e-Statistics

Test for Homogeneity

Here the homogeneity (equality) of the population variances $\sigma_1^2, \sigma_2^2, \ldots, \sigma_k^2$ from k groups is tested. The hypothesis testing problem evaluates the null hypothesis

$\displaystyle H_0: \sigma_1^2 = \sigma_2^2 = \cdots = \sigma_k^2$

Here we introduce two test procedures—Hartley's maximum F-ratio test and Levine's test, and choose in the following procedure.

Let $X_{i1}, X_{i2}, \ldots, X_{in_i}$ be the data from the i-th group, and let be the sample median of the i-th group. In the test procedure we set $Z_{ij} = X_{ij}$ for the Hartley's test, whereas, we have to set $Z_{ij} = \vert X_{ij} - m_i\vert$ when the Levine's test is considered. Then the test procedure uses the sample mean $\displaystyle \bar{Z}_{i\cdot} = \frac{1}{n_i} \sum_{j=1}^{n_i} Z_{ij}$ and the sample variance $\displaystyle s_i^2 = \frac{1}{n_i-1} \sum_{j=1}^{n_i} (Z_{ij} - \bar{Z}_{i\cdot})^2$ within group for every factor level $i = 1,\ldots,k$ .

The data from groups are arranged in multiple columns each of whom represents a factor level. Or, Here a single measurement column (specified above) is grouped by another column (specified here) indicating factor levels.

The Hartley's test requires the normality assumption and the same sample size $n_1 = n_2 = \cdots = n_k$ (but the test can be performed even if the sample sizes are not equal). The test statistic is built on the maximum F-ratio $s_{\mbox{max}}^2/s_{\mbox{min}}^2$ of the largest $s_{\mbox{max}}^2$ to the smallest $s_{\mbox{min}}^2$ of k sample variances.
The Levine's test does not require the same sample size, and works reasonably even if the normality assumption does not hold. The Levine's test uses the test statistic constructed for analysis of variance in essence.

The above summary statistics are used to calculate the test statistic and the p-value . By rejecting we can find evidence of heterogeneous variances (that is, we can support the alternative hypothesis that the population variances are not equal).