In this section we extend the concept of statistical testing for the central values for one population to that of comparing central values from two independent populations. This simple extension produces a set of statistical tests that are probably used in 50% of all research studies. This is the first real statistical method for your tool chest. The two-sample t-test (page 271) is still one of the most often used of all statistical procedures.

A nonparametric alternative to the two-sample t-test, the Wilcoxon Rank Sum Test, is discussed in section 6.3. It is one of many statistical tests based on some form of ranking of the data instead of the actual measured values.

In section 6.4 we discuss the study situation in which two related measurements are taken on each individual in the population. Here we have one population but two measurements (random variables) that we wish to compare. We call this situation the paired data case and present a t-test for these data as well. A nonparametric alternative to the paired data t-test, Wilcoxon Signed-Rank test, is presented in section 6.5.

At the end of this section we again discuss approaches to determining sample size needs in the two independent sample case. Our goal here is to figure out how many samples we need to be test the difference between the means of the two populations when we can specify how big of a difference is truly significant and what level of confidence we wish in the final test results.

By this section you will see a pattern beginning to emerge for the material presented in this course. We build on basic ideas. The two population case follows the one- population case. In the next Unit we will extend the methods to the multiple (greater than two) population case.

A final observation. A couple of nonparametric tests have been included in this section. The use of parametric or nonparametric tests will often depend on a lot of factors. It may be that the journals you wish to publish in mandate the use of nonparametric tests procedures. This often stems from the fact that the data being analyzed may not have nice distribution characteristics and low sample sizes. The combination of these two events tell us that parametric test procedures may not meet the confidence targets (specifically Type I error rates) specified. Nonparametric methods are more conservative than parametric test in this situation and hence provide a more believable test result. In the past, computational costs dictated that you perform either a parametric or nonparametric test. Today, computational costs are rarely an issue, suggesting that you should perform and examine both approaches for your data. In most cases, the conclusions from the tests will be identical. When they are not, it will greatly benefit your analysis if you figure out why they are different. Here is an excellent place for graphical assessment of your data.

Two Independent Populations (PowerPoint, PDF)

Paired Data Situation (PowerPoint, PDF)