# Statistics by Groups Results - ANOVA & Tests tab

Breakdown: Descriptive Statistics by Groups

Select the ANOVA & tests tab in the Statistics by Groups Results dialog box to access options to perform ANOVAs and test ANOVA assumptions.

Analysis of Variance. Click this button to produce a spreadsheet with the results of univariate analyses of variance for each dependent variable. If statistically significant, it can be concluded that the means across the groups are different in magnitude; use the options on the Post-hoc tab to identify significant differences between individual groups (means). Note that if multiple categorical factors are specified, Statistica computes  the one-way analysis of variance treating the interaction of the multiple factors as a single group. This is equivalent to investigating the interaction among multiple factors. Use the General Linear Model (GLM) or ANOVA/MANOVA module to compute complete univariate and multivariate analysis of variance tables (see also Methods for analysis of variance). However, note that the one-way ANOVA available from this dialog box is particularly suited for quickly analyzing one-way univariate designs with very many groups.

Perform Welch's F-Test. Select this check box to include the computation of Welch's F statistic to test for equality of means when the variances are unequal.

Tests of homog. of variances. Two tests for the homogeneity of variance assumption are available in this group box. For more information on the importance of the homogeneity of variance assumption, see Homogeneity of Variances in the ANOVA/MANOVA module.

Levene test. Click the Levene test button to produce a spreadsheet containing the Levene test for each selected dependent variable. The significance tests reported by the Analysis of variance option (see above) are based on the assumption that the variances in the different groups are the same (homogeneous). A powerful statistical test of this assumption is Levene's test (however, see also the description of the Brown-Forsythe modification of this test below). For each dependent variable, an analysis of variance is performed on the absolute deviations of values from the respective group means. If the Levene test is statistically significant, the hypothesis of homogeneous variances should be rejected. Note that the F statistic (in ANOVA) provides a robust test for mean differences as long as 1) the N per group is greater than 10 (and, in particular, in the case of equal N), and 2) the means across groups are not correlated with the standard deviations across groups. (The assumption of uncorrelated means and standard deviations can easily be checked by producing the Plot of means vs. standard deviations from this tab.) Thus, a significant Levene test does not necessarily call into question the validity of the ANOVA results. Also, in the case of unbalanced designs (i.e., unequal N per group), the Levene test is itself not very robust, as has recently been pointed out in, for example, Glass and Hopkins (1996; see also the description of the Brown-Forsythe tests option).

Brown-Forsythe tests. Click the Brown & Forsythe tests button to produce a spreadsheet containing the Brown & Forsythe test for each selected dependent variable. The significance tests reported by the Analysis of Variance option are based on the assumption that the variances in the different groups are the same (homogeneous). A powerful statistical test of this assumption is provided via the Levene's test (homogeneity of variances) option. Recently, some authors (e.g., Glass and Hopkins, 1996) have called into question the power of the Levene test for unequal variances. Specifically, the absolute deviation (from the group means) scores can be expected to be highly skewed; thus, the normality assumption for the ANOVA of those absolute deviation scores is usually violated. This poses a particular problem when there is unequal N in the two (or more) groups that are to be compared. A more robust test that is very similar to the Levene test has been proposed by Brown and Forsythe (1974). Instead of performing the ANOVA on the deviations from the mean, you can perform the analysis on the deviations from the group medians. Olejnik and Algina (1987) have shown that this test gives quite accurate error rates even when the underlying distributions for the raw scores deviate significantly from the normal distribution. However, recently, Glass and Hopkins (1996, p. 436) have pointed out that both the Levene test and the Brown-Forsythe modification suffer from what those authors call a "fatal flaw," namely, that both tests rely on the homogeneity of variances assumption (of the absolute deviations from the means or medians), and hence, it is not clear how robust these tests are in the presence of significant variance heterogeneity and unequal N.

p-value for highlighting. The default p-value for highlighting is .05. You can adjust this p-value by entering a new value in the edit box or using the microscrolls. For more details on p-value, see Elementary Concepts.

Categorized normal prob. plots. Click this button to produce a cascade of normal probability plots for the selected dependent variables, categorized by the grouping variables. The standard variable selection dialog box is first displayed if more than one dependent variable has been selected via the Variables option. See also the ANOVA overview for an explanation of the normality assumption.

Categorized half-normal p-plots. Click this button to produce a cascade of half-normal probability plots for the selected dependent variables, categorized by the grouping variables. The standard variable selection dialog box is first displayed if more than one dependent variable has been selected via the Variables option.

Categorized detrended p-plots. Click this button to produce a cascade of detrended normal probability plots for the selected dependent variables, categorized by the grouping variables. The standard variable selection dialog box is first displayed if more than one dependent variable has been selected via the Variables option.

Plot of means vs. std. devs. Click this button to plot the means for selected variables across groups against the respective standard deviations. The standard variable selection dialog box is first displayed if more than one dependent variable has been selected via the Variables option. This plot is useful in order to spot potential outliers among the means that may contribute to an erroneous conclusion of statistically significant differences between means. One of the most common and most serious violations of assumptions for ANOVA is when the means are correlated with the standard deviations across groups. For example, suppose there are 5 groups of 10 observations each. If in one group, two observations are extreme outliers, the variance in that group will be much larger and the mean will be very different (larger or smaller) from the grand mean. However, for the overall F test (ANOVA) the pooled (averaged) within-group variance is taken as an estimate of the error variance. Thus, the reliability of the outlier mean will be overestimated, and the ANOVA may erroneously yield a significant F statistic.

Interaction plots. Click this button to produce an interaction plot of means by groups. Interaction plots are produced according to the following specifications, within the levels of any additional grouping factors.

More than one dependent variable was selected for the current analysis. In this case, when you click on the Interaction plots button, the resulting Select the Variables for Interaction Plot dialog box displays each of the previously selected dependent variables. You can select only those variables that you want plotted, or click OK to accept the default selection (all variables in the list); each variable in the resulting interaction plots is represented by a different line color or pattern. Clicking OK in this dialog box displays the Arrangement of Factors dialog box (see below).

One dependent variable was selected for the current analysis. Since you have already selected the dependent variable to plot, when you click the Interaction plots button, the Arrangement of Factors dialog box is displayed (see below). In this interaction plot, the grouping variables are represented by different line colors and patterns.

Arrangement of factors dialog box. Once you select the variables (see above) to be plotted, the Arrangement of Factors dialog box is displayed, where you can assign two (if two or more dependent variables are selected) or three (if one dependent variable is selected) grouping variables to different aspects of the interaction plot (line pattern, lower x-axis, upper x-axis).

Plot confidence intervals for the mean. Select this check box to display the confidence interval for the mean (error bars) on the Interactions plots. Use the corresponding field to specify the exact confidence interval for the mean to be used on the plot. The default value is 95%.