IndependenceTests (version 0.5)

A.dep.tests: Tests for mutual or serial independence between categorical variables

Description

The tests are constructed from the M<U+00F6>bius transformation applied to the probability cells in a multi-way contingency table. The Pearson chi-squared test of mutual independence is partitioned into A-dependence statistics over all subsets A of variables. The goal of the partition is to identify subsets of dependent variables when the mutual independence hypothesis is rejected by the Pearson chi-squared test. The methodology can be directly adapted to test for serial independence of d successive observations of a stationary categorical time series.

For categorical time series, especially those of a nominal (non ordinal) nature, the user should be aware that tests of serial independence obtained by methods suited to quantitative sequences by quantification of the labels are not invariant to permutation of the labels contrary to the test described here.

Usage

A.dep.tests(Xmat, choice = 1, d = 0, m = d, freqname = "", type = "text")

Arguments

Xmat

Table, matrix or data-frame of the contingency table data, if choice = 1. Vector of the time series data, if choice = 2.

choice

Integer. 1 for mutual independence, 2 for serial independence.

d

Integer. Used only if choice = 2 for the number of successive obervations.

m

Integer. Maximum cardinality of subsets \(A\) for which an \(A\)-dependence statistic is required. This option is particularly useful for large values of d.

freqname

Character. Used only if choice = 1 and when Xmat is a matrix or a data-frame to identify the variable for the counts (frequencies).

type

"text" or "html"

Value

Returns an object of class list containing the following components:

TA

\(A\)-dependence statistics for each subset \(A\) of variables.

fA

degrees of freedom of the \(A\)-dependence statistics.

pvalA

\(p\)-values of the \(A\)-dependence statistics.

X

summary of the results.

X2

test statistic for mutual independence obtained by the sum of the \(A\)-dependence statistics, if choice = 1.

Y2

test statistic for serial independence obtained by the sum of the \(A\)-dependence statistics, if choice = 2.

f

number of degrees of freedom associated with the test statistic X2 or Y2.

pval

the \(p\)-value associated with the test statistic X2 or Y2.

References

Bilodeau M., Lafaye de Micheaux P. (2009). A-dependence statistics for mutual and serial independence of categorical variables, Journal of Statistical Planning and Inference, 139, 2407-2419.

Agresti A. (2002). Categorical data analysis, Wiley, p. 322

Whisenant E.C., Rasheed B.K.A., Ostrer H., Bhatnagar Y.M. (1991). Evolution and sequence analysis of a human Y-chromosomal DNA fragment, J. Mol. Evol., 33, 133-141.

Examples

# NOT RUN {
# Test of  mutual independence between 3 independent Bernoulli variables.

n <- 100
data <- data.frame(X1 = rbinom(n, 1, 0.3), X2 = rbinom(n, 1, 0.3) , X3 =
                   rbinom(n, 1, 0.3))
X <- table(data)
A.dep.tests(X)

# Test of mutual independence between 4 variables which are
# 2-independent and 3-independent, but are 4-dependent.

n <- 100
W <- sample(x = 1:8, size = n, TRUE)
X1 <- W %in% c(1, 2, 3, 5)
X2 <- W %in% c(1, 2, 4, 6)
X3 <- W %in% c(1, 3, 4, 7)
X4 <- W %in% c(2, 3, 4, 8)
data <- data.frame(X1, X2, X3, X4)
X <- table(data)
A.dep.tests(X)

# Test of serial independence of a nucleotide sequence of length
# 4156 described in Whisenant et al. (1991).

data(dna)
x2 <- dna[1]
for (i in 2:length(dna)) x2 <- paste(x2, dna[i], sep = "")
x <- unlist(strsplit(x2, ""))
x[x == "a" | x == "g"] <- "r"
x[x == "c" |  x== "t"] <- "y"
# }
# NOT RUN {
out <- A.dep.tests(x, choice = 2, d = 1501, m = 2)$TA[[1]]
plot(100:1500, out[100:1500], xlab = "lag j", ylab = "T(1,j+1)", pch = 19)
abline(h = qchisq(.995, df = 1))
# }
# NOT RUN {
# Analysis of a contingency table in Agresti (2002) p. 322 

data(highschool)
A.dep.tests(highschool, freqname = "count")

# }