Heiman G. Basic Statistics for the Behavioral Sciences

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178 CHAPTER 8 / Linear Regression

We also use when comparing different relationships to see which is more informa-

tive. Say that we find a relationship between the length of a person’s hair and his or her

creativity, but is only . Yes, this indicates a relationship, but such a weak one is

virtually useless. The fact that indicates that knowing someone’s hair length

improves our knowledge about their creativity by only four-hundredths of one percent!

However, say that we also find a relationship between age and creativity, and here is

. This relationship is more important, at least in a statistical sense, because

. Age is the more important variable because knowing participants’ ages gets

us 16% closer to accurately predicting their creativity. Knowing their hair length gets

us only closer to predicting their creativity.

The logic of is applied to any relationship. For example, in the previous chapter,

we discussed . Squaring this coefficient also indicates the proportion of variance

accounted for. (It is as if we performed the appropriate regression analysis, computed

and so on.) Likewise, in later chapters, we will determine the proportion of variance

accounted for in experiments. In all cases the answer indicates how useful the relation-

ship is.

REMEMBER Computing the proportion of variance accounted for is the way

to evaluate the scientific importance of any relationship.

.04%

5 .16

1.40

5 .0004

r1.02r

■

is the proportion of variance accounted for: the

proportional improvement in accuracy produced by

using a relationship to predict scores.

■

The larger the proportion of variance accounted

for, the greater the scientific importance of the

relationship.

MORE EXAMPLES

We correlate students’ scores on the first exam with

their scores on the final exam: , so

To predict final exam scores, we can ig-

nore the relationship and predict that everyone scored

at the mean of the final exam. Or we can use this rela-

tionship and regression techniques to compute for

each student. By using the relationship, we will be “on

average” 16% closer to each student’s actual final

exam score.

For Practice

Relationship A has ; relationship B has

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5 .16.

5r 51.40

1. The relationship with the scatterplot that hugs the

regression line more is _____.

2. The relationship with scores closest to the

scores that we’ll predict for them is ______.

3. As compared to predicting the for participants,

relationship A produces scores that are ______

closer to the scores.

4. As compared to predicting the for participants,

relationship B produces scores that are ______

closer to the scores.

5. Using relationship B to predict scores will improve

our accuracy by ______ times as much as will

using relationship A.

Answers

1. B

2. B

5. 4

.36

.09

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A QUICK REVIEW

Statistics in Published Research: Linear Regression 179

A WORD ABOUT MULTIPLE CORRELATION AND REGRESSION

Sometimes we discover several variables that each help us to more accurately predict

a variable. For example, a positive correlation exists between height and ability to

shoot baskets in basketball: The taller people are, the more baskets they tend to make.

Also, a positive correlation exists between how much people practice basketball and

their ability to shoot baskets: The more they practice, the more baskets they tend to

make. Obviously, to be as accurate as possible in predicting how well people will shoot

baskets, we should consider both how tall they are and how much they practice. This

example has two predictor variables (height and practice) that predict one criterion

variable (basket shooting). When we wish to simultaneously consider multiple predic-

tor variables for one criterion variable, we use the statistical procedures known as mul-

tiple correlation and multiple regression. Although the computations involved in these

procedures are beyond this text, understand that the multiple correlation coefficient,

called the multiple , indicates the strength of the relationship between the multiple

predictors taken together, and the criterion variable. The multiple regression equation

allows us to predict someone’s score by simultaneously considering his or her scores

on all variables. The squared multiple R is the proportion of variance in the vari-

able accounted for by using the relationship to predict scores.

STATISTICS IN PUBLISHED RESEARCH: LINEAR REGRESSION

In addition to multiple correlation, you may encounter studies that use other, advanced

versions of correlation and regression. Understand that the basic approach in these pro-

cedures is also to summarize the strength and type of relationship that is present, and to

use an score to predict a central, summary score.

In reports of a regression analysis, you will sometimes see our , but you may also

see the symbol . Our other symbols are generally also found in publications, but a vari-

ation of the slope—b—may be referred to as “beta” and “ ” Also, when researchers

discuss measures of prediction error, they often use the term residual. Finally, you may

not always see scatterplots in published reports. Instead a graph showing only the

regression line may be included, and from the size of and , you must envision the

general scatterplot.

