Tests That the Variances Are Equal Report

The Tests That the Variances Are Equal report shows the differences between group means to the grand mean and to the median, and gives a summary of testing procedures.

If the equal variances test reveals that the group variances are significantly different, use Welch’s test instead of the regular ANOVA test. The Welch statistic is based on the usual ANOVA F test. However, the means are weighted by the reciprocal of the group mean variances (Welch 1951; Brown and Forsythe 1974; Asiribo and Gurland 1990). If there are only two levels, the Welch ANOVA is equivalent to an unequal variance t-test.

Description of the Tests that the Variances Are Equal Report

Level

Lists the factor levels occurring in the data.

Count

Records the frequencies of each level.

Std Dev

Records the standard deviations of the response for each factor level. The standard deviations are equal to the means of the O’Brien variable. If a level occurs only once in the data, no standard deviation is calculated.

MeanAbsDif to Mean

Records the mean absolute difference of the response and group mean. The mean absolute differences are equal to the group means of the Levene variable.

MeanAbsDif to Median

Records the absolute difference of the response and group median. The mean absolute differences are equal to the group means of the Brown-Forsythe variable.

Test

Lists the names of the tests performed.

F Ratio

Records a calculated F statistic for each test. See Tests That the Variances Are Equal.

DFNum

Records the degrees of freedom in the numerator for each test. If a factor has k levels, the numerator has k - 1 degrees of freedom. Levels occurring only once in the data are not used in calculating test statistics for O’Brien, Brown-Forsythe, or Levene. The numerator degrees of freedom in this situation is the number of levels used in calculations minus one.

DFDen

Records the degrees of freedom used in the denominator for each test. For O’Brien, Brown-Forsythe, and Levene, a degree of freedom is subtracted for each factor level used in calculating the test statistic. If a factor has k levels, the denominator degrees of freedom is n - k.

p-Value

Probability of obtaining, by chance alone, an F-ratio value larger than the one calculated if in reality the variances are equal across all levels.

Note: A warning appears if any level of the X variable contains fewer than 5 observations. For more information about the performance of the above tests with small sample sizes, see Brown and Forsythe (1974) and Miller (1972).

Description of the Welch’s Test Report

F Ratio

Shows the F test statistic for the equal means test.

DFNum

Records the degrees of freedom in the numerator of the test. If a factor has k levels, the numerator has k - 1 degrees of freedom. Levels occurring only once in the data are not used in calculating the Welch ANOVA. The numerator degrees of freedom in this situation is the number of levels used in calculations minus one.

DFDen

Records the degrees of freedom in the denominator of the test. See Tests That the Variances Are Equal.

Prob>F

Probability of obtaining, by chance alone, an F value larger than the one calculated if in reality the means are equal across all levels. Observed significance probabilities of 0.05 or less are considered evidence of unequal means across the levels.

t Test

Shows the relationship between the F ratio and the t Test. Calculated as the square root of the F ratio. Appears only if the X factor has two levels.