Two- and K-Sample Location Tests

Description

Testing the equality of the distributions of a numeric response variable in two or more independent groups against shift alternatives.

Usage

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
## S3 method for class 'formula'
oneway_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'IndependenceProblem'
oneway_test(object, ...)

## S3 method for class 'formula'
wilcox_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'IndependenceProblem'
wilcox_test(object, conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula'
kruskal_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'IndependenceProblem'
kruskal_test(object, ...)

## S3 method for class 'formula'
normal_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'IndependenceProblem'
normal_test(object, ties.method = c("mid-ranks", "average-scores"),
            conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula'
median_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'IndependenceProblem'
median_test(object, mid.score = c("0", "0.5", "1"),
            conf.int = FALSE, conf.level = 0.95, ...)

## S3 method for class 'formula'
savage_test(formula, data, subset = NULL, weights = NULL, ...)
## S3 method for class 'IndependenceProblem'
savage_test(object, ties.method = c("mid-ranks", "average-scores"),
            conf.int = FALSE, conf.level = 0.95, ...)

Arguments

formula

a formula of the form y ~ x | block where y is a numeric variable, x is a factor 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 class "IndependenceProblem".

conf.int

a logical indicating whether a confidence interval for the difference in location should be computed. Defaults to FALSE.

conf.level

a numeric, confidence level of the interval. Defaults to 0.95.

ties.method

a character, the method used to handle ties: the score generating function either uses mid-ranks ("mid-ranks", default) or averages the scores of randomly broken ties ("average-scores").

mid.score

a character, the score assigned to observations exactly equal to the median: either 0 ("0", default), 0.5 ("0.5") or 1 ("1"); see ‘Details’.

...

further arguments to be passed to independence_test.

Details

oneway_test, wilcox_test, kruskal_test, normal_test, median_test and savage_test provide the Fisher-Pitman permutation test, the Wilcoxon-Mann-Whitney test, the Kruskal-Wallis test, the van der Waerden test, the Brown-Mood median test and the Savage test. A general description of these methods is given by Hollander and Wolfe (1999). For the adjustment of scores for tied values see Hájek, Šidák and Sen (1999, pp. 133–135).

The null hypothesis of equality, or conditional equality given block, of the distribution of y in the groups defined by x is tested against shift alternatives. In the two-sample case, the two-sided null hypothesis is H_0: mu = 0, where μ = Y_1 - Y_2 and Y_s is the median of the responses in the sth sample. In case alternative = "less", the null hypothesis is H_0: mu >= 0. When alternative = "greater", the null hypothesis is H_0: mu <= 0. Confidence intervals for the difference in location are available (except for oneway_test) and computed according to Bauer (1972).

If x is an ordered factor, the default scores, 1:nlevels(x), can be altered using the scores argument (see independence_test); this argument can also be used to coerce nominal factors to class "ordered". In this case, a linear-by-linear association test is computed and the direction of the alternative hypothesis can be specified using the alternative argument.

The Brown-Mood median test offers a choice of mid-score, i.e., the score assigned to observations exactly equal to the median. In the two-sample case, mid-score = "0" implies that the linear test statistic is simply the number of subjects in the second sample with observations greater than the median of the pooled sample. Similarly, the linear test statistic for the last alternative, mid-score = "1", is the number of subjects in the second sample with observations greater than or equal to the median of the pooled sample. If mid-score = "0.5" is selected, the linear test statistic is the mean of the test statistics corresponding to the first and last alternatives and has a symmetric distribution, or at least approximately so, under the null hypothesis (see Hájek, Šidák and Sen, 1999, pp. 97–98).

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". Confidence intervals can be extracted by confint.

Note

Starting with version 1.1-0, oneway_test no longer allows the test statistic to be specified; a quadratic form is now used in the K-sample case. Please use independence_test if more control is desired.

References

Bauer, D. F. (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67(339), 687–690.

Hájek, J., Šidák, Z. and Sen, P. K. (1999). Theory of Rank Tests, Second Edition. San Diego: Academic Press.

Hollander, M. and Wolfe, D. A. (1999). Nonparametric Statistical Methods, Second Edition. New York: John Wiley & Sons.

Examples

  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
## Tritiated Water Diffusion Across Human Chorioamnion
## Hollander and Wolfe (1999, p. 110, Tab. 4.1)
diffusion <- data.frame(
    pd = c(0.80, 0.83, 1.89, 1.04, 1.45, 1.38, 1.91, 1.64, 0.73, 1.46,
           1.15, 0.88, 0.90, 0.74, 1.21),
    age = factor(rep(c("At term", "12-26 Weeks"), c(10, 5)))
)

## Exact Wilcoxon-Mann-Whitney test
## Hollander and Wolfe (1999, p. 111)
## (At term - 12-26 Weeks)
(wt <- wilcox_test(pd ~ age, data = diffusion,
                   distribution = "exact", conf.int = TRUE))

