ANOVA

This dialog calculates one-way Analysis of Variance (ANOVA) and related statistics for a single variable that is measured in two or more groups.

Both parametric and non-parametric test statistics are reported. Supplementary statistics provide information on whether the subgroups meet the distributional requirements for a one-way ANOVA.

The ANOVA dialog prompts for:

• The numeric variable for which to carry out the ANOVA.

• Whether to log₁₀ transform the data.

• The categorical variable that distinguishes the different groups to be compared.

• Whether to use all data or only the subset that has been selected in the data table.

The results are shown in two tables below the data selection options. The two tables are on separate tabs. The first table contains ANOVA statistics and the second table contains distribution statistics for the numeric variable in each subgroup.

At the bottom of the dialog, the “Source Data” button shows all of the selected values for the numeric and categorical variables. The “Group Data” button shows the same data, but with the numeric values for each value of the categorical variable in a separate column. When these tables are displayed, the keystroke Ctrl-S can be used to save these tables to a CSV file or spreadsheet.

The output from this analysis tool is strictly tabular, but graphical contrasts between the subgroups can be viewed using boxplots, stripcharts, and other types of plots that can be produced from the Plot/General menu.

ANOVA Statistics

Test statistics and p values are tabulated for all of the tests listed in the following subsections.

One-way ANOVA

The one-way ANOVA evaluates whether the means of all of the subgroups are equivalent. The test results are summarized as the calculated F statistic and its corresponding p value.

The ANOVA analysis requires that the data in every subgroup be normally distributed and have equivalent variances. An F statistic and p value will be calculated even if these requirements are not met. The results of Levene’s test indicate whether the subgroups have equivalent variances, and the results of normality tests are reported for all subgroups with 20 or more samples.

Kruskal-Wallis test

The Kruskal-Wallis test evaluates whether the medians of all of the subgroups are equivalent. The Kruskal-Wallis test is a non-parametric alternative to the one-way ANOVA, and does not require that the subgroups have normal distributions and equivalent variances. The test results are summarized as the calculated H statistic and its corresponding p value.

Alexander-Govern test

This test (Alexander and Govern 1994) evaluates whether the means of all the subgroups are equivalent. Unlike the one-way ANOVA, this test does not require that all subgroups have equivalent variance. All subgroups must be normally distributed, however. The results of this test are summarized as the test statistic A and its corresponding p value.

This test is not performed if any of the subgroups have only a single observation.

Levene’s test

This test evaluates whether all the subgroups have equivalent variances about the mean. The results of this tests are summarized as the test statistic W and its corresponding p value. Levene’s test is most applicable to distributions that are symmetric about the mean.

If the null hypothesis of equal variances is rejected (e.g., p < 0.05), then the one-way ANOVA result are not relevant and the Kruskal-Wallis or Alexander-Govern test results should be relied on instead.

Brown-Forsythe test

This test evaluates whether all the subgroups have equivalent variances about the median. It is equivalent to Levene’s test but may be more robust when the data distribution is not symmetric about the mean. The results of this test are summarized as the test statistic F and its corresponding p value.

Bartlett’s test

This test evaluates whether all the subgroups have equivalent variances about the mean. The results of this tests are summarized as the test statistic X² and its corresponding p value. Bartlett’s test is most applicable when the individual subgroups do not depart greatly from normality.

This test is not performed if any of the subgroups have only a single observation.

Distribution Statistics

For every subgroup, the second output table displays the following distribution statistics:

• Number of observations

• Mean

• Median

• Variance

• Skewness

• Kurtosis

• The p value for the Anderson-Darling test of normality

• The p value for the Lilliefors test of normality

• The p value for the omnibus test of normality (D’Agostino 1971).

Other distribution statistics can be seen in the display produced by the Statistics/Univariate statistics menu option. Distributions can be visualized using several different types of plots that can be produced from the Plot/General menu option.