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This and the previous chapter introduced many new symbols and concepts. However,

they boil down to four major topics:

1. The correlation coefficient: The correlation coefficient communicates the type and

strength of a relationship. The larger the coefficient, the stronger is the relationship:

the more consistently one value of is paired with one value of and the closer the

data come to forming a perfect straight-line relationship.

2. The regression equation: The regression equation allows you to draw the regression

line through the scatterplot and to use the relationship with to predict any individ-

ual’s score.

3. Errors in prediction: The standard error of the estimate indicates the “average”

amount your predictions will be in error when using a particular relationship.

PUTTING IT

ALL TOGETHER

180 CHAPTER 8 / Linear Regression

4. The proportion of variance accounted for: By squaring a correlation coefficient,

you know how much smaller the errors in predicting scores are when you use the

relationship, compared to if you do not use the relationship.

Using the SPSS Appendix Appendix B.4 explains how to compute the components

of the linear regression equation as well as the standard error of the estimate (although

it is an estimated population version—as if in our defining formula for we divided

by instead of ).

CHAPTER SUMMARY

1. Linear regression is the procedure for predicting unknown scores based on

correlated scores. It produces the linear regression line, which is the best-fitting

straight line that summarizes a linear relationship.

2. The linear regression equation includes the slope, indicating how much and in

what direction the regression line slants, and the intercept, indicating the value

of when the line crosses the axis.

3. For each , the regression equation produces , which is the predicted score for

that

4. The standard error of the estimate is interpreted as the “average error” when

using to predict scores. This error is also summarized by the variance of the

scores around

5. With regression we assume that (1) the scores are homoscedastic, meaning that

the spread in the scores around all scores is the same, and (2) the scores at

each are normally distributed around their corresponding value of

6. The stronger the relationship, the smaller are the values of and because then

the scores are closer to and so the smaller the difference between and

7. The proportion of variance accounted for is the proportional improvement in

accuracy that is achieved by using the relationship to predict scores, compared

to using to predict scores. This coefficient of determination is computed by

squaring the correlation coefficient.

8. The proportion of variance not accounted for—the coefficient of alienation—is

the proportion of the prediction error that is not eliminated when is the

predicted score instead of

9. The proportion of variance accounted for indicates the statistical importance of a

relationship.

10. Multiple correlation and multiple regression are procedures for describing the

relationship when multiple predictor variables are simultaneously used to

predict scores on one criterion variable. 1Y2

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Application Questions 181

KEY TERMS

coefficient of alienation 177

coefficient of determination 177

criterion variable 162

heteroscedasticity 172

homoscedasticity 171

linear regression equation 163

linear regression line 161

multiple correlation coefficient 179

abY¿

multiple regression equation 179

predicted score 161

predictor variable 162

proportion of variance accounted 174

slope 163

standard error of the estimate 170

variance of the scores around 169

intercept 163Y

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REVIEW QUESTIONS

(Answers for odd-numbered questions are in Appendix D.)

1. What is the linear regression line?

2. What is the linear regression procedure used for?

3. What is and how do you obtain it?

4. What is the general form of the linear regression equation? Identify its

component symbols.

5. (a) What does the intercept indicate? (b) What does the slope indicate?

6. Distinguish between the predictor variable and the criterion variable in linear

regression.

7. (a) What is the name for ? (b) What does tell you about the spread in the

scores? (c) What does tell you about your errors in prediction?

8. (a) What two assumptions must you make about the data in order for the standard

error of the estimate to be accurate, and what does each mean? (b) How does

heteroscedasticity lead to an inaccurate description of the data?

9. How is the value of related to the size of ? Why?

10. When are multiple regression procedures used?

11. (a) What are the two statistical names for ? (b) How do you interpret ?

APPLICATION QUESTIONS

12. What research steps must you go through to use the relationship between a

person’s intelligence and grade average in high school so that, if you know a

person’s IQ, you can more accurately predict the person’s grade average?