## Extract observed Wilcoxon statistic
## Note: this is the sum of the ranks for age = "12-26 Weeks"
statistic(wt, "linear")

## Expectation, variance, two-sided pvalue and confidence interval
expectation(wt)
covariance(wt)
pvalue(wt)
confint(wt)

## For two samples, the Kruskal-Wallis test is equivalent to the W-M-W test
kruskal_test(pd ~ age, data = diffusion,
             distribution = "exact")

## Asymptotic Fisher-Pitman test
oneway_test(pd ~ age, data = diffusion)

## Approximative (Monte Carlo) Fisher-Pitman test
pvalue(oneway_test(pd ~ age, data = diffusion,
                   distribution = approximate(B = 10000)))

## Exact Fisher-Pitman test
pvalue(ot <- oneway_test(pd ~ age, data = diffusion,
                         distribution = "exact"))

## Plot density and distribution of the standardized test statistic
op <- par(no.readonly = TRUE) # save current settings
layout(matrix(1:2, nrow = 2))
s <- support(ot)
d <- sapply(s, function(x) dperm(ot, x))
p <- sapply(s, function(x) pperm(ot, x))
plot(s, d, type = "S", xlab = "Test Statistic", ylab = "Density")
plot(s, p, type = "S", xlab = "Test Statistic", ylab = "Cum. Probability")
par(op) # reset


## Example data
ex <- data.frame(
    y = c(3, 4, 8, 9, 1, 2, 5, 6, 7),
    x = factor(rep(c("no", "yes"), c(4, 5)))
)

## Boxplots
boxplot(y ~ x, data = ex)

## Exact Brown-Mood median test with different mid-scores
(mt1 <- median_test(y ~ x, data = ex, distribution = "exact"))
(mt2 <- median_test(y ~ x, data = ex, distribution = "exact",
                    mid.score = "0.5"))
(mt3 <- median_test(y ~ x, data = ex, distribution = "exact",
                    mid.score = "1")) # sign change!

## Plot density and distribution of the standardized test statistics
op <- par(no.readonly = TRUE) # save current settings
layout(matrix(1:3, nrow = 3))
s1 <- support(mt1); d1 <- dperm(mt1, s1)
plot(s1, d1, type = "h", main = "Mid-score: 0",
     xlab = "Test Statistic", ylab = "Density")
s2 <- support(mt2); d2 <- dperm(mt2, s2)
plot(s2, d2, type = "h", main = "Mid-score: 0.5",
     xlab = "Test Statistic", ylab = "Density")
s3 <- support(mt3); d3 <- dperm(mt3, s3)
plot(s3, d3, type = "h", main = "Mid-score: 1",
     xlab = "Test Statistic", ylab = "Density")
par(op) # reset


## Length of YOY Gizzard Shad
## Hollander and Wolfe (1999, p. 200, Tab. 6.3)
yoy <- data.frame(
    length = c(46, 28, 46, 37, 32, 41, 42, 45, 38, 44,
               42, 60, 32, 42, 45, 58, 27, 51, 42, 52,
               38, 33, 26, 25, 28, 28, 26, 27, 27, 27,
               31, 30, 27, 29, 30, 25, 25, 24, 27, 30),
    site = gl(4, 10, labels = as.roman(1:4))
)

## Approximative (Monte Carlo) Kruskal-Wallis test
kruskal_test(length ~ site, data = yoy,
             distribution = approximate(B = 10000))

## Approximative (Monte Carlo) Nemenyi-Damico-Wolfe-Dunn test (joint ranking)
## Hollander and Wolfe (1999, p. 244)
## (where Steel-Dwass results are given)
it <- independence_test(length ~ site, data = yoy,
                        distribution = approximate(B = 50000),
                        ytrafo = function(data)
                            trafo(data, numeric_trafo = rank_trafo),
                        xtrafo = mcp_trafo(site = "Tukey"))

## Global p-value
pvalue(it)

## Sites (I = II) != (III = IV) at alpha = 0.01 (p. 244)
pvalue(it, method = "single-step") # subset pivotality is violated


## Asymptotic Jonckheere-Terpstra test for ordered groups
pieces <- data.frame(
    control = c(40, 35, 38, 43, 44, 41),
    rough = c(38, 40, 47, 44, 40, 42),
    accurate = c(48, 40, 45, 43, 46, 44)
)
pieces <- stack(pieces)
pieces$ind <- ordered(pieces$ind,
                      levels = c("control", "rough", "accurate"))

## Look at K: the second line just sums up.
ff <- function(x) {
    K <- multcomp::contrMat(table(x), "Tukey")[, x]
    as.vector(rep(1, nrow(K)) %*% K)
}

independence_test(values ~ ind, data = pieces,
                  alternative = "greater",
                  ytrafo = function(data)
                      trafo(data, numeric_trafo = rank_trafo),
                  xtrafo = function(data)
                      trafo(data, ordered_trafo = ff))

Want to suggest features or report bugs for rdrr.io? Use the GitHub issue tracker.