Levene's test

Levene's test has been used in the past before a comparison of means to inform the decision on whether to use a pooled t-test or the Welch's t-test for two sample tests or analysis of variance or Welch's modified oneway ANOVA for multi-level tests. However, it was shown that such a two-step procedure may markedly inflate the type 1 error obtained with the t-tests and thus is not recommended. Instead, the preferred approach is to just use Welch's test in all cases.

Levene's test may also be used as a main test for answering a stand-alone question of whether two sub-samples in a given population have equal or different variances.

Levene's test was developed by and named after American statistician and geneticist Howard Levene.

Definition

Levene's test is equivalent to a 1-way between-groups analysis of variance (ANOVA) with the dependent variable being the absolute value of the difference between a score and the mean of the group to which the score belongs (shown below as ${\displaystyle Z_{ij}=|Y_{ij}-{\overline{Y}}_{i\cdot }|}$ ). The test statistic, ${\displaystyle W}$, is equivalent to the ${\displaystyle F}$ statistic that would be produced by such an ANOVA, and is defined as follows:

${\displaystyle W=\frac{(N-k)}{(k-1)}\cdot \frac{\sum _{i=1}^k{N}_i(Z_{i\cdot }-Z_{\cdot \cdot }{)}^2}{\sum _{i=1}^k\sum _{j=1}^{N_i}(Z_{ij}-Z_{i\cdot }{)}^2},}$

where

• ${\displaystyle k}$ is the number of different groups to which the sampled cases belong,
• ${\displaystyle N_i}$ is the number of cases in the ${\displaystyle i}$th group,
• ${\displaystyle N}$ is the total number of cases in all groups,
• ${\displaystyle Y_{ij}}$ is the value of the measured variable for the${\displaystyle j}$th case from the ${\displaystyle i}$th group,
• ${\displaystyle Z_{ij}=\{\begin{array}{ll}|Y_{ij}-{\overline{Y}}_{i\cdot }|,& {\overline{Y}}_{i\cdot }\text{ is a mean of the }i\text{-th group},\\ |Y_{ij}-{\tilde{Y}}_{i\cdot }|,& {\tilde{Y}}_{i\cdot }\text{ is a median of the }i\text{-th group}.\end{array}}$

(Both definitions are in use though the second one is, strictly speaking, the Brown–Forsythe test – see below for comparison.)

• ${\displaystyle Z_{i\cdot }=\frac{1}{N_i}\sum _{j=1}^{N_i}{Z}_{ij}}$ is the mean of the ${\displaystyle Z_{ij}}$ for group ${\displaystyle i}$,
• ${\displaystyle Z_{\cdot \cdot }=\frac{1}{N}\sum _{i=1}^k\sum _{j=1}^{N_i}{Z}_{ij}}$ is the mean of all ${\displaystyle Z_{ij}}$.

The test statistic ${\displaystyle W}$ is approximately F-distributed with ${\displaystyle k-1}$ and ${\displaystyle N-k}$ degrees of freedom, and hence is the significance of the outcome ${\displaystyle w}$ of ${\displaystyle W}$ tested against ${\displaystyle F(1-\alpha ;k-1,N-k)}$ where ${\displaystyle F}$ is a quantile of the F-distribution, with ${\displaystyle k-1}$ and ${\displaystyle N-k}$ degrees of freedom, and ${\displaystyle \alpha }$ is the chosen level of significance (usually 0.05 or 0.01).