IndependenceTest

General Independence Test

Testing the independence of two sets of variables measured on arbitrary scales.

Usage

# S3 method for formula
independence_test(formula, data, subset = NULL, weights = NULL, …)
# S3 method for table
independence_test(object, …)
# S3 method for IndependenceProblem
independence_test(object, teststat = c("maximum", "quadratic", "scalar"),
                  distribution = c("asymptotic", "approximate",
                                   "exact", "none"),
                  alternative = c("two.sided", "less", "greater"),
                  xtrafo = trafo, ytrafo = trafo, scores = NULL,
                  check = NULL, …)

Details

independence_test provides a general independence test for two sets of variables measured on arbitrary scales. This function is based on the general framework for conditional inference procedures proposed by Strasser and Weber (1999). The salient parts of the Strasser-Weber framework are elucidated by Hothorn et al. (2006) and a thorough description of the software implementation is given by Hothorn et al. (2008).

The null hypothesis of independence, or conditional independence given block, between y1, …, yq and x1, …, xp is tested.

A vector of case weights, e.g., observation counts, can be supplied through the weights argument and the type of test statistic is specified by the teststat argument. Influence and regression functions, i.e., transformations of y1, …, yq and x1, …, xp, are specified by the ytrafo and xtrafo arguments respectively; see trafo for the collection of transformation functions currently available. This allows for implementation of both novel and familiar test statistics, e.g., the Pearson \(\chi^2\) test, the generalized Cochran-Mantel-Haenszel test, the Spearman correlation test, the Fisher-Pitman permutation test, the Wilcoxon-Mann-Whitney test, the Kruskal-Wallis test and the family of weighted logrank tests for censored data. Furthermore, multivariate extensions such as the multivariate Kruskal-Wallis test (Puri and Sen, 1966, 1971) can be implemented without much effort (see ‘Examples’).

If, say, y1 and/or x1 are ordered factors, the default scores, 1:nlevels(y1) and 1:nlevels(x1) respectively, can be altered using the scores argument; this argument can also be used to coerce nominal factors to class "ordered". For example, when y1 is an ordered factor with four levels and x1 is a nominal factor with three levels, scores = list(y1 = c(1, 3:5), x1 = c(1:2, 4)) supplies the scores to be used. For ordered alternatives the scores must be monotonic, but non-montonic scores are also allowed for testing against, e.g., umbrella alternatives. The length of the score vector must be equal to the number of factor levels.

The conditional null distribution of the test statistic is used to obtain \(p\)-values and an asymptotic approximation of the exact distribution is used by default (distribution = "asymptotic"). Alternatively, the distribution can be approximated via Monte Carlo resampling or computed exactly for univariate two-sample problems by setting distribution to "approximate" or "exact" respectively. See asymptotic, approximate and exact for details.

Note

Starting with coin version 1.1-0, maximum statistics and quadratic forms can no longer be specified using teststat = "maxtype" and teststat = "quadtype" respectively (as was used in versions prior to 0.4-5).

References

Hothorn, T., Hornik, K., van de Wiel, M. A. and Zeileis, A. (2006). A Lego system for conditional inference. The American Statistician 60(3), 257--263. 10.1198/000313006X118430

Hothorn, T., Hornik, K., van de Wiel, M. A. and Zeileis, A. (2008). Implementing a class of permutation tests: The coin package. Journal of Statistical Software 28(8), 1--23. 10.18637/jss.v028.i08

Johnson, W. D., Mercante, D. E. and May, W. L. (1993). A computer package for the multivariate nonparametric rank test in completely randomized experimental designs. Computer Methods and Programs in Biomedicine 40(3), 217--225. 10.1016/0169-2607(93)90059-T

Puri, M. L. and Sen, P. K. (1966). On a class of multivariate multisample rank order tests. Sankhya A 28(4), 353--376.

Puri, M. L. and Sen, P. K. (1971). Nonparametric Methods in Multivariate Analysis. New York: John Wiley & Sons.

Strasser, H. and Weber, C. (1999). On the asymptotic theory of permutation statistics. Mathematical Methods of Statistics 8(2), 220--250.

Aliases

• independence_test
• independence_test.formula
• independence_test.table
• independence_test.IndependenceProblem

Examples

# NOT RUN {
## One-sided exact van der Waerden (normal scores) test...
independence_test(asat ~ group, data = asat,
                  ## exact null distribution
                  distribution = "exact",
                  ## one-sided test
                  alternative = "greater",
                  ## apply normal scores to asat$asat
                  ytrafo = function(data)
                      trafo(data, numeric_trafo = normal_trafo),
                  ## indicator matrix of 1st level of asat$group
                  xtrafo = function(data)
                      trafo(data, factor_trafo = function(x)
                          matrix(x == levels(x)[1], ncol = 1)))

## ...or more conveniently
normal_test(asat ~ group, data = asat,
            ## exact null distribution
            distribution = "exact",
            ## one-sided test
            alternative = "greater")


## Receptor binding assay of benzodiazepines
## Johnson, Mercante and May (1993, Tab. 1)
benzos <- data.frame(
      cerebellum = c( 3.41,  3.50,  2.85,  4.43,
                      4.04,  7.40,  5.63, 12.86,
                      6.03,  6.08,  5.75,  8.09,  7.56),
       brainstem = c( 3.46,  2.73,  2.22,  3.16,
                      2.59,  4.18,  3.10,  4.49,
                      6.78,  7.54,  5.29,  4.57,  5.39),
          cortex = c(10.52,  7.52,  4.57,  5.48,
                      7.16, 12.00,  9.36,  9.35,
                     11.54, 11.05,  9.92, 13.59, 13.21),
    hypothalamus = c(19.51, 10.00,  8.27, 10.26,
                     11.43, 19.13, 14.03, 15.59,
                     24.87, 14.16, 22.68, 19.93, 29.32),
        striatum = c( 6.98,  5.07,  3.57,  5.34,
                      4.57,  8.82,  5.76, 11.72,
                      6.98,  7.54,  7.66,  9.69,  8.09),
     hippocampus = c(20.31, 13.20,  8.58, 11.42,
                     13.79, 23.71, 18.35, 38.52,
                     21.56, 18.66, 19.24, 27.39, 26.55),
       treatment = factor(rep(c("Lorazepam", "Alprazolam", "Saline"),
                          c(4, 4, 5)))
)

## Approximative (Monte Carlo) multivariate Kruskal-Wallis test
## Johnson, Mercante and May (1993, Tab. 2)
independence_test(cerebellum + brainstem + cortex +
                  hypothalamus + striatum + hippocampus ~ treatment,
                  data = benzos,
                  teststat = "quadratic",
                  distribution = approximate(nresample = 10000),
                  ytrafo = function(data)
                      trafo(data, numeric_trafo = rank_trafo)) # Q = 16.129
# }