Group Variation Analyst

Technicalities

It is frequently of interest to compare differences in results among several groups. When the outcome measurements across the groups are continuous, and certain assumptions are met, a methodology known as Analysis of Variance, or ANOVA, may be employed to compare the means of the groups. The term Analysis of Variance may be misleading since the objective of the analysis is to compare means. However, through an analysis of the variation in the data we will be able to draw conclusions about possible differences in the group means.

Group Variation Analyst computes the one-factor ANOVA F-test for difference for pairs of group means rather than for all groups at once.

The assumptions to be met in order for the ANOVA to produce reliable results are three:

1. Error terms must be random and independent. That is, the difference (or error) for one observation should not be related to the difference for any other observation. Most often this assumption is violated when data are collected over a period of time, because measurements made at adjacent time points may be more alike than those made at very different time, for instance when measuring air temperature.
2. Values in each group must be normally distributed. As long as the distributions are not extremely different from a normal distribution (bell shaped curve), the level of significance of the ANOVA test is fairly robust, particularly for large samples.
3. Homogeneity of variance. This means that the variance within each group must be equal (very close) for all groups. This assumption is often violated when analyzing groups with different sample sizes. Thus, for computational effectiveness, robustness, and power, there should be groups of equal sample size whenever possible.

If the normality and homogeneity assumptions are violated, an appropriate data transformation may be used for normalizing data and reducing the difference in variances.

In ANOVA the null hypothesis is that no difference exists in the means of the groups. The test splits the total variation in Within and Between group variation and it computes the test statistics F. The F-value follows an F-distribution, and for a given level of confidence we may reject the null hypothesis if the F-value exceeds the F-critical, which is the upper tail of the F-distribution.

Unequal Sample Size

Although we have written that there should be groups of equal sample size whenever possible, we can still obtain a suitable test statistic when the sample sizes are not all equal. Specifically, we use the following formula as a test statistic and reject the null hypothesis when the computed value exceeds F-critical. See Madsen pg 530 for details.

Other ANOVA methods

The analysis-of-variance method we have discussed so far is the one-factor model. Two or more factors can also be tested, and MarketingStat is willing to develop further methods as our clients make the request.