Durbin and Skillings-Mack tests

What is the Durbin and Skillings-Mack test?

The goal of the test proposed by Durbin (1951) is to allow analyzing rigorously the results of a study carried out within the framework of a balanced incomplete block design (BIBD), using a nonparametric procedure – thus not making any assumption on the distribution of the measurements. Skillings and Mack (1981) suggested an extension of this approach for more general incomplete block designs.

The Durbin and Skillings-Mack tests are an extension of the Friedman test (1937) which can only be used in the case of complete block designs.

As for the Friedman test, the null and alternative hypotheses used in these tests are:
•    H0 : The t treatments are not different.
•    Ha : At least one of the treatments is different from another.

The Durbin and Skillings-Mack test in XLSTAT

To compute the p-values corresponding to the various statistics, XLSTAT offers two alternative methods:

• Asymptotic method: The p-value is obtained using the asymptotic approximation of the distribution of the Q and F statistics. The reliability of the approximation depends on the number of treatments and on the number of blocks.
• Monte Carlo method: The computation of the p-value is based on random resamplings. The user must set the number of resamplings. A confidence interval on the p-value is provided. The more resamplings are performed, the better the estimation of the p-value.

In order to avoid freezing Excel because of too long computations, it is possible with the two latter methods to set the maximum time that should be spent computing the p-value.

If the p-value is such that the H0 hypothesis has to be rejected, then at least one treatment is different from another. To identify which treatments are responsible for rejecting H0, a multiple comparison procedure can be used. XLSTAT allows to use for the Durbin test the procedure suggested by Conover (1999). In the case of non balanced incomplete block designs no procedure is available, but the average ranks are displayed so that treatments can be ranked.