Hypotheses: they relate to the equality of the sum of ranks for successive measurements or are simplified to medians:

where \(\theta_1,\theta_2,...\theta_k\) – medians of the analysed feature at successive measurements in the population studied.

Formulas: two test statistics are determined: the Friedman statistic

and its Iman-Davenport modification

where:
\(n\) – sample size,
\(R_{ij}\) – ranks assigned to successive measurements \((j=1,2,...k)\), separately for each object studied \((i=1,2,...n)\),
\(\displaystyle C=1-\frac{\sum(t^3-t)}{n(k^3-k)}\) – correction for ties,
\(t\) – number of cases included in a tie. The correction \(C\) is used when ties occur (otherwise \(C=1\)).

The \(T_1\) statistic asymptotically (for large sample sizes) has the \(\chi^2\) distribution with \(df=k-1\) degrees of freedom. The \(T_2\) statistic has the F Snedecor distribution with \(df_1=k-1\) and \(df_2=(n_j-1)(k-1)\) degrees of freedom.

The \(p\) value, determined on the basis of the test statistic, is compared with \(\alpha\): if \(p \le \alpha\), we reject \(\mathcal{H}_0\) in favor of \(\mathcal{H}_1\); if \(p > \alpha\), there is no reason to reject \(\mathcal{H}_0\).

POST-HOC tests

An introduction to contrasts and POST-HOC tests is provided in section Contrasts and POST-HOC tests, which concerns the one-way analysis of variance.

The Dunn test
Used for simple comparisons (the counts in each measurement are always the same). The Dunn test (Dunn, 1964[53]) is corrected for multiple testing – most often with the Bonferroni or Sidak correction (other, newer corrections are described in more detail in the Multiple comparisons section).

• \(\mathfrak{(i)}\)

  The critical difference is calculated using the formula:

  \[ CD=Z_{\alpha(corrected)}\sqrt{\frac{k(k+1)}{6n}}, \]

  where \(Z_{\alpha(corrected)}\) is the critical value (statistic) of the normal distribution for the significance level \(\alpha\) adjusted for the number of possible simple comparisons \(c\) according to the selected correction.

• \(\mathfrak{(ii)}\)

  The test statistic is defined by:

  \[ Z=\frac{\sum_{j=1}^k c_j\overline{R}_j}{\sqrt{\frac{k(k+1)}{6n}}}, \]

  where \(\overline{R}_j\) – mean rank of measurement \(j\), for \((j=1,2,...k)\). The test statistic asymptotically (for large sample sizes) has the normal distribution, and the \(p\) value is adjusted for the number of possible simple comparisons according to the selected correction.

The Conover-Inman test
The non-parametric equivalent of LSD Fisher[48], used for simple comparisons (the counts in each measurement are always the same).

• \(\mathfrak{(i)}\)

  The critical difference is calculated using the formula:

  \[ CD=\sqrt{F_{\alpha,1,df_2}}\cdot\sqrt{\frac{2\left(n_jA-\sum_{j=1}^tR_j^k\right)}{(n_j-1)(k-1)}}, \]

  where \(\displaystyle A=\sum_{i=1}^{n_j}\sum_{j=1}^kR_{ij}^2\) – sum of squared ranks, and \(F_{\alpha,1,df_2}\) is the critical value (statistic) of the F Snedecor distribution for a given significance level \(\alpha\) and for degrees of freedom 1 and \(df_2\), respectively.

• \(\mathfrak{(ii)}\)

  The test statistic is defined by:

  \[ t=\frac{\sum_{j=1}^k c_jR_j}{\sqrt{\frac{2\left(n_jA-\sum_{j=1}^tR_j^k\right)}{(n_j-1)(k-1)}}}, \]

  where \(R_j\) – sum of ranks of measurement \(j\), for \((j=1,2,...k)\). The test statistic has the \(t\)-Student distribution with \(df_2\) degrees of freedom.