# Comparing Three or More Means

Overview

This chapter continues the progression of analysis of quantitative data. In particular, we continue the analysis of means.  In Chapter 10, we conducted hypothesis tests on a single mean.  Then, in Chapter 11, we conducted hypothesis tests in which we compared two independent means.  The material in this chapter presents hypothesis tests in which we compare three or more means.  This parallels the presentation we followed for comparing proportions (one proportion-Chapter 10; two proportions-Chapter 11; three or more proportions-Chapter 12).

What to Emphasize

Be sure to emphasize that the material in this section is used to analyze a quantitative response variable in which we are comparing three or more populations.  The data is obtained through k independent simple random samples or through a completely randomized design with k treatments. This material is an extension of the two sample t-test for comparing two independent means.  However, we must use a new sampling distribution: the F-distribution.

• Comparing Three or More Means: One-Way Analysis of Variance – Be sure to discuss why the method is called analysis of variance. We are comparing two different estimates of a variance.  One estimate is based on a weighted average of the sample variances from each of the k  This estimate is an unbiased estimate of the sample variance regardless of whether the k samples come from populations with the same mean or not.  The other estimate is based on the difference between each sample mean and the overall mean.  This estimate of the variance is unbiased only if the k samples come from populations with the same mean.
• Discuss why we cannot conduct a test of the equality of three or more means by performing various individual two sample t-tests.
• The first objective presents a discussion of the requirements to perform a one-way analysis of variance (ANOVA). Be sure to emphasize that verifying model requirements is very important for all statistical procedures.
• It is very important for students to conceptually understand why the methods we are using work. Consider spending time explaining the content of the conceptual understanding of one-way ANOVA at the beginning of Objective 2.
• While computing the F-test statistic by hand is extremely labor intensive, it is worthwhile to go through a by-hand computation for a small data set so students get a feel for how the F-test statistic may be used to judge the equality of the means from the k
• However, the emphasis in one-way ANOVA should be the interpretation of the P-value obtained via technology.
• Also, it is always a good idea to provide graphical evidence as support of the data analysis. Encourage students to construct side-by-side boxplots of the raw data.
• Post Hoc Tests on One-Way Analysis of Variance – Once the null hypothesis of equal population means for k treatments is rejected, we perform a post hoc (after the fact) test. This test allows us to identify which means differ.
• There are a variety of post hoc tests for identifying which means differ. We use Tukey’s Test in this text. Tukey’s Test only allows for comparison of two means. The by-hand computation for this test is rather challenging, so we recommend that technology be used to obtain the results of the test.

• The Randomized Complete Block Design – This section introduces a more elaborate design in which a single factor is manipulated at various levels (the treatment), but a second variable is controlled a fixed at certain levels as well (the block). The researcher is not interested in determining whether the block is significant in explaining variation in the response variable, but does believe the block variable may be used to reduce experimental error.
• The randomized complete block design is a generalization of the matched-pairs design (analyzed in Section 11.2). In Section 11.2 the blocks were the individuals upon which measures were obtained for each treatment level.  Blocks included items such as husband/wife, before/after treatment, same geographic location, twins, and so on.
• Emphasize that blocks are used to reduce experimental error. If a researcher believes a certain variable may play a role in the value of the response variable, but is not necessarily interested in the significance of that variable, it makes sense to block. Provide some simple examples of this situation.  For example, in comparing three or more sunscreens for effectiveness, we might block by individual.   Or, in comparing three or more car rental agencies for price, we might block by location.
• If the null hypothesis of equal means is rejected, we once again perform Tukey’s test to determine which of the pairwise means differ.

• Two-Way Analysis of Variance – This section introduces the analysis required to compare k means when two factors are controlled, manipulated, and set at different levels. In general, we call the factors A and B.  If Factor A may be set at 3 levels (high, medium, low) and Factor B may be set at 2 levels (Yes or No), then we have a 3 x 2 factorial design.
• When analyzing data obtained from a two-way ANOVA, it is very important that students understand the ideas of main effects and interaction effect. Spend time discussing the visual approach to understanding these effects.