The Skillings–Mack Test (Friedman Test when There are Missing Data)

Abstract

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 x2 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.