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"),
...
)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 |
dist |
the distribution for determining the p-value.
Defaults to |
formula |
a formula of the form |
data |
an optional matrix or data frame (or similar: see
|
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 |
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}: θ_i(x) = θ_j(x) is tested in the two-tailed test against the alternative A_{ij}: θ_i(x) \ne θ_j(x), ~~ i \ne j.
Let R_{ij} be the rank of X_{ij}, where X_{ij} is jointly ranked from ≤ft\{1, 2, …, N \right\}, ~~ N = ∑_{i=1}^k n_i, then the test statistic under the absence of ties is calculated as
SEE PDF
with \bar{R}_j, \bar{R}_i the mean rank of the i-th and j-th group and the expected variance as
SEE PDF
A pairwise difference is significant, if |t_{ij}|/√{2} > q_{kv}, with k the number of groups and v = ∞ 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
SEE PDF
The function provides two different dist
for p-value estimation:
A list with class "PMCMR" containing the following components:
a character string indicating what type of test was performed.
a character string giving the name(s) of the data.
lower-triangle matrix of the estimated quantiles of the pairwise test statistics.
lower-triangle matrix of the p-values for the pairwise tests.
a character string describing the alternative hypothesis.
a character string describing the method for p-value adjustment.
a data frame of the input data.
a string that denotes the test distribution.
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.
## 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)Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.