Error Term for Post-hoc Tests in GLM, GRM, and ANOVA

Post-hoc comparisons tests are available on the GLM, GRM, and ANOVA More Results dialog boxes - Post-hoc tab.

Error term. Use the options in the Error term group box to select an error term (estimate of the error variance sigma) for the post-hoc comparisons for the effect specified in the Effect drop-down box on the GLM, GRM, and ANOVA More Results - Post-hoc tab. For effects including between factors only, you can choose the Between error option button or a user-defined value (select the MS, df option button); for effects including within-subject (repeated measures) factors, you can also choose the Within error option button (i.e., the error term for the respective effect involving the repeated measures factor) or an estimate based on the pooled between and within error term (select the Between; within; pooled option button). This latter estimate of sigma is computed using the suggestions by Winer, Brown, and Michels (1991, pp. 526-531), and Milliken and Johnson (1992, pp. 322-350); note that the results based on this estimate are biased, and the amount of the bias will depend (1) on the degrees of freedom for the between error (the fewer, the less bias), and (2) on the magnitude of the ratio of the between error to the within error (the closer this ratio is to 1, the smaller the bias). See Winer, Brown, and Michels (1991, pp. 526) for additional details.

Between error. Select the Between error option button to use for the estimate of the error variance the mean square error for the between group portion of the current effect; for example, if the current effect is A*B*R, where R is a within-subjects (repeated measures) factor, then the Between error option will take the mean squares error (mean squares residual) for the between-only effects (e.g., the A*B interaction) as the error term for the post-hoc comparisons. In designs and for effects involving between factors only, and without random effects in the design, this would be the appropriate error term.

Within error. Select the Within error option button to use for the estimate of the error variance the mean squares error for the current effect (which involves at least one within-subject (repeated measures) factor; hence, STATISTICA will use the error term for the current within effect). This error term would be appropriate for effects that involve within-subjects (repeated measures) factors only (and no between effects). Note that this option is only available if the current effect involves within-subjects (repeated measures) factors.

Between; within; pooled. Select the Between; within; pooled option button to use for the estimate of the error different error terms, depending on the nature of the comparisons for the respective pair of means:

  • For comparisons of means at different levels of the within (repeated measures) factors, and at the same levels of the between-group factors, Statistica uses the respective Within error;

  • For comparisons of means at different levels of the between and within (repeated measures) factors or at the same level of the within factor, Statistica uses the pooled mean squares error for the current (within) effect and the between portion of the current effect. For example, if the current effect is A*B*R, where A and B are between factors, and R is a within factor, then this option would pool for the error term (1) the mean squares residual for the between design, and (2) the mean squares error for the current effect (which involves the repeated measures factor R). Note that this procedure yields a biased estimate of sigma, and the results thus obtained should be interpreted with caution (see the first paragraph in this topic above; for details see also the discussion in Winer, Brown, and Michels, 1991, pp. 526-531; Milliken and Johnson (1992, pp. 322-350).

Note that this option is only available if the current effect involves within-subjects (repeated measures) factors and between factors.

MS, df. Select the MS, df option button to specify a user-defined estimate of the error mean squares and respective degrees of freedom. This option is particularly useful in custom designs involving random effects and other designs where the error terms are estimated from other effects in the design (e.g., cross-over designs with carry-over effects; see, for example, Milliken and Johnson, 1992).

Specifically, the standard error of the difference of the mean will be computed using the formula:

where MS is the user-defined value and ni and nj are the sample sizes for the two groups that are being compared. The user-defined degrees of freedom will be used in the subsequent test of significance.