| IndependenceTest {coin} | R Documentation |
General Independence Test
Description
Testing the independence of two sets of variables measured on arbitrary scales.
Usage
## S3 method for class 'formula'
independence_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'table'
independence_test(object, ...)
## S3 method for class '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, ...)
Arguments
formula |
a formula of the form |
data |
an optional data frame containing the variables in the model formula. |
subset |
an optional vector specifying a subset of observations to be used. Defaults
to |
weights |
an optional formula of the form |
object |
an object inheriting from classes |
teststat |
a character, the type of test statistic to be applied: either a maximum
statistic ( |
distribution |
a character, the conditional null distribution of the test statistic can be
approximated by its asymptotic distribution ( |
alternative |
a character, the alternative hypothesis: either |
xtrafo |
a function of transformations to be applied to the variables |
ytrafo |
a function of transformations to be applied to the variables |
scores |
a named list of scores to be attached to ordered factors; see
‘Details’. Defaults to |
check |
a function to be applied to objects of class
|
... |
further arguments to be passed to or from other methods (currently ignored). |
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-monotonic 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.
Value
An object inheriting from class "IndependenceTest".
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. doi: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. doi: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. doi: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.
Examples
## 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