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Levene's original paper only proposed using the mean. By power, we mean the ability of the test to detect unequal variances when the variances are in fact unequal. By robustness, we mean the ability of the test to not falsely detect unequal variances when the underlying data are not normally distributed and the variables are in fact equal. "The three choices for defining Zij determine the robustness and power of Levene's test.
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I will quoute NIST here, from the URL provided above: The 3 variants of the Levene's test (where in JMP's fourth method can be classified as being the same as the first method) are applicable for different situations. But mind also that both of them can also be wrong. (of course! or else they would be severely criticized). So which is correct? You may have guessed this right. Minitab on the other hand uses the median and the square of the (Xi-Xmean). JMP uses the mean and absolute value of (Xi-Xmean). Minitab vs JMPWe alredy described JMP's Levene's test method. For the details of the formula, you may refer to this link while the fourth method is described in JMP's Statistics documentation. It uses the mean, but in addition, instead of using the classical squared (Xi-Xmean), it uses the absolute value of the (Xi-Xmean) in its computation of the data spread. There is a fourth variation that is being used by JMP. One uses the median, and the other one uses the trimmed mean. It is a formula for a test statistic where it uses an averaging of the data. Now originally there is only one Levene's test. Non-parametric test means it does not assume a distribution for it to be usable. Levene's test is one of the non-parametric tests available to statisticians. However when the data is not from a normal distribution, we should use Levene's test. That is why given the choice,and your data is tested to be normally distributed, these tests should be the first choice. The tests Chi-square, F-test,and Bartlett's test are very powerful, but only for normally distributed data. A less powerful test means that it is more conservative in saying that there is no difference between the samples. A more powerful test means that it can discern even slight differences,and give the conclusion that there is a difference if it sees one. By power we mean the ability to detect differences. We do a more generalized test which is applicable for other distributions, but in return we lose power. So what do we do when the data are not from a normal distribution? We do a trade off. It is in this backdrop that the Levene's test comes in. The alpha, or probability of False Negatives (you conclude as no difference but in reality there is), would be much higher than what we expect it to be. By sensitive we mean that if the data are not from a normal population, the test would give inaccurate results. These tests however are very sensitive in the assumption that the sample data being examined are coming from a population that are normally distributed. If more than two samples are involved and we want to simultaneously check the homogeneity of their variances, we use Bartlett's Test. If there are two sample variances we wish to compare, we use the F-test for two variances. The Levene's Test of Homogeniety of VariancesTo test whether a sample variance is equal to some hypothesized value, we use the Chi-square test for variance. But before I tell you about the details of what I found, it is necessary that I give you enough background of the Levene's Test. Good thing is that both of them have superb documentations. Now I have no choice but to examine the internal formula and algoritms used by the softwares. Minitab agreed with my intuition, but that does not mean we are correct and JMP is wrong. And guess what? The conclusion disagrees with JMP! As i have an immediate access only to Minitab that time, I used Minitab 15. So the first thing I did was to try to replicate JMP's Levene's test result using another statistical software. Being a logical person and a practitioner of applied statistics, I refuse to accept the results in its face value without understanding first why it contradicts my personal understanding of the statistical problem. I once used Levene's test in JMP and found that the results are obviously counter-intuitive to my understanding of the data im testing.