Computes simultaneous confidence intervals on all pairwise differences of means.

Routine SCIPM computes simultaneous confidence intervals on all k* = k(k ‑ 1)/2 pairwise comparisons of k means μ1, μ2, …, μk in the one-way analysis of variance model. Any of several methods can be chosen. A good review of these methods is given by Stoline (1981). Also the methods are discussed in many elementary statistics texts, e.g., Kirk (1982, pages 114‑127).

Let s2 (input in S2) be the estimated variance of a single observation. Let ν be the degrees of freedom (input in DFS2) associated with s2: Let α =1 ‑ CONPER/100.0. The methods are summarized as follows:

Tukey method: The Tukey method gives the narrowest simultaneous confidence intervals for all pairwise differences of means μi ‑ μj in balanced (n1 = n2 = … = nk = n) one-way designs. The method is exact and uses the Studentized range distribution. The formula for the difference μi ‑ μj is given by

where q1-α;k,ν is the (1 ‑ α)100 percentage point of the Studentized range distribution with parameters k and ν.

Tukey-Kramer method: The Tukey-Kramer method is an approximate extension of the Tukey method for the unbalanced case. (The method simplifies to the Tukey method for the balanced case.) The method always produces confidence intervals narrower than the Dunn-Sidak and Bonferroni methods. Hayter (1984) proved that the method is conservative, i.e., the method guarantees a confidence coverage of at least (1 ‑ α)100%. Hayter’s proof gave further support to earlier recommendations for its use (Stoline 1981). (Methods that are currently better are restricted to special cases and only offer improvement in severely unbalanced cases, see, e.g., Spurrier and Isham 1985). The formula for the difference μi ‑ μj is given by

Dunn-Šidák method: The Dunn-Šidák method is a conservative method. The method gives wider intervals than the Tukey-Kramer method. (For large ν and small α and k, the difference is only slight.) The method is slightly better than the Bonferroni method and is based on an improved Bonferroni (multiplicative) inequality (Miller, pages 101, 254‑255). The method uses the t distribution (see IMSL routine TIN, in Chapter 17, “Probability Distribution Function and Inverses”). The formula for the difference μi ‑ μj is given by

where tf;ν is the 100f percentage point of the t distribution with ν degrees of freedom.

Bonferroni method: The Bonferroni method is a conservative method based on the Bonferroni (additive) inequality (Miller, page 8). The method uses the t distribution. The formula for the difference μi ‑ μj is given by

Scheffé method: The Scheffé method is an overly conservative method for simultaneous confidence intervals on pairwise difference of means. The method is applicable for simultaneous confidence intervals on all contrasts, i.e., all linear combinations

The method can be recommended here only if a large number of confidence intervals on contrasts in addition to the pairwise differences of means are to be constructed. The method uses the F distribution (see IMSL routine FIN, in Chapter 17, “Probability Distribution Function and Inverses”. The formula for the difference μi ‑ μj is given by

where F1-α;k-1,ν is the (1 ‑ α)100 percentage point of the F distribution with k ‑1 and ν degrees of freedom.

One-at-a-time t method (Fisher’s LSD): The one-at-a-time t method is the method appropriate for constructing a single confidence interval. The confidence percentage input is appropriate for one interval at a time. The method has been used widely in conjunction with the overall test of the null hypothesis μ1 = μ2 = … = μk by the use of the F statistic. Fisher’s LSD (least significant difference) test is a two-stage test that proceeds to make pairwise comparisons of means only if the overall F test is significant.

Milliken and Johnson (1984, page 31) recommend LSD comparisons after a significant F only if the number of comparisons is small and the comparisons were planned prior to the analysis. If many unplanned comparisons are made, they recommend Scheffe’s method. If the F test is insignificant, a few planned comparisons for differences in means can still be performed by using either Tukey, Tukey-Kramer, Dunn-Šidák or Bonferroni methods. Because the F test is insignificant, Scheffe’s method will not yield any significant differences. The formula for the difference μi ‑ μj is given by

Simultaneous 99% confidence intervals are computed for all pairwise comparisons of 5 means from a one-way analysis of variance design. In order to compare the results of each method, all the options for IMETH are used for input. The data are given by Kirk (1982, Table 3.5-1, page 117). In the output, pairs of means declared not equal are indicated by the letter N. The other pairs of means (for which there is insufficient evidence from the data to declare the means are unequal) are indicated by an equal sign (=).