Durbin test
In the analysis of designed experiments, the Friedman test is the most common non-parametric test for complete block designs. The Durbin test is a nonparametric test for balanced incomplete designs that reduces to the Friedman test in the case of a complete block design.
In a randomized block design, k treatments are applied to b blocks. In a complete block design, every treatment is run for every block and the data are arranged as follows:
For some experiments, it may not be realistic to run all treatments in all blocks, so one may need to run an incomplete block design. In this case, it is strongly recommended to run a balanced incomplete design. A balanced incomplete block design has the following properties:
The Durbin test is based on the following assumptions:
Let R(Xij) be the rank assigned to Xij within block i (i.e., ranks within a given row). Average ranks are used in the case of ties. The ranks are summed to obtain
The Durbin test is then
The test statistic is
where
where t is the number of treatments, k is the number of treatments per block, b is the number of blocks, and r is the number of times each treatment appears.
For significance level α, the critical region is given by
where Fα, k − 1, bk − b − t + 1 denotes the α-quantile of the F distribution with k − 1 numerator degrees of freedom and bk − b − t + 1 denominator degrees of freedom. The null hypothesis is rejected if the test statistic is in the critical region. If the hypothesis of identical treatment effects is rejected, it is often desirable to determine which treatments are different (i.e., multiple comparisons). Treatments i and j are considered different if
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