7.4.2. Assuming the observations are normal, do the processes have the same variance?

7. Product and Process Comparisons
7.4. Comparisons based on data from more than two processes

7.4.2. Assuming the observations are normal, do the processes have the same variance?

Before comparing means, test whether the variances are equal

Techniques for comparing means of normal populations generally assume the populations have the same variance. Before using these ANOVA techniques, it is advisable to test whether this assumption of homogeneity of variance is reasonable. This page describes two procedures (Bartlett's test and Levene's test) that are used for this purpose.

A test for equal variances due to Bartlett is commonly used

Bartlett's Test for Homogeneity of Variances

Let's examine the null and alternative hypotheses.

Assume we have samples of size n_i from the i-th population, i = 1, 2, . . . , k, and the usual variance estimates from each sample:

Now introduce the following notation: n_j= n_j- 1 (the n_jare the degrees of freedom) and

When none of the degrees of freedom is small, Bartlett showed that M is distributed approximately as c²_k-1. The chi square approximation is generally acceptable if all the n_i are at least 5.

This is a slightly biassed test, according to Bartlett. It can be improved by dividing M by the factor

Instead of M, it is suggested to use M/C for the test statistic.

This test is not robust, it is very sensitive to departures from normality.

An alternative description of Bartlett's test, which also describes how DATAPLOT implements the test, appears in Chapter 1,

An illustrative example of Bartlett's test

Gear Data Example (from Chapter 1)

Gear diameter measurements were made on 10 batches of product. The complete set of measurments appears in Chapter 1. Bartlett's test was applied to this dataset leading to a rejection of the assumption of equal batch variances at the .05 critical value level.

The Levene test for equality of variances

The Levene Test for Homogeneity of Variances

Levene's test offers a more robust alternative to Bartlett's procedure. That means it will be less likely to reject a true hypothesis of equality of variance just because the distributions of the sampled populations are not normal. When non-normality is suspected, Levene's procedure is a better choice than Bartlett's.

Levene's test, and its implementation in DATAPLOT, was described in Chapter 1. This description also includes an example where the test is applied to the gear data. Levene's test does not reject the assumption of equality of batch variances for this data. This differs from the conclusion drawn from Barlett's test and is a better answer if, indeed, the batch population distributions are non-normal. .