# What is the difference between test to check homogenity of variance and ANOVA?

I found many tests like: Fligner, Levene etc etc to check homogeneity of variances, my question is: What is the difference between those tests and ANOVA ?

Fligner-Killeen's and Levene's tests are two ways to test the ANOVA assumption of "equal variances in the population" before conducting the ANOVA test. Levene's is widely used and is typically the default in programs like SPSS, but either test (or even Brown-Forsythe) is acceptable. ANOVA is the omnibus test of mean differences among groups. While, in name, ANOVA analyzes the variance (between, within, and overall) among three or more groups, its hypotheses actually make statements about the equality of means versus there being "at least two means different."

• i have a doubt regarding "ANOVA is the omnibus test of mean differences among groups" AND "ANOVA analyzes the variance (between, within, and overall) among three or more groups" I really do not understand this aspect. ANOVA will test if the means are different OR if the variances are different? Thank you for the clarificatio – Dail Nov 9 '11 at 14:28
• ANOVA tests for differences between means. It will tell you if the means are different overall, but you typically have to use a post-hoc test (e.g., Tukey Honestly Significant Difference) to determine which of your three or more groups are different from each other. Mathematically, ANOVA "analyzes the variance" by computing sums of squares and mean squares values in the background (that are like variances), but it is still a test about mean differences in the end. – Garrett Nov 9 '11 at 14:36
• ok perfect, so what is the test to check if the VARIANCES are different from each other? – Dail Nov 9 '11 at 14:37
• You can take your pick among Levene's, Fligner-Killeen's or Brown-Forsythe. Levene's is pretty straightforward and easy to run. – Garrett Nov 9 '11 at 14:39
• To compare mean differences between/among groups by, let's say, just subtracting the averages is not sufficient to establish statistical significance. Rather, it is important to capture the variability of each of your groups that you're comparing, as well. To do this, ANOVA compares the raw data of each group to corresponding group means and an overall (total) mean. These comparisons yield "sums of squares" that when divided by degrees of freedom give you "mean squares." These mean squares are, for all intents and purposes, variances. In the end, you compared means but you analyzed variances. – Garrett Nov 9 '11 at 16:52

Just thought I'd post a little more about the Fligner Killeen test. It is a nonparametric way of comparing the variances of more than two groups that is very robust against non-normal data. Essentially, it starts off the same way as a Brown Forsythe test for the ANOVA, obtaining the absolute deviations of each observation from its respective group median. Rather than performing an ANOVA on these residuals, the FK test ranks these residuals from low to high (where a rank of 1 is given to the lowest data point), assigning the average value of any tied ranks. By dividing each of these resulting ranks by the value 2(n+1), where n is the total number of data points across all groups, and then adding 0.5 to each result, each of the ranked residuals is "normalized" into an area under the normal curve.

Using the inverse normal distribution, we then convert these areas back into z-scores, taking the absolute value of any negative z-scores. We obtain the average z-score for each group, as well as the overall average z-score, and the overall variance of the z-scores. We then find a "mean square" for each group by taking its average z-score and subtracting the overall z-score, squaring the difference, and multiplying by the respective sample size of the group. Do this for all the groups, add them up, and divide by the total variance of all the z-scores. This is your FK statistic that is evaluated against a chi-square distribution with degrees of freedom equal to (number of groups - 1). If the result is significant, the group's have statistically different variances.

Hope this helps!

Don't know about Fligner, but Levene's test is actually an ANOVA of absolute deviations from group means (or group medians, this would be Brown-Forsythe test).

• so what is more accurate Levene or ANOVA? – Dail Nov 9 '11 at 13:43
• ? Don't understand you. ANOVA checks deviations from the mean. Levene checks deviations from the mean of absolute deviations about the mean. – ttnphns Nov 9 '11 at 15:43

ANOVA is called "analysis of variance" because it decomposes the total variance into variance within groups (the "error") and variance among the group means. So it tests whether group means are equal by comparing the variance among them to that expected based solely on the within-group variance: is the variation among group means "greater than expected by chance alone" i.e. purely from sampling variability.

This is totally different than Levene's or other such tests which test whether the variances of the groups are equal. Heuristically, Levene's and Brown-Forsythe's tests (I'm not sure about Fligner; sorry Mike) are like ANOVA on the squares or absolute values of the within-group residuals, so they test, roughtly, whether the mean magnitude of the residuals -- thus the within-group variability -- differs among groups.