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
$$
t_{ij} = \frac{\bar{R}_j - \bar{R}_i}
{\sigma_R \left(1/n_i + 1/n_j\right)^{1/2}} \qquad \left(i \ne j\right),
$$
with \(\bar{R}_j, \bar{R}_i\) the mean rank of the
\(i\)-th and \(j\)-th group and the expected variance as
$$
\sigma_R^2 = N \left(N + 1\right) / 12.
$$
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
$$
C = 1 - \frac{\sum_{i=1}^r t_i^3 - t_i}{N^3 - N}.
$$
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.