GLM, GRM, and ANOVA Results - Means Tab

Select the Means tab of the GLM Results, GLZ Results, GRM Results, or the ANOVA Results dialog boxes to access options to display the means for any effect containing categorical predictor variables only, or for repeated measures effects. If there are no categorical effects or repeated measures effects in the model, these options are not available.

Effect

Select the desired effect in the Effect drop-down list, and then select to display or plot either the Observed, unweighted; Observed, weighted; or Least squares means. You can also display the means (unweighted, weighted, or least squares) for all categorical effects by clicking the respective All marginal tables buttons (see below).

Observed, unweighted

Click the Observed, unweighted button to produce a spreadsheet of the observed unweighted means for the selected Effect (see above). These are computed by averaging the means across the levels and combinations of levels of the factors not used in the marginal means table (or plot), and then dividing 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). 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), and the standard errors for these means are estimated from the pooled within-cell variances.

Plot

Click the Plot button to create a graph of the observed unweighted means for the selected Effect (see above). Depending upon your design, when you click this button, the Dependent Vars for the Plot dialog box will be displayed, which allows you to specify the dependent variables to use in the means plot. Next, the Specify the Arrangement of the Factors in the Plot dialog box may be displayed, which allows you to specify the arrangement of factors that STATISTICA will use in the means plot.

All marginal tables, observed unweighted. Click the All marginal tables, observed unweighted button to produce spreadsheets of the observed unweighted means for all of the categorical effects (regardless of what is selected in the Effect field).

Observed, weighted

Click the Observed, weighted button to produce a spreadsheet of the observed weighted means for the selected Effect (see above). These are computed as the standard means for the respective combinations of factor levels, directly from the data. Thus, the resulting means are weighted marginal means, since they are weighted by the number of observations in each cell of the design (in full factorial designs, you could also compute the weighted marginal means by averaging the cell means involved in each marginal mean, weighted by the respective number of observations in the respective cells). 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), and the standard errors for these means are estimated from the respective cell variances for each respective mean (i.e., the respective actual observed standard deviations in each cell).

Plot

Click the Plot button to create a graph of the observed weighted means for the selected Effect (see above). Depending upon your design, when you click this button, the Dependent Vars for the Plot dialog box will be displayed, where you can specify the dependent variables to use in the means plot. Next, the Specify the Arrangement of the Factors in the Plot dialog box may be displayed, where you can specify the arrangement of factors that STATISTICA will use in the means plot.

All marginal tables, observed weighted. Click the All marginal tables, observed weighted button to produce spreadsheets of the observed weighted means for all of the categorical effects (regardless of what is selected in the Effect field).

Least squares means

Click the Least squares means button to produce a spreadsheet of the least squares means for the selected Effect. Least squares means are the expected population marginal means, given the current model. Thus, these are usually the means of interest when interpreting significant effects from the ANOVA or MANOVA table. Note that for full factorial designs without missing cells, the Least squares means are identical to the Observed, unweighted means (see above). Least squares means are also sometimes called predicted means, because they are the predicted values when all factors in the model are either held at their means or the factor levels for the respective means. Note that if there are continuous predictors (covariates) in the model, the least squares means are computed from the values for those predictors as set in the Covariate values group box (see below). For details concerning the computation of least squares means refer to Milliken and Johnson (1992), Searle, Speed, and Milliken (1980), or Searle (1987). Note that when you are in the GLZ module, STATISTICA does not compute the least squares means, rather the equivalent expected values for the respective non-linear (generalized linear) model, i.e., the predicted means are computed.

Plot

Click the Plot button to create a graph of the least squares means for the selected Effect. Depending upon your design, when you click this button, the Dependent Vars for the Plot dialog box will be displayed, where you can specify the dependent variables to use in the means plot. Next, the Specify the Arrangement of the Factors in the Plot dialog box may be displayed, where you can specify the arrangement of factors that STATISTICA will use in the means plot.

All marginal tables, least squares means. Click the All marginal tables, least squares means button to produce spreadsheets of the least squares means for all of the categorical effects (regardless of what is selected in the Effect field).

Covariate values

The options in the Covariate values group box determine at what values the continuous predictor variables (covariates) will be set for the computation of least squares means. By default, the values for any continuous predictors (covariates) in the model will be held at their respective overall Covariate means. You can also specify User-defined values for the covariates; after selecting this option button, click the Define button to display the Values for Covariates dialog box and specify the values. Finally, you can set the values for the continuous predictor variables so as to compute the Adjusted means, these are the predicted values (means) after "adjusting" for the variation of the means of the continuous predictor variables over the cells in the current Effect (see above). Adjusted means are widely discussed in the traditional analysis of covariance (ANCOVA) literature; see, for example, Finn (1974), Pedhazur (1973), or Winer, Brown, and Michels, K. M. (1991). The Adjusted means option button is only available in full factorial designs. Note that the Covariate values group box will not be available when you are using the ANOVA module.

Show std errs

Select the Show std errs check box to display standard errors and confidence limits for the means in the spreadsheet or plot of means (see the above buttons). The plot of means will show the confidence limits as error bars around the respective means. The actual confidence limits are based on the current setting in the Confidence limits field available on the GLM Results - Quick tab.

Note: standard errors for unweighted marginal means. The standard errors for the observed unweighted means are computed 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: standard errors for weighted marginal means. 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.

Show means +/- std errs

Select this check box to show in the tables and plots of means the plus or minus standard error range around each mean. These will only be shown if the Show std errs check box is also selected. By default, when the Show means +/- std errs check box is cleared, the (95%) confidence intervals will be computed instead (or any other confidence interval, consistent with the specification in the Confidence limits field of the Quick tab).

See also GLM - Index.