13. We find that the correlation between math ability (X) and musical aptitude scores

(Y ) is . The standard error of the estimate is . Bubbles has a

math score of 60 and Foofy a score of 72. (a) Based on their math scores, who is

predicted to have the higher music score and why? (b) What procedure would we

use to predict their music scores? (c) If our predictions are wrong, what is the

“average” amount we expect to be “off”? (d) How much smaller is this error than

if we don’t use math scores to predict music scores? (e) What is you answer in

part (d) called?

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182 CHAPTER 8 / Linear Regression

14. (a) Explain conceptually why the proportion of variance accounted for equals

1.0 with a perfect correlation. (b) Why should you expect most relationships to

account for only about 9% to 25% of the variance?

15. What do you know about a research project when you read that it employed

multiple correlation and regression procedures?

16. Poindexter finds when correlating number of hours studied and number

of errors made on a statistics test. He also finds between speed of taking

the test and number of errors on the test. He concludes that hours studied forms

twice as strong a relationship and is therefore twice as important as the speed of

taking the test. (a) Why is he correct or incorrect? (b) Compare these relationships

and draw the correct conclusion.

17. (a) In question 16 what advanced statistical procedures can Poindexter employ to

improve his predictions about test errors even more? (b) Say that the resulting

correlation coefficient is . Using the proportion of variance accounted for,

explain what this means.

18. A researcher finds that the correlation between variable A and variable B is

. She also finds that the correlation between variable C and variable B is

. Which relationship is scientifically more useful and by how much?

19. You measure how much people are initially attracted to a person of the opposite

sex and how anxious they become during their first date. For the following ratio

data, answer the questions below.

Attraction Anxiety

Participant XY

128

2614

315

438

5610

6915

768

868

947

10 2 6

(a) Compute the statistic that describes the relationship here. (b) Compute the

linear regression equation. (c) What anxiety score do you predict for a person who

has an attraction score of 9? (d) When using this relationship, what is the

“average” amount of error you should expect in your predictions?

20. (a) For the relationship in question 19, what is the proportion of variance

accounted for? (b) What is the proportion of variance not accounted for?

21. In question 19 of the Application Questions in Chapter 7, we correlated “burnout”

scores with absenteeism scores Using those data: (a) Compute the linear

regression equation. (b) Compute the standard error of the estimate. (c) For a

burnout of 4, what absence score is predicted? (d) How useful is knowing burnout

scores for predicting absenteeism?

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Integration Questions 183

22. A researcher measures how positive a person’s mood is and how creative he or she

is, obtaining the following interval scores:

Mood Creativity

Participant XY

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286

3911

464

555

637

774

825

946

10 1 4

(a) Compute the statistic that summarizes this relationship. (b) What is the

predicted creativity score for anyone scoring 3 on mood? (c) If your prediction is in

error, what is the amount of error you expect to have? (d) How much smaller will

your error be if you use the regression equation than if you merely used the overall

mean creativity score as the predicted score for all participants?

23. Dorcas complains that it is unfair to use SAT scores to determine college admittance

because she might do much better in college than predicted. (a) What statistic(s) will

indicate whether her complaint is likely to be correct? (b) In reality, the positive cor-

relation coefficient between SAT scores and college performance is only moderate.

What does this tell you about how useful the SAT is for predicting college success?

INTEGRATION QUESTIONS

24. (a) How are experiments and correlational designs similar in their purpose?

(b) What is the distinguishing characteristic between an experiment and a

correlational study? (Chs. 2, 7)

25. In the typical experiment, (a) Do we group the scores to summarize them, and if

so how? (b) What are the two descriptive statistics we compute for the summary?

(Chs. 4, 5)

26. In the typical correlational design, (a) Do we group the scores to summarize them,

and if so how? (b) What are the three descriptive statistics we compute for the

summary? (Chs. 7, 8)

27. In a typical experiment, a researcher says she has found a good predictor (a) What

name do we give to this variable? (b) Scores on which variable are being predicted?

