Applied Statistics for Public Policy

Brian P. Macfie, Philip M. Nufrio

• Publisher: Routledge
• Year: 2017
• ISBN:
  9781317363521
• Edition: 1
• Topic: Business
• Subtopic: Business Mathematics
• Index: Business

This practical text provides students with the statistical tools needed to analyze data, and shows how statistics can be used as a tool in making informed, intelligent policy decisions. The authors' approach helps students learn what statistical measures mean and focus on interpreting results, as opposed to memorizing and applying dozens of statistical formulae. The book includes more than 500 end-of-chapter problems, solvable with the easy-to-use Excel spreadsheet application developed by the authors. This template allows students to enter numbers into the appropriate sheet, sit back, and analyze the data. This comprehensive, hands-on textbook requires only a background in high school algebra and has been thoroughly classroom-tested in both undergraduate and graduate level courses. No prior expertise with Excel is required. A disk with the Excel template and the data sets is included with the book, and solutions to the end-of-chapter problems will be provided on the M.E. Sharpe website.

Unit IV
Measures of Association

14 Comparing More Than Two Population Means with ANOVA

Chapter Objectives

• Discuss the general idea of analysis of variance (ANOVA).
• Assumptions required to test hypothesis for more than two means using ANOVA.
• Development and applications of the one-way ANOVA model.
• Development and applications of the two-way ANOVA model.
• Performing one-way and two-way ANOVA tests with POLYSTAT.

Introduction

In Chapter 10, we showed how to test a hypothesis about a population mean by using a single sample mean drawn from that population. In this instance, we tested if the hypothesized population mean (i.e., the parameter) was statistically different from the sample mean (i.e., the statistic). In Chapter 11, the analysis was extended to comparing whether two population means were statistically different by comparing two sample means—one drawn from each sample. Depending on the circumstance, the appropriate test statistic was either the z-score or t-statistic.

If we now take the next step, often a researcher wants to know whether more than two groups differ on a specific variable. In this chapter, we expand what was already covered and now consider the case where we test whether more than two population means are statistically different. In a sense, we are asking whether there is an association among a group of means. These groups or samples can either be naturally occurring or can be set up by the researcher for the purpose of study. In the case of the latter, we call this the experimental design approach (Chapter 19 will have more on this). In a true experimental design, the researcher can assign subjects to different groups, control the environment (i.e., when are the subjects exposed to a treatment, who is exposed to a treatment, etc.), and manipulate the independent variable in order to bring about a change in the dependent variable. The effect of the treatment is then analyzed. As an example, if one were examining the effects of four different medical treatments on Alzheimer's patients, individuals with similar attributes would be randomly assigned to groups and subjected to the different treatments. The results would then be analyzed for differences.¹

What Is Analysis of Variance?

The procedure that is used to test a hypothesis about a difference between more than two means is called analysis of variance (ANOVA). As implied by the description of the test, we are analyzing the variances between these samples. There is nothing mysterious about the term ANOVA; it is simply an acronym derived from analysis of variance. Recall from Chapter 11, the test statistic for comparing differences between variances from different samples is the F-test and uses the F-distribution.

At this point, the first question many students ask is "Why can't we just continue to use the t-statistic to test if the means are different?" The answer is rather obvious. The t-statistic can only be calculated between two phenomena (i.e., a population mean and a sample mean, two sample means, etc.). Although it is technically possible to compare multiple means by conducting separate t-tests, it is not very practical. Imagine the problem we are faced with in the case of just three means. We would calculate a t-statistic between the first and second sample, then we would calculate a t-statistic between the first and third sample, and then we would calculate a t-statistic between the second and third sample. These multiple t-statistics may not tell us whether the three means are statistically different. As an example, suppose the first t-test is statistically significant, while the other two are not. We cannot determine whether this tells us the group of the means is different or the same.

The problem only gets worse in the case of comparing more than three means. Conducting multiple f-tests can lead to severe statistical problems tha...