Bartlett’s Test

The assumption of equal variances across treatment groups may cause serious problems if violated in one-way analysis of variance models. A common test for homogeneity of variances is Bartlett’s test. This statistical test checks whether the variances from different groups (or samples) are equal.

Suppose that there are r treatment groups and we want to test

$$\begin{array}{l}{H}_0:{\sigma }_1^2={\sigma }_2^2=\cdots ={\sigma }_r^2\\ \text{versus}\\ H_1:{\sigma }_m^2\ne {\sigma }_k^2\text{for}\text{some}m\ne k.\end{array}$$

In this context, we assume that we have independently chosen random samples of size $n_i,i=1,\dots ,r$ from each of the r independent populations. Let $X_{ij}~N\left({\mu }_i,{\sigma }_i^2\right)$ be independently distributed with a normal distribution having mean μi and variance ${\sigma }_i^2$ for each $j=1,\dots ,n_i$ and each $i=1,\dots ,r$. Let ${\overline{X}}_i$ be the sample mean and $S_i^2$ the sample variance of the sample taken from the $i\text{th}$ group or population. The uniformly most powerful unbiased parametric test ...