(d) What statistic describes the “average” error in predictions? (Chs. 4, 5)

28. In a typical correlational study, a researcher says he has found a good predictor

(a) What name do we give to this variable? (b) Scores on which variable are being

predicted? (c) For a particular participant, what score will be the predicted score?

(d) What statistic describes the “average” error in predictions? (Chs. 7, 8)

29. You know that a relationship accounts for a substantial amount of variance. What

does this tell you about (a) its strength? (b) the consistency that a particular is

paired with only one , (c) the variability in scores at each ? (d) how closely

the scatterplot hugs the regression line? (e) the size of the correlation coefficient?

(f) how well we can predict scores? (Chs. 7, 8)Y

XYX

184 CHAPTER 8 / Linear Regression

■ ■ ■ SUMMARY OF

FORMULAS

1. The formula for the linear regression equation is

2. The formula for the slope of the linear

regression line is

3. The formula for the intercept of the linear

regression line is

4. The formula for the variance of scores around

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2 1b21X2

b 5

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5. The formula for the standard error of the

estimate is

6. The formula for the proportion of variance

accounted for is

7. The formula for the proportion of variance not

accounted for is

Coefficient of determination 5 1 2 r

Coefficient of determination 5 r

5 S

21 2 r

You now know the common descriptive statistics used in behavioral research. There-

fore, you are ready to learn the other type of procedures, called inferential statistics.

Later chapters will show you the different procedures that are used with different

research designs. This chapter sets the foundation for all inferential procedures by

introducing you to the wonderful world of probability. Don’t worry, though. The

discussion is rather simple, and there is little in the way of formulas. However, you

do need to understand the basics of probability. In the following sections, we discuss

(1) what probability is, (2) how we determine probability, and (3) how to use probabil-

ity to draw conclusions about samples.

NEW STATISTICAL NOTATION

In daily conversation, the words chances, odds, and probability are used interchange-

ably. In statistics, however, they have different meanings. Odds are expressed as

fractions or ratios (“The odds of winning are 1 in 2”). Chance is expressed as a per-

centage (“There is a 50% chance of winning”). Probability is expressed as a decimal

(“The probability of winning is .50”). For inferential procedures, always express the

answers you compute as probabilities.

185

Using Probability to Make

Decisions about Data

GETTING STARTED

To understand this chapter, recall the following:

■

From Chapter 3, that relative frequency is the proportion of time that scores

occur.

■

From Chapter 6, that by using z-scores we can determine the proportion of the

area under the normal curve for particular scores and thereby determine their

relative frequency.

■

Also from Chapter 6, that by using a sampling distribution of means and

z-scores we can determine the proportion of the area under the normal curve

for particular sample means and thereby determine their relative frequency.

Your goals in this chapter are to learn

■

What probability is and how it is computed.

■

How to compute the probability of raw scores and sample means using z-scores.

■

How random sampling and sampling error may or may not produce

representative samples.

■

How to use a sampling distribution of means to determine whether a sample

represents a particular population.

186 CHAPTER 9 / Using Probability to Make Decisions about Data

The symbol for probability is p. The probability of a particular event—such as event

A—is p(A), which is pronounced “p of A” or “the probability of A.”

WHY IS IT IMPORTANT TO KNOW ABOUT PROBABILITY?

Recall that ultimately we want to draw inferences about relationships in nature—in

what we call the population. However, even though we may find a relationship in our

sample data, we can never “prove” that this relationship is found in the population. We

must measure the entire population to know what occurs, but if we could measure the

(infinite) population, we wouldn’t need samples to begin with. Thus, we will always

have some uncertainty about whether the relationship in our sample is found in the pop-

ulation. The best that we can do is to describe what we think is likely to be found. This

is where inferential statistics come into play. Recall from Chapter 2 that inferential

statistics are for deciding whether our sample data accurately represent the relation-

ship found in the population (in nature). Essentially, we use probability and inferential

procedures to make an intelligent bet about whether we would find the sample’s rela-

tionship if we could study everyone in the population. Therefore, it is important that

you understand the basics of probability so that you understand how we make these

bets. This is especially so, because we always have this uncertainty about the popula-

tion, so we perform inferential statistics in every study. Therefore, using probability—

and making bets—is an integral part of all behavioral research.

