## Friday, 11 December 2009

### Derivations and Motivations

The following post will set the derivations and basis for the models presented in subsequent posts.

The hypothesis used when testing for homogeneity of variances is:

(F 2.1)

representing the null hypothesis to be tested, versus

(F 2.2)

as the alternate hypothesis.

The null hypothesis for the test of homogeneity of variance is based on the assertion that the variance of the dependent variable is equal across groups defined by the independent variable. This is that the variance is homogeneous. Where the probability associated with the homogeneity test statistic is less than or equal to the level of significance being used in the test, the null hypothesis is rejected and a conclusion that the variance is not homogeneous is asserted. Conversely, where the test statistic is greater than the level of significance being used, the null hypothesis is not rejected and the variance is held to be homogeneous.

#### ANOVA F Test

If we take to be the ith value of the kth group and define the following values, and and we assume that is normally distributed and are independent for all values of i and j with the expected normal values for the mean of and the standard deviation of , we can obtain the best unbiased linear estimate of and . These are defined to be:

is estimated by:

(F 2.3)

is estimated by:

(F 2.4)

The F test value (for the ANOVA F Test) is defined (Casella, 2002, p 534) as:

(F 2.5)

And we reject

In this calculation, it is generally assumed that the population variance is equal (variance homeostasis or the homogeneity of variances). In this event, the value for can be written in a simplified form as:

(F 2.6)

This results in an F distribution with a central F variable with (K-1) and (N-K) degrees of freedom. As a consequence, the requirement for a suitable test of variances is necessary to ensure that homogeneity of variances exists.

##### Robustness of F- statistic under unequal variances

Figure 1 displays how the rejection probability is decreases as the variation among the groups increases. As a consequence, the probability of the correct decision in which the Null hypothesis should be rejected decreases as the variances of the datasets become more and more heterogeneous. Thus the capacity or efficiency of the test in detecting the difference decreases as the variances of the three groups become more and more heterogeneous. This implies that the F-test is not robust for datasets that have large heterogeneous variances.

Figure 1 Rejection probability as heteroscedasticity increases

The variances that we start with and increments are arbitrary and we can expect the same result as long as we maintain some reasonable relation between the initial value of the means and standard deviations. The magnitude of increments can also be arbitrary.

Similarly we compare the rejection probabilities by increasing and .

As the probability of rejection increases with heteroscedasticity, the F-ratio is non-robust with non- homogeneous datasets.

#### Other Tests

In this series of posts, a number of alternatives to the ANOVA F test have been evaluated.

##### Bartlett χ2 test

Bartlett’s test is the one most frequently taught tests of variance homogeneity (Conover et al., 1981). The ease of calculation and general simplicity of many aspects of this test make it a staple in introductory statistics classes (Lim & Loh, 1996; Ott, 1998; Zar, 1999). The test statistic B involves a comparison of the separate within-group sums-of-squares to the pooled within-group sum-of-squares. The test statistic is given by:

where;

and

such that:

(F 2.7)

Bartlett’s statistic (B) can generally be used to test the null hypothesis (F 2.1 for Ho) assuming that the distribution function (F) follows a standard normal cumulative distribution function (CDF). Bartlett demonstrated (Bartlett,1946 ) that where the variances are equal () B will generally follow a distribution where the approximation holds well for small sample sizes. It was also demonstrated that where holds as valid, the convergence in the distribution of B can be expressed as . Consequently, the null hypothesis will be rejected where B exceeds the percentile of a chi-squared distribution with (K-1) degrees of freedom.