Newman–Keuls method

The Newman–Keuls method was introduced by Newman in 1939 and developed further by Keuls in 1952. This was before Tukey presented various definitions of error rates (1952a, 1952b, 1953). The Newman–Keuls method controls the Family-Wise Error Rate (FWER) in the weak sense but not the strong sense: the Newman–Keuls procedure controls the risk of rejecting the null hypothesis if all means are equal (global null hypothesis) but does not control the risk of rejecting partial null hypotheses. For instance, when four means are compared, under the partial null hypothesis that μ1=μ2 and μ3=μ4=μ+delta with a non-zero delta, the Newman–Keuls procedure has a probability greater than alpha of rejecting μ1=μ2 or μ3=μ4 or both. In that example, if delta is very large, the Newman–Keuls procedure is almost equivalent to two Student t tests testing μ1=μ2 and μ3=μ4 at nominal type I error rate alpha, without multiple testing procedure; therefore the FWER is almost doubled. In 2006, Shaffer showed (by extensive simulation) that the Newman–Keuls method controls the FDR with some constraints. ==Required assumptions==

The assumptions of the Newman–Keuls test are essentially the same as for an independent groups t-test: normality, homogeneity of variance, and independent observations. The test is quite robust to violations of normality. Violating homogeneity of variance can be more problematic than in the two-sample case since the MSE is based on data from all groups. The assumption of independence of observations is important and should not be violated. ==Procedures==

The Newman–Keuls method employs a stepwise approach when comparing sample means. Prior to any mean comparison, all sample means are rank-ordered in ascending or descending order, thereby producing an ordered range (p) of sample means. Because the number of means within a range changes with each successive pairwise comparison, the critical value of the q statistic also changes with each comparison, which makes the Neuman-Keuls method more lenient and hence more powerful than Tukey's range test. Thus, if a pairwise comparison was found to be significantly different using the Newman–Keuls method, it may not necessarily be significantly different when analyzed with Tukey's range test. Conversely, if the pairwise comparison was found not to be significantly different using the Newman–Keuls method, it cannot be significantly different with Tukey's range test either. ==Limitations==

The Newman–Keuls procedure cannot produce a confidence interval for each mean difference, or for multiplicity adjusted exact p-values due to its sequential nature. Results are somewhat difficult to interpret since it is difficult to articulate what are the null hypotheses that were tested. ==See also==