Marginal Means in Experiments with Three-Level Factors

Both the Quick tab and the Means tab of the Analysis of an Experiment with Three-Level Factors dialog box contain options to compute the marginal means for the design (e.g., the means for Factor 1 by Factor 2, collapsed across Factor 3 and Factor 4). When you click the Display button, the Marginal means spreadsheet will be produced (once you have specified which means to compute in the Compute marginal means for dialog box).

By default, Statistica computes unweighted marginal means, that is, it averages the means across the settings and combinations of settings of the factors not used in the marginal means table (or plot), and then divides by the number of means in the average. Thus, each mean that is averaged to compute a marginal mean is implicitly assigned the same weight, regardless of the number of observations on which the respective mean is based. The resulting estimate is an unbiased estimate of m -bar (mu-bar), the population marginal mean. If the design is not balanced, and some means are based on different numbers of observations, then you can also compute the weighted marginal means (weighted by the respective cell n's; select the Display/plot weighted means check box on the Quick tab or the Means tab) by selecting this option. Note that the weighted mean is an unbiased estimate of the weighted population marginal mean (for details, see, for example, Milliken and Johnson, 1984, page 132).

Standard errors for unweighted marginal means

For the unweighted marginal means, Statistica will compute the standard errors based on the current error term from the ANOVA table:

Std.Err.(m -bar) = sest /t * sqrt[S(1/ ni)]

In this formula, sest is the estimated sigma (computed as the square root of the estimated error variance from the current ANOVA table), t is the number of means that is averaged to compute the respective marginal mean, and ni refers to the number of observations in the t experimental conditions from which the respective unweighted marginal mean is computed. Note that this estimate of the standard errors of the marginal means is dependent on the current error term from the ANOVA table, and hence it is dependent on the current model that is fit to the data and the choice of error term (selected in the ANOVA error term group box on the Model tab).

Standard errors for weighted marginal means

If the Display/plot weighted means check box on the Quick tab or the Means tab is selected, the standard errors for the marginal means are computed as if you had ignored the other factors, (those not in the marginal means table). Thus, for weighted marginal means the standard error is not dependent on the estimate of the error variance from the current ANOVA table, and hence, it is not dependent on the current model that is fit to the data, or the current choice of error term (in the ANOVA error term group box on the Model tab).