# 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`

where`y1`

, …,`yq`

and`x1`

, …,`xp`

are measured on arbitrary scales (nominal, ordinal or continuous with or without censoring) and`block`

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 to`NULL`

, 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 functions`asymptotic`

or`approximate`

can be used. For univariate two-sample problems,`"exact"`

or use of the function`exact`

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 in`formula`

; see ‘Details’. Defaults to`trafo`

.- ytrafo
a function of transformations to be applied to the variables

`y1`

, …,`yq`

supplied in`formula`

; see ‘Details’. Defaults to`trafo`

.- 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 to`NULL`

.- …
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
# }
```

*Documentation reproduced from package coin, version 1.3-1, License: GPL-2*