Levene's test

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 ${\displaystyle Z_{ij}=|Y_{ij}-{\bar {Y}}_{i\cdot }|}$ ). 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,
• ${\displaystyle Z_{ij}={\begin{cases}|Y_{ij}-{\bar {Y}}_{i\cdot }|,&{\bar {Y}}_{i\cdot }{\text{ is a mean of the }}i{\text{-th group}},\\|Y_{ij}-{\tilde {Y}}_{i\cdot }|,&{\tilde {Y}}_{i\cdot }{\text{ is a median of the }}i{\text{-th group}}.\end{cases}}}$

(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).

The Brown–Forsythe test uses the median instead of the mean in computing the spread within each group (${\displaystyle {\bar {Y}}}$ vs. ${\displaystyle {\tilde {Y}}}$, above). Although the optimal choice depends on the underlying distribution, the definition based on the median is recommended as the choice that provides good robustness against many types of non-normal data while retaining good statistical power.[3] If one has knowledge of the underlying distribution of the data, this may indicate using one of the other choices. Brown and Forsythe performed Monte Carlo studies that indicated that using the trimmed mean performed best when the underlying data followed a Cauchy distribution (a heavy-tailed distribution) and the median performed best when the underlying data followed a chi-squared distribution with four degrees of freedom (a heavily skewed distribution). Using the mean provided the best power for symmetric, moderate-tailed, distributions.

• Bartlett's test
• F-test of equality of variances
• Box's M test

• Parametric and nonparametric Levene's test in SPSS
• http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm