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Statistics for the Social Sciences A General Linear Model Approach

Russell T. Warne

Table of Contents

Preface

ix

Acknowledgements List of Examples

xiii xv

Chapter 1 Chapter 2

Statistics and Models Levels of Data

1 21

Chapter 3 Chapter 4

Visual Models Models of Central Tendency and Variability

35 77

Chapter 5 Chapter 6

Linear Transformations and z-Scores Probability and the Central Limit Theorem

103 127

Chapter 7

Null Hypothesis Statistical Significance Testing and z-Tests

151

Chapter 8 Chapter 9

One-Sample t-Tests Paired-Samples t-Tests

181 215

Chapter 10 Unpaired Two-Sample t-Tests Chapter 11 Analysis of Variance Chapter 12

Correlation

Chapter 13 Regression Chapter 14 Chi-Squared Test Chapter 15

Applying Statistics to Research, and Advanced Statistical Methods

241 275 329 369 403 443

Appendix A1

485

Appendix A2 Appendix A3

494 497

Appendix A4 Appendix A5

500 503

Appendix A6

506

Glossary Answer Key

508 524

References Name Index

548 564

Subject Index

569

AN ACCESSIBLE WRITING STYLE TO RELIEVE STUDENTS’ ANXIETIES ABOUT STATISTICS

1

Statistics and Models

“Why do I need to take statistics?”

Every semester that I teach statistics I have students asking this question. It’s a fair question to ask. Most of my students want to work as therapists, social workers, psychologists, anthropologists, sociologists, teachers, or in other jobs that seem far removed from formulas and numbers. As students majoring in the social sciences, almost all of them are more interested in people than they are in numbers. For many students (including me, a long time ago), statistics is something to get out of the way so that they can devote their attention to the stuff they really want to do (Rajecki, Appleby, Williams, Johnson, & Jeschke, 2005).

The purpose of this chapter is to show you why students majoring in the social sciences have to take statistics. This chapter will also explore some introductory concepts and terminology that are necessary to understand the book. The most important concept introduced in this chapter is models, which many experts use to understand their statistical results. Because my goal is to help you think about statistics like a professional, I will use that perspective throughout the book.

Learning Goals • Explain why students in the behavioral sciences need statistics to be successful in their majors.

• Outline the general process for designing and executing a research study.

• Describe the benefits of using models to understand reality.

• Demonstrate why using a model – even if it is not completely accurate – is useful for people working in the social sciences.

• Identify the three types of models and describe their characteristics.

Why Statistics Matters Although many students would not choose to take a statistics course, nearly every social science department requires its students to take a statistics course (e.g., Norcross et al., 2016; Stoloff et al., 2010). Why? Apparently, the professors in these departments think that statistics is essential to their students’ education, despite what their students may think.

The main reason that many students must take statistics is that research in the social sciences is dominated by methodologies that are statistics-based; this family of methods is called quantitative research. Researchers who use quantitative research convert their data into numbers for the purpose of analysis, and the numbers are then analyzed by statistical methods. Numerical

The book is full of interesting real-life EX AMPLES from the social sciences, including examples from autism studies, education, psychology, criminology, family studies, organizational behavior, and more.

Examples Chapter(s)

Example

Field

1

Baby naming

Sociology

1

Birth dates and hockey teams

Sports

1

Freud’s theories were not scientific

Psychology/philosophy of science

1

Working memory theory is falsifiable

Psychology

1

Personality and job performance

Psychology/business

1, 3, 4, 9

Sibling relationships and autism

Psychology/family studies

1

Parental spending on children

Psychology/sociology/family studies

1

Why do people commit crimes?

Sociology/criminology

1

Changing schools and dropout rates

Education

1

Mate selection (matching hypothesis)

Psychology/sociology/family science

1

Darwin’s theory of evolution

Biology

1

Newton’s laws of motion

Physics

2

Affection

Psychology/family science

2

Perception of stimuli and actual stimuli intensity

Psychology

2

Levels of data examples (temperature, weight, gender, etc.)

Social sciences

2

Educational and psychological test scores and rating scales

Education/psychology

3

Murder rate in Canadian provinces

Sociology/criminology

3

Deceptive truck ad

Business/marketing

3

Deceptive political poll

Political science/journalism

3

Advanced placement tests (interactive map link)

Education

3

Baby Name Voyager (interactive chart link)

Sociology

4

Outliers in math ability

Education/psychology

4

Resilient children in poverty

Psychology/sociology

4

65% of households have below-average income

Economics/sociology

5

Quetelet’s data

Biology

5

Normally distributed variables

Biology/psychology/education

Basic Probability

133

Sidebar 6.3 The gambler’s fallacy and the hot-hand fallacy As I was collecting the data for Figure 6.2a and 6.2b, there was a series of 32 consecutive die throws where the outcome was not a 5. This is an extremely improbable series of outcomes; the joint probability would be p = (5/6)32 = .0029. Therefore, it seems likely that rolling a 5 was “due.” Right?

Wrong. The belief that a drought of a particular outcome means that the outcome will probably occur soon is what statisticians call the “gambler’s fallacy.” In reality, dice rolls, lottery numbers, roulette wheel spins, and wins on a slot machine are all independent random events. Because the outcomes are independent, each one has no influence on other outcomes. The roulette wheel does not INSERTS remember what explain the last outcome was – and therefore that outcome cannot influence what happens when thecommon wheel is spun again. Mathematically, a way to think about this is that the joint probability of a sequence of trials is misconceptions not relevant to the actual probability of any particular trial. Therefore, the fact that 31 trials in a row did not inofstatistics produce a die roll of 5 was irrelevant to the 32nd trial. The 32nd trial still had a 1/6 probability producing with a 5 – regardless of the outcomes of previous trials. depth and rigour.

Do people actually believe in the gambler’s fallacy? Anecdotal evidence suggests they do.

A Wall Street Journal article (Copeland, 2014) mentioned a slot machine at the MGM Grand casino in Las Vegas, Nevada, that had not paid out its major jackpot in 20 years. As a result, the machine was – by far – the casino’s most popular slot machine. Scientific research also provides evidence that people truly do believe the gambler’s fallacy. Clotfelter and Cook (1993) studied a state-run lottery in Maryland that people could win by buying a ticket that matched three randomly selected one-digit numbers in the same order. The joint probability of winning this game is 1 in 1,000 because p = (.10)(.10)(.10) = .001. The researchers found that after a three-digit combination was selected, bets on that three-digit combination dropped by approximately one-third and did not fully recover for nearly three months. Apparently, lottery players believe that after a combination of numbers has won that it is time for some other combination of numbers to win – behavior consistent with the gambler’s fallacy.

An idea related to the gambler’s fallacy is the “hot-hand fallacy,” which suggests that the occurrence of an outcome makes the outcome more likely in the future. In other words, people who believe in the hot-hand fallacy believe that “streaks” are more likely to continue than to be broken. In the data used to create Figures 6.2a and 6.2b, there were five die rolls that were all 1’s. Again, this is an extremely unlikely event, with a joint probability of p = (.167)(.167)(.167)(.167)(.167) = .00013. To some people, such an unlikely streak seems lucky, and it appears that the dice rolls are not really random.

Again, this is incorrect reasoning for the same reason that the gambler’s fallacy is faulty: random events do not “remember” what occurred before and earlier events have no impact on later ones. As with the gambler’s fallacy, people really do believe in “hot streaks,” as research on roulette players’ behavior has shown (Sundali & Croson, 2006).

To convert the histogram in Figure 6.2a into a probability distribution, it is necessary to divide each frequency count by the total number of trials – in this case, 600. This produces the probability estimates for each outcome. These are diagrammed in Figure 6.2b.

Notice how Figure 6.2b looks very similar to Figure 6.2a. This is because dividing every frequency by the total number of trials is a linear transformation, which, as Chapter 3 showed, does not change the overall pattern of results in a visual model. Linear transformations only change the axis of a visual model (in this case, the y-axis). Therefore, the most common outcome in the 600 trials (a die roll of 2, which was the outcome of 107 trials) also has the highest empirical

Additional Thoughts on ANOVA

297

GENERAL LINEAR • The membership of the groups is separate and non-overlapping. In other words, the MOMENTSindependent place variable that divides your data into three or more groups must be a nominallevel variable with three or more mutually exclusive and exhaustive categories. statistical methods dependent variable is interval- or ratio-level so that means, variances, and residuals can • The in the General Linear be calculated with its data.

