kwAllPairsNemenyiTest function

Nemenyi's All-Pairs Rank Comparison Test

Performs Nemenyi's non-parametric all-pairs comparison test for Kruskal-type ranked data.

kwAllPairsNemenyiTest(x, ...)

## Default S3 method:
kwAllPairsNemenyiTest(x, g, dist = c("Tukey", "Chisquare"), ...)

## S3 method for class 'formula'
kwAllPairsNemenyiTest(
  formula,
  data,
  subset,
  na.action,
  dist = c("Tukey", "Chisquare"),
  ...
)

Arguments

• x: a numeric vector of data values, or a list of numeric data vectors.
• ...: further arguments to be passed to or from methods.
• g: a vector or factor object giving the group for the corresponding elements of "x". Ignored with a warning if "x" is a list.
• dist: the distribution for determining the p-value. Defaults to "Tukey".
• formula: a formula of the form response ~ group where response gives the data values and group a vector or factor of the corresponding groups.
• data: an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
• subset: an optional vector specifying a subset of observations to be used.
• na.action: a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Returns

A list with class "PMCMR" containing the following components:

• method: a character string indicating what type of test was performed.
• data.name: a character string giving the name(s) of the data.
• statistic: lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
• p.value: lower-triangle matrix of the p-values for the pairwise tests.
• alternative: a character string describing the alternative hypothesis.
• p.adjust.method: a character string describing the method for p-value adjustment.
• model: a data frame of the input data.
• dist: a string that denotes the test distribution.

Details

For all-pairs comparisons in an one-factorial layout with non-normally distributed residuals Nemenyi's non-parametric test can be performed. A total of $m = k(k-1)/2$

hypotheses can be tested. The null hypothesis H$_{ij}: \theta_i(x) = \theta_j(x)$ is tested in the two-tailed test against the alternative A$_{ij}: \theta_i(x) \ne \theta_j(x), ~~ i \ne j$.

Let $R_{ij}$ be the rank of $X_{ij}$, where $X_{ij}$ is jointly ranked from $\left\{1, 2, \ldots, N \right\}, ~~ N = \sum_{i=1}^k n_i$, then the test statistic under the absence of ties is calculated as

with $\bar{R}_j, \bar{R}_i$ the mean rank of the $i$-th and $j$-th group and the expected variance as

A pairwise difference is significant, if $|t_{ij}|/\sqrt{2} > q_{kv}$, with $k$ the number of groups and $v = \infty$

the degree of freedom.

Sachs(1997) has given a modified approach for Nemenyi's test in the presence of ties for $N > 6, k > 4$

provided that the kruskalTest indicates significance: In the presence of ties, the test statistic is corrected according to $\hat{t}_{ij} = t_{ij} / C$, with

The function provides two different dist

for $p$-value estimation:

• Tukey: The $p$-values are computed from the studentized range distribution (alias Tukey), $\mathrm{Pr} \left\{ t_{ij} \sqrt{2} \ge q_{k\infty\alpha} | mathrm{H} \right\} = \alpha$.
• Chisquare: The $p$-values are computed from the Chisquare distribution with $v = k - 1$ degree of freedom.

Examples

## Data set InsectSprays
## Global test
kruskalTest(count ~ spray, data = InsectSprays)

## Conover's all-pairs comparison test
## single-step means Tukey's p-adjustment
ans <- kwAllPairsConoverTest(count ~ spray, data = InsectSprays,
                             p.adjust.method = "single-step")
summary(ans)

## Dunn's all-pairs comparison test
ans <- kwAllPairsDunnTest(count ~ spray, data = InsectSprays,
                             p.adjust.method = "bonferroni")
summary(ans)

## Nemenyi's all-pairs comparison test
ans <- kwAllPairsNemenyiTest(count ~ spray, data = InsectSprays)
summary(ans)

## Brown-Mood all-pairs median test
ans <- medianAllPairsTest(count ~ spray, data = InsectSprays)
summary(ans)

References

Nemenyi, P. (1963) Distribution-free Multiple Comparisons. Ph.D. thesis, Princeton University.

Sachs, L. (1997) Angewandte Statistik. Berlin: Springer.

Wilcoxon, F., Wilcox, R. A. (1964) Some rapid approximate statistical procedures. Pearl River: Lederle Laboratories.

See Also

Tukey, Chisquare, p.adjust, kruskalTest, kwAllPairsDunnTest, kwAllPairsConoverTest