third variable Z.

Usage

cond.fun.chisq.test(x, y, z=NULL, data=NULL, log.p = FALSE,
                    method = c("fchisq", "nfchisq"))

Arguments

x

vector or character; either a discrete random variable (cause) represented as vector or a character column name in data.

y

vector or character; either a discrete random variable (effect) represented as vector or a character column name in data.

z

vector or character; either a discrete random variable (condition) represented as vector or a character column name in data. In case of NULL, a fun.chisq.test on a contingency table, with x as row variable and y as column variable, is returned. See fun.chisq.test for details. The default is NULL.

data

a data frame containing three or more columns whose names can be used as values for x, y and z. In case of NULL, x, y and z must be vectors. The default is NULL.

log.p

logical; if TRUE, the p-value is given as log(p). Taking the log improves numerical precision when the p-value is close to zero. The default is FALSE.

method

a character string to specify the method to compute the conditional functional chi-squared test statistic and its p-value. The options are "fchisq" (default) and "nfchisq". See Details.

Details

The conditional functional chi-squared test introduces the concept of conditional functional depedency, where the functional association between two variables (x and y) is tested conditioned on a third variable (z) (Zhang 2014). Two methods are provided to compute the chi-squared statistic and its p-value. When method = "fchisq", the p-value is computed using the chi-squared distribution; when method = "nfchisq", a normalized statistic is obtained by shifting and scaling the original chi-squared statistic and a p-value is computed using the standard normal distribution (Box et al. 2005). The normalized test is more conservative on the degrees of freedom.

Value

A list with class "htest" containing the following components:

statistic

the conditional functional chi-squared statistic if method = "fchisq"; or the normalized conditional functional chi-squared statistic if method = "nfchisq".

parameter

degrees of freedom for the conditional functional chi-squared statistic.

p.value

p-value of the conditional functional test. If method = "fchisq", the p-value is computed by an asymptotic chi-squared distribution; if method = "nfchisq", the p-value is computed by the standard normal distribution.

estimate

an estimate of the conditional function index between 0 and 1. The value of 1 indicates strong functional dependency between x and y, given z. It is asymmetrical with respect to whether x was chosen as the cause of effect y or vice versa.

Author(s)

Sajal Kumar and Mingzhou Song

References

Box GE, Hunter JS, Hunter WG (2005). Statistics for Experimenters: Design, Innovation and Discovery, 2nd edition. Wiley-Interscience, New York.

Zhang Y (2014). Nonparametric Statistical Methods for Biological Network Inference. Ph.D. thesis, Department of Computer Science, New Mexico State University, Las Cruces, NM, USA.

See Also

See (unconditional) functional chi-squared test fun.chisq.test.

Examples

# Generate a relationship between variables X and Z
xz = matrix(c(30,2,2, 2,2,40, 2,30,2),ncol=3,nrow=3,
            byrow = TRUE)
# Re-construct X
x = rep(c(1:nrow(xz)),rowSums(xz))
# Re-construct Z
z = c()
for(i in 1:nrow(xz))
  z = c(z,rep(c(1:ncol(xz)),xz[i,]))

# Generate a relationship between variables Z and Y
# Make sure Z retains its distribution
zy = matrix(c(4,30, 30,4, 4,40),ncol=2,nrow=3,
            byrow = TRUE)
# Re-construct Y
y = rep(0,length(z))
for(i in unique(z))
  y[z==i] = rep(c(1:ncol(zy)),zy[i,])

# Tables
table(x,z)
table(z,y)
table(x,y)

# Conditional functional dependency
# Y = f(X) | Z should be false
cond.fun.chisq.test(x=x,y=y,z=z)
# Z = f(X) | Y should be true
cond.fun.chisq.test(x=x,y=z,z=y)
# Y = f(Z) | X should be true
cond.fun.chisq.test(x=z,y=y,z=x)