Framework, providing • The homogeneity of variance assumption applies to ANOVA. In Chapter 10 we saw that an overarching the homogeneity of variance assumption requires the variance of the groups to be similar. in an ANOVA need to have similar sample sizes. cognitive schema • Groups to make understanding If these assumptions look familiar to you it is because these are the exact same assumptions for the individual conceptst-test. This should be unsurprising because ANOVA was created to perform unpaired-samples comparisons simpler. of several group means without inflating Type I error and without conducting multiple NHSTs. Indeed, both the unpaired two-sample t-test and ANOVA are members of the GLM. General Linear Moment 11.1 in this text explains the relationship between the two procedures.

General Linear Moment 11.1 If you have been paying close attention to these General Linear Moment boxes, you know that all null hypothesis tests are closely related because they are all members of the GLM. The same is true with ANOVA because it is a generalization of the unpaired two-sample t-test.

Therefore, an unpaired two-sample t-test is just an ANOVA with two groups. This is demonstrated when conducting an ANOVA with only two groups. In the previous chapter we became familiar with the example from Fox and Anderegg’s (2014) study about appropriate behavior on social media. When we performed an unpaired two-sample t-test in Chapter 10, we decided to reject the null hypothesis (H 0 : X 1 ¼ X 2 Þ. Performing an ANOVA with the same data also results in a rejected null hypothesis – because they have the same null hypothesis when there are only two groups in an ANOVA. The decision to reject the null hypothesis for both NHSTs is not unique to this example. When there are two groups, ANOVA and the unpaired two-sample t-test will always produce the same decision about the null hypothesis. This is because when there are only two groups, the ANOVA will simplify to an unpaired two-sample t-test.

There is one important difference: in the unpaired two-sample t-test, d = .219, but in ANOVA, η2 = .0118. However, these effect sizes are not directly comparable because of the two different ways they are calculated and two different interpretations of them. However, there is a formula (modified from Durlak, 2009, p. 928) that converts one effect size to the other: 2

η ¼

!2 d pffiffiffiffiffiffiffiffiffiffiffiffiffi d2 þ 4

Plugging the Cohen’s d value into this equation produces: !2 � �2 � �2 � �2 :219 :219 :219 :219 2 ¼ :1088492 η ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 2:011955 :047961 þ 4 4:047961 :2192 þ 4 η2 ¼ :1088492 ¼ :0118

SOFTWARE GUIDES explain how to conduct basic statistical tests in the two most common programs for statistical analysis in the social sciences: Excel and SPSS.

Reflection Questions: Application

319

Software Guide Because ANOVA is merely an extension of the unpaired two-sample t-test, it should be unsurprising that both Microsoft Excel and SPSS can perform an ANOVA. This software guide provides step-bystep instructions for conducting an ANOVA in each program.

Excel With the default settings, Microsoft Excel cannot conduct an ANOVA quickly. However, with the Analysis ToolPak add-in installed, it is easy to conduct an ANOVA. To install the Toolpak, click the “File” ribbon and then select “Options” (see Figure 11.9).

Figure 11.9 First step in installing the Analysis ToolPak in Microsoft Excel. This is the file ribbon. After clicking this, the user should then click “Options.” In this version of Microsoft Excel, it is the second item from the bottom.

Written by a quantitative psychologist, this textbook explains complex statistics in accessible language to undergraduates in all branches of the social sciences.

Built around the central framework of the General Linear Model (GLM), Statistics for the Social Sciences teaches students how different statistical methods are interrelated to one another. With the GLM as a basis, students with varying levels of background are better equipped to interpret statistics and learn more advanced methods in their later courses. The author makes statistics relevant to students’ varying majors by using fascinating real-life examples from the social sciences.

Features • Written in an accessible way so students of varying backgrounds can comprehend the material.

• Strong pedagogical features include learning goals, guided practices, and reflection questions in comprehension and application.

• Includes software guides in Excel and SPSS for students and instructors.

Russell T. Warne is an assistant professor of psychology at Utah Valley University.

Online Resources www.cambridge.org/warne Power Points Test banks

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