ANOVA and T-Tests

In the last Statistical Sidebar, "Distribution of Scores Around the Mean and the Purpose of z-scores," I explained how data might be distributed around a mean. In this edition, I will discuss how the mean itself forms the central piece of some fundamental statistical tests. In the article, "Improving the Audio Game Playing Performances of the Visually Impaired People Through Multimodal Training," by Balan and colleagues, the differences between a pretest and a posttest measure are reported by the results of two different tests: an ANOVA and a t-test. Although the exact statistical results for the ANOVAs are not reported, the exact results for the t-tests are given, and the general significance level is given for both. (For a review of significance or p-value, please refer to the Statistical Sidebar that was published in the January-February 2016 issue of the Journal of Visual Impairment & Blindness.) Since these authors report both ANOVA and t-test results for some of the same comparisons, I would like to explain why this is possible. Note that it is not usual that an author would report both results for one comparison. Later, I will also explain why this is so. Both an ANOVA (which is short for ANalysis Of VAriance) and a t-test are statistical tests that compare the mean scores for certain groups. A t-test is designed to compare the mean of two groups and the ANOVA compares the means of more than two groups. However, it is possible to use the ANOVA to compare two groups as well. Both t-tests and ANOVAs have certain assumptions about the data they are used to compare. They both assume that the data fall on a normal distribution (I mentioned the normal distribution in the January-February 2017 sidebar) and that the individual scores that go into a mean are independent of each other. The ANOVA also has the assumption that the groups being compared have similar variances or spreads in their scores (this is called homoscedasticity). With these assumptions being assessed and satisfied, a researcher can compare two groups (as was done in the article referred to in this sidebar) using either statistical test. In this manuscript, the ANOVA is used as a preliminary check for significance, and the t-test is used to make a more detailed comparison. This approach is usually done when there are more than two groups. The ANOVA indicates some differences among groups, but it does not specify where exactly those differences are, and t-tests are used to make several group-to-group comparisons in order to ferret out exactly which groups are significantly different. With only two groups being compared, the ANOVA becomes superfluous. The calculation for a t-test is slightly different, depending on whether the two groups are paired or not. Paired groups or paired samples mean that the same participants make up the scores in each group that is being compared. In the referenced article, the authors compared the same people at pretest and at posttest, and these two sets of scores are paired. If two entirely different groups of people are being compared, like a group of men and a group of women being compared on their heights, then a researcher would use an unpaired calculation. …