GLM, GRM, and ANOVA More Results - Assumptions Tab

Select the Assumptions tab of the GLM More Results or the ANOVA More Results dialog box to access options to test various assumptions for the validity of ANOVA/MANOVA methods, either by performing explicit statistical tests of the assumptions (e.g., Levene's test), or by reviewing graphs that summarize certain aspects of the data (e.g., Plot of means vs. std. deviations). Some of the options will only be available in multivariate designs (with more than a single dependent variable).

Variables

Click the Variables button to display a standard variable selection dialog box. Select the dependent variables and/or continuous predictor variables that you want to include in the spreadsheets and graphs produced by the options in this dialog box.

Effect

Select the between effect for subsequent tests or graphs chosen from this dialog box in the Effect drop-down box. Specifically, subsequent spreadsheets and graphs are computed for the observations in the respective marginal cells for the effect, ignoring all higher order effects. For example, if in a three-way full factorial design for factors A, B, and C, you select the A*B effect, and then compute the Levene's test (in the Homogeneity of variances/covariances group box), an analysis of variance is performed on the absolute deviations of the observations from the within-cell weighted marginal means (see also the Means tab) for the A*B between effect; if you select the Histogram options (in the Distribution ... group boxes) you could review the distribution of the continuous variable values within the combinations of levels of the A*B interaction, and so on.

Homogeneity of variances/covariances

The options in the Homogeneity of variances/covariances group box allow you to test the homogeneity of variances/covariances assumption. One of the assumptions of univariate ANOVA is that the variances are equal (homogeneous) across the cells of the between-groups design. In the multivariate case (MANOVA), this assumption applies to the variance/covariance matrix of dependent variables (and covariates). Those assumptions, and the effect of their violation, are also discussed in the Overview for the ANOVA/MANOVA module.

Cochran C, Hartley, Bartlett

Click the Cochran C, Hartley, Bartlett button to produce a spreadsheet with the Hartley F-max statistic, Cochran C statistic, and the Bartlett Chi-square test (with appropriate degrees of freedom and p-values). All of these statistics test the homogeneity of variances assumption in the univariate case. If there are multiple dependent variables or covariates, the tests are displayed for each dependent variable and for each covariate. All of these tests are described in most standard ANOVA textbooks (e.g., see Winer, 1962, p. 94, Winer, Brown, & Michels, 1991). As described in the Overview for the ANOVA/MANOVA module, the consequences of even quite major violations of the homogeneity of variances assumption are not that critical. Lindeman (1974, p. 33) summarizes the results of various studies of this issue, and shows that only under the most severe violations do we need to be concerned about the validity of the F statistic. It is, however, very important to examine any correlations between the means and the variances [or standard deviations; use option Plot means vs. std. deviations (Variances) (see below) to produce those correlations].

Box M test (cov. matrix).

Click the Box M test button to produce the results of the Box M test. This is a multivariate test of the homogeneity of variances and covariances for multiple dependent variables or covariates. The Box M test is very sensitive to deviations from the normal distribution and its results should be viewed with some skepticism. If this test is significant, it means that the variance/covariance matrices in the different between-group cells in the current Effect (see above) are significantly different from each other (see Anderson, 1958). In that case, you should probably examine the within-group variance/covariance matrices for any major heterogeneity problems; however, violations of this homogeneity of variances/covariances assumption usually do not seriously threaten the validity of the multivariate results.

Levene's test (ANOVA)

Click the Levene's test button to produce the results of the Levene test. The Levene test (as well as the Brown-Forsythe modification of this test, available via the Statistics by Groups Results - ANOVA & tests tab in Basic Statistics) for homogeneity of variances amounts to performing a one-way ANOVA on the absolute deviation scores (from the respective cell means). The logic of this test is that the greater the variance in a cell, the larger are the absolute deviations from the respective cell mean. This test is discussed in Milliken and Johnson (1984). However, note that the Levene test is not necessarily very robust itself against violations of the homogeneity of variances assumption (e.g., Glass and Hopkins, 1996, p. 436, call this test "fatally flawed").

Distribution of vars within groups

With the plots available in the Distribution of variables within groups group box, you can review the distribution (univariate and bivariate) of the continuous variables (dependent and predictors) in the current analysis. After choosing an option, the Select Groups dialog box will be displayed where you can select to review the respective graphs either within one or more of the combinations of levels for the current effect, or for all groups. If you select All groups in the Select Groups dialog box, the raw data are plotted; to review the within-cell residuals (deviations from the respective means) for all groups, use the Plots of within-cell residuals options (see Distribution of within-cell residuals below). Depending on which button you click, you can create Histograms, Normal p-plots (probability plots), or Detrended normal probability plots of the selected continuous Variables (see above). If at least two variables are selected, you can produce Scatterplots or a scatterplot Matrix for pairs of the selected continuous Variables. As described above, these plots can be produced for all groups combined (effectively ignoring the between design), or within one or more of the combinations of levels for the between factors in the current effect.

Distribution of within-cell residuals

The plots available in the Distribution of within-cell residuals group box allow you to review the distribution (univariate and bivariate) of the within-cell deviations from the respective weighted means for the continuous variables (dependent and predictors) in the current analysis. Specifically, STATISTICA will compute the deviations of the observed values from the respective weighted marginal means (see the Means tab) for the current effect, and use those within-cell residuals for the plots. To review the raw data values, use the Distribution of variables within groups options (see above). Depending on which button you click, you can create Histograms, Normal p-plots (probability plots), or Detrended normal probability plots of the within-cell residuals for the selected continuous Variables (see above). If at least two variables are selected, you can produce a Scatterplot of the within-cell residuals for pairs of the selected continuous Variables.

Half-normal plot of z-transf. within corr's.

Click this button to display a half-normal plot of the z-transformed within-cell correlations. This plot is useful for exploring the distribution of the within-cell (group) correlations between variables. If the correlations are homogeneous across groups in the population (a MANOVA assumption), then we would expect the correlations to be normally distributed across the samples in the study. Because correlation coefficients generally do not follow the normal distribution, the z-transformed correlations are used for the plot instead. The half-normal probability plot is constructed from those z-transformed within-cell correlations. This option is only available if multiple Variables (see above) were selected.

Plot means vs. std. deviations.

Click the Plot means vs. std. deviations button to plot the (unweighted) marginal means (see also the Means tab) for the selected Variables against the standard deviations, across the marginal cells for the current between-groups Effect (see above). The Overview section for the ANOVA/MANOVA module discusses how the overall F tests in the ANOVA table can be very misleading if the means are correlated with the variability. Briefly, if extreme (large or small) means occur in cells with larger variances, then there are usually outliers present in those cells. As a result, the confidence limits of the respective means would be much larger if estimated from those cells alone, while the overall ANOVA result makes those means appear more reliable, leading to statistically significant results. This happens quite often in actual research, and you are well-advised to produce this plot before accepting critical results. If the means are correlated with the variability across the cells of the design, one can (1) try to identify outliers and exclude them from the analysis, (2) use nonparametric tests, or (3) use transformations (such as a log transformation) on the data to "pull in the long tails" of the distributions with the larger variances.

Variances

Click the Variances button to plot the (unweighted) marginal means (see also the Means tab) for the selected Variables against the variances, across the marginal cells for the current between-groups Effect (see above). For further details on this plot, please read the Plot means vs. std. deviations section above.

See also GLM - Index.