# 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 |

`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 class |

`conf.int` |
a logical indicating whether a confidence interval for the difference in
location should be computed. Defaults to |

`conf.level` |
a numeric, confidence level of the interval. Defaults to |

`ties.method` |
a character, the method used to handle ties: the score generating function
either uses mid-ranks ( |

`mid.score` |
a character, the score assigned to observations exactly equal to the median:
either 0 ( |

`...` |
further arguments to be passed to |

### 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 *s*th 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))
``` |