Marginal Means for Screening (Plackett-Burman) Experiments

Both the Quick tab and the Means tab of the Analysis of a Screening Experiment with Two-Level Factors dialog box contain options for displaying (and plotting) the marginal means for a given factor (e.g., the means for Factor 1 by Factor 2, collapsed across Factor 3 and Factor 4). When you choose to Display the marginal means, the Marginal means spreadsheet is produced (once you have specified which means to compute in the Compute marginal means for dialog box).

Weighted and unweighted means

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). 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 computes the standard errors based on the ANOVA error term (accessed on the Model tab) 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 ANOVA error term from the ANOVA table, and hence it is dependent on the current model that is fitted 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 being fit to the data, or the current choice of error term (in the ANOVA error term group box on the Model tab).

Confidence limits for marginal means

The confidence limits for the marginal means are computed based on the respective estimates of the standard errors. The percentile used for the confidence limits (e.g., 95%, 90%, etc.) depends on the setting in the Confidence interval box on the ANOVA/Effects tab.