The Skillings–Mack test (Friedman test when there are missing data)

The Skillings-Mack statistic (Skillings and Mack, 1981, Technometrics 23: 171 – 177) is a general Friedman-type statistic that can be used in almost any block design with an arbitrary missing-data structure. The missing data can be either missing by design, for example, an incomplete block design, or missing completely at random. The Skillings-Mack test is equivalent to the Friedman test when there are no missing data in a balanced complete block design, and the Skillings – Mack test is equivalent to the test suggested in Durbin (1951, British Journal of Psychology, Statistical Section 4: 85-90) for a balanced incomplete block design. The Friedman test was implemented in Stata by Goldstein (1991, Stata Technical Bulletin 3: 26-27) and further developed in Goldstein (2005, Stata Journal 5: 285). This article introduces the skilmack command, which performs the Skillings-Mack test. The skilmack command is also useful when there are many ties or equal ranks (N.B. the Friedman statistic compared with the χ2 distribution will give a conservative result), as well as for small samples; appropriate results can be obtained by simulating the distribution of the test statistic under the null hypothesis.