How to Perform Levene’s Test?
The null hypothesis for Levene’s test is that the variance among groups is equal.
The alternative hypothesis is that the variance among different groups is not equal (for at least one pair the variance is not equal to others).
The test statistics for Levene’s test are:
where,
- k: number of different groups to which the sampled cases belong.
- Ni: Number of elements in different groups.
- N: total number of cases in all groups
where,
- Yij: the value of the jth case and ith group.
There are three types of Levene’s statistics available
- If a distribution has a longer-tailed distribution like the Cauchy distribution then we use a trimmed mean.
- For skewed distribution, if the distribution is not clear we will use the median for test statistics.
- For the symmetric distribution and moderately tailed distribution, we use mean value for distribution.
- Decide the level of significance (alpha). Generally, we take it as 0.05.
- Find the critical value in the F-distribution table for the given level of significance, (N-k), and (k-1) parameters.
- If W > F∝, k-1, N-k, then we reject the null hypothesis.
- else, we do not reject the null hypothesis.
Suppose there are 2 groups of students containing their scores in a maths test are below:
Group 1 | Group 2 |
---|---|
14 | 34 |
34 | 36 |
16 | 44 |
43 | 18 |
45 | 42 |
36 | 39 |
42 | 16 |
43 | 35 |
16 | 15 |
27 | 33 |
Here, our null hypothesis is defined as:
and the alternate hypothesis is
And our level of significance is:
Now, calculate the test statistics using the above formula
Group 1 | Group 2 | G1 (Y) : (Xi – Mean) | G2 (Z) : (Xi – Mean) | (Yi– meanVar)2 | (Zi– meanVar)2 | |
---|---|---|---|---|---|---|
14 | 34 | 2.8 | 17.6 | 49 | 60.84 | |
34 | 36 | 4.8 | 2.4 | 25 | 54.76 | |
16 | 44 | 12.8 | 15.6 | 9 | 33.64 | |
43 | 18 | 13.2 | 11.4 | 11.56 | 2.56 | |
45 | 42 | 10.8 | 13.4 | 1 | 12.96 | |
36 | 39 | 7.8 | 4.4 | 4 | 29.16 | |
42 | 16 | 15.2 | 10.4 | 29.16 | 0.36 | |
43 | 35 | 3.8 | 11.4 | 36 | 2.56 | |
16 | 15 | 16.2 | 15.6 | 40.96 | 33.64 | |
27 | 33 | 1.8 | 4.6 | 64 | 27.04 | |
Average | 31.6 | 31.2 | 8.92 | 10.68 |
|
|
where meanVar is,
- and k – 1 = Num of groups -1 =1
- N – k = 20 – 2 = 18.
- By solving the test statistics using the following parameters
- Since, W < F0.05,1,19, hence we do not reject the null hypothesis.
Levene’s test
In this article, we will learn about Levene’s test which is generally used to assess the equality of variances between two or more groups or samples.