Description Usage Arguments Details Value Note References Examples

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

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, ...)
``` |

`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 |

`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.

An object inheriting from class `"IndependenceTest"`

.
Confidence intervals can be extracted by confint.

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.

Bauer, D. F. (1972). Constructing confidence sets using rank statistics.
*Journal of the American Statistical Association* **67**(339),
687–690. doi: 10.1080/01621459.1972.10481279

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.

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## 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, type = "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(nresample = 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 <- dperm(ot, s)
p <- pperm(ot, s)
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(nresample = 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(nresample = 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
``` |

```
Loading required package: survival
Exact Wilcoxon-Mann-Whitney Test
data: pd by age (12-26 Weeks, At term)
Z = -1.2247, p-value = 0.2544
alternative hypothesis: true mu is not equal to 0
95 percent confidence interval:
-0.76 0.15
sample estimates:
difference in location
-0.305
12-26 Weeks 30
12-26 Weeks
40
12-26 Weeks
12-26 Weeks 66.66667
[1] 0.2544123
95 percent confidence interval:
-0.76 0.15
sample estimates:
difference in location
-0.305
Exact Kruskal-Wallis Test
data: pd by age (12-26 Weeks, At term)
chi-squared = 1.5, p-value = 0.2544
Asymptotic Two-Sample Fisher-Pitman Permutation Test
data: pd by age (12-26 Weeks, At term)
Z = -1.5225, p-value = 0.1279
alternative hypothesis: true mu is not equal to 0
[1] 0.1307
99 percent confidence interval:
0.1221453 0.1396087
[1] 0.1318681
Exact Two-Sample Brown-Mood Median Test
data: y by x (no, yes)
Z = 0.28284, p-value = 1
alternative hypothesis: true mu is not equal to 0
Exact Two-Sample Brown-Mood Median Test
data: y by x (no, yes)
Z = 0, p-value = 1
alternative hypothesis: true mu is not equal to 0
Exact Two-Sample Brown-Mood Median Test
data: y by x (no, yes)
Z = -0.28284, p-value = 1
alternative hypothesis: true mu is not equal to 0
Approximative Kruskal-Wallis Test
data: length by site (I, II, III, IV)
chi-squared = 22.852, p-value < 1e-04
[1] 6e-04
99 percent confidence interval:
0.0003553848 0.0009440240
II - I 0.94922
III - I 0.00842
IV - I 0.00686
III - II 0.00072
IV - II 0.00060
IV - III 0.99996
Warning message:
In .local(object, ...) : p-values may be incorrect due to violation
of the subset pivotality condition
```

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