Hypotheses:

where ”incompatible” observed frequencies are the observed frequencies calculated when the value of the analysed feature differs across successive measurements.

Formulas:
The test statistic is defined by:

where:
\(T=\sum_{i=1}^n\sum_{j=1}^kx_{ij}\),
\(R=\sum_{i=1}^n\left(\sum_{j=1}^kx_{ij}\right)^2\),
\(C=\sum_{j=1}^k\left(\sum_{i=1}^nx_{ij}\right)^2\),
\(x_{ij}\) – the value of the \(j\)-th measurement for the \(i\)-th object (i.e. 0 or 1).

This statistic asymptotically (for large sample sizes) has the \(\chi^2\) distribution with \(df=k-1\) degrees of freedom.

The \(p\) value, determined on the basis of the test statistic, is compared with the significance level \(\alpha\): 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).

Example – simple comparisons (for the difference in proportion for one selected pair of measurements).
Hypotheses:

• \(\mathfrak{(i)}\)

  The critical difference is calculated using the formula:

  \[ CD=Z_{\frac{\alpha}{c}}\sqrt{2\frac{kT-R}{n^2k(k-1)}}, \]

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

• \(\mathfrak{(ii)}\)

  The test statistic is defined by:

  \[ Z=\frac{\sum_{j=1}^k c_jp_j}{\sqrt{2\frac{kT-R}{n^2k(k-1)}}}, \]

  where \(p_j\) – proportion of the \(j\)-th measurement \((j=1,2,...k)\). This statistic asymptotically (for large sample sizes) has the normal distribution, and the \(p\) value is adjusted for the number of possible simple comparisons \(c\).