When the variances across groups are not equal, the usual analysis of variance assumptions are not satisfied and the anova F test is not valid. JMP provides four tests for equality of group variances and an anova that is valid when the group sample variances are unequal. The concept behind the first three tests of equal variances is to perform an analysis of variance on a new response variable constructed to measure the spread in each group. The fourth test is Bartlett’s test, which is similar to the likelihood-ratio test under normal distributions.
Shows the F test from an anova where the response is the absolute value of the difference of each observation and the group median (Brown and Forsythe 1974).
Shows the F test from an anova where the response is the absolute value of the difference of each observation and the group mean (Levene 1960). The spread is measured as (as opposed to the SAS default ).
Compares the weighted arithmetic average of the sample variances to the weighted geometric average of the sample variances. The geometric average is always less than or equal to the arithmetic average with equality holding only when all sample variances are equal. The more variation there is among the group variances, the more these two averages differ. A function of these two averages is created, which approximates a χ2-distribution (or, in fact, an F distribution under a certain formulation). Large values correspond to large values of the arithmetic or geometric ratio, and therefore to widely varying group variances. Dividing the Bartlett Chi-square test statistic by the degrees of freedom gives the F value shown in the table. Bartlett’s test is not very robust to violations of the normality assumption (Bartlett and Kendall 1946).
If there are only two groups tested, then a standard F test for unequal variances is also performed. The F test is the ratio of the larger to the smaller variance estimate. The p-value from the F distribution is doubled to make it a two-sided test.
Note: If you have specified a Block column, then the variance tests are performed on data after it has been adjusted for the Block means.
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 1974b; Asiribo, Osebekwin, and Gurland 1990). If there are only two levels, the Welch anova is equivalent to an unequal variance t-test.
Records a calculated F statistic for each test.
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.
Probability of obtaining, by chance alone, an F value larger than the one calculated if in reality the variances are equal across all levels.
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.
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.
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