IndependenceTest
General Independence Test
Testing the independence of two sets of variables measured on arbitrary scales.
- Keywords
- htest
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, …)
Arguments
- formula
a formula of the form
y1 + ... + yq ~ x1 + ... + xp | block
wherey1
, …,yq
andx1
, …,xp
are measured on arbitrary scales (nominal, ordinal or continuous with or without censoring) andblock
is an optional factor for stratification.- 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
NULL
.- weights
an optional formula of the form
~ w
defining integer valued case weights for each observation. Defaults toNULL
, implying equal weight for all observations.- object
an object inheriting from classes
"table"
or"'>IndependenceProblem"
.- teststat
a character, the type of test statistic to be applied: either a maximum statistic (
"maximum"
, default), a quadratic form ("quadratic"
) or a standardized scalar test statistic ("scalar"
).- distribution
a character, the conditional null distribution of the test statistic can be approximated by its asymptotic distribution (
"asymptotic"
, default) or via Monte Carlo resampling ("approximate"
). Alternatively, the functionsasymptotic
orapproximate
can be used. For univariate two-sample problems,"exact"
or use of the functionexact
computes the exact distribution. Computation of the null distribution can be suppressed by specifying"none"
. It is also possible to specify a function with one argument (an object inheriting from"'>IndependenceTestStatistic"
) that returns an object of class"'>NullDistribution"
.- alternative
a character, the alternative hypothesis: either
"two.sided"
(default),"greater"
or"less"
.- xtrafo
a function of transformations to be applied to the variables
x1
, …,xp
supplied informula
; see ‘Details’. Defaults totrafo
.- ytrafo
a function of transformations to be applied to the variables
y1
, …,yq
supplied informula
; see ‘Details’. Defaults totrafo
.- scores
a named list of scores to be attached to ordered factors; see ‘Details’. Defaults to
NULL
, implying equally spaced scores.- check
a function to be applied to objects of class
"'>IndependenceTest"
in order to check for specific properties of the data. Defaults toNULL
.- …
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-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.
Value
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.
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
# }