Levene's test

In statistics, Levene's test is an inferential statistic used to assess the equality of variances in different samples. Some common statistical procedures assume that variances of the populations from which different samples are drawn are equal. Levene's test assesses this assumption. It tests the null hypothesis that the population variances are equal. If the resulting p-value of Levene's test is less than some critical value (typically 0.05), the obtained differences in sample variances are unlikely to have occurred based on random sampling. Thus, the null hypothesis of equal variances is rejected and it is concluded that there is a difference between the variances in the population.

Procedures which typically assume homogeneity of variance include analysis of variance and t-tests. One advantage of Levene's test is that it does not require normality of the underlying data. Levene's test is often used before a comparison of means. When Levene's test is significant, modified procedures are used that do not assume equality of variance.

Levene's test may also test a meaningful question in its own right if a researcher is interested in knowing whether population group variances are different.

Definition

The test statistic, W, is defined as follows:

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

where

• $W$ is the result of the test;hhhhhhhhhhhhhhhhhh
• $k$ is the number of different groups to which the samples belong,
• $N$ is the total number of samples,
• $N_{i}$ is the number of samples in the $i$th group,
• $Y_{ij}$ is the value of the $j$th sample from the $i$th group,
• $Z_{ij}=\left\{{\begin{matrix}|Y_{ij}-{\bar {Y}}_{i\cdot }|,&{\bar {Y}}_{i\cdot }{\mbox{ is a mean of i-th group }}\\|Y_{ij}-{\tilde {Y}}_{i\cdot }|,&{\tilde {Y}}_{i\cdot }{\mbox{ is a median of i-th group }}\end{matrix}}\right.$

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

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

The significance of $W$ is tested against $F(\alpha ,k-1,N-k)$ where $F$ is a quantile of the F test distribution, with $k-1$ and $N-k$ its degrees of freedom, and $\alpha$ is the chosen level of significance (usually 0.05 or 0.01).

Comparison with the Brown–Forsythe test

The Brown–Forsythe test uses the median instead of the mean. 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. 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-square distribution with four degrees of freedom (a heavily skewed distribution). Using the mean provided the best power for symmetric, moderate-tailed, distributions.

See also

• Bartlett's test

External links

http://www.itl.nist.gov/div898/handbook/eda/section3/eda35a.htm