THE LOGIC OF PROBABILITY

Probability is used to describe random, or chance, events. People often mistakenly believe

that the definition of random is that events occur in a way that produces no pattern. How-

ever, the events may form a pattern, so, for example, a lottery might produce the random

sequence of 1, 2, 3, 4, 5. Rather, an event is random if it is uncontrolled, so that the events

occur naturally without any influence from us. That is, we allow nature to be fair, with no

bias toward one event over another (no rigged roulette wheels or loaded dice). Thus, a ran-

dom event occurs or does not occur merely because of the luck of the draw. Probability is

our way of mathematically describing how luck operates to produce an event.

Using statistical terminology, the event that does occur in a given situation is a sam-

ple. The larger collection of all possible events that might occur in this situation is the

population. Thus, the event might be drawing a particular playing card from the popu-

lation of a deck of cards, or tossing a coin and obtaining a particular sample of heads

and tails from the population of possible heads and tails. In research, the event is

obtaining a particular sample that contains particular participants or scores.

Because probability deals only with random events, we compute the probability only

of samples obtained through random sampling. Random sampling is selecting a sam-

ple in such a way that all elements or individuals in the population have an equal

chance of being selected. Thus, in research, random sampling is anything akin to

blindly drawing names from a large hat. In all cases, a particular random sample occurs

or does not occur simply because of the luck of the draw.

But how can we describe an event that happens only by chance? By paying attention

to how often the event occurs when chance is operating. The probability of any event is

based on how often the event occurs over the long run. Intuitively, we use this logic all

the time: If event A happens frequently over the long run, then we think it is likely to

The Logic of Probability 187

happen again now, and we say that it has a high probability. If event B happens infre-

quently, then we think it is unlikely, and we say that it has a low probability.

When we decide that an event happens frequently, we are making a relative judgment,

describing the event’s relative frequency. This is the proportion of time that the event

occurs out of all events that might occur from the population. This is also the event’s

probability. The probability of an event is equal to the event’s relative frequency in the

population of possible events that can occur. An event’s relative frequency is a number

between 0 and 1, so the event’s probability is from 0 to 1.

REMEMBER The probability of an event equals the event’s relative frequency

in the population.

Probability is essentially a system for expressing our confidence that a particular ran-

dom event will occur. First, we assume that an event’s past relative frequency will con-

tinue over the long run in the future. Then we express our confidence that the event will

occur in any single sample by expressing the relative frequency as a probability

between 0 and 1. For example, I am a rotten typist, and I randomly make typos 80% of

the time. This means that in the population of my typing, typos occur with a relative

frequency of .80. We expect the relative frequency of my typos to continue at a rate of

.80 in anything I type. This expected relative frequency is expressed as a probability:

the probability is .80 that I will make a typo when I type the next woid.

Likewise, all probabilities communicate our confidence. If event A has a relative fre-

quency of zero in a particular situation, then the probability of event A is zero. This

means that we do not expect A to occur in this situation because it never does. But if

event A has a relative frequency of .10 in this situation, then A has a probability of .10:

Because we expect it to occur in only 10% of our samples, we have some—but not

much—confidence that A will occur in the next sample. On the other hand, if A has a

probability of .95, we are confident that it will occur: It occurs 95% of the time in the

population, we expect it in 95% of our samples, and so our confidence is at .95 that it

will occur now. At the most extreme, an event’s relative frequency can be 1: It is 100%

of the population, so its probability is 1. Here, we are positive it will occur in this situa-

tion because it always does.

All possible events together constitute 100% of the population. This means that the

relative frequencies of all events must add up to 1, so the probabilities must also add up

to 1. Thus, if the probability of my making a typo is .80, then because ,

the likelihood that I will type error-free words is

Understand that except when equals either 0 or 1, it is up to chance whether a partic-

ular sample contains the event. For example, even though I make typos 80% of the time,

I may go for quite a while without making one. That 20% of the time I make no typos has

to occur sometime. Thus, although the probability is that I will make a typo in each

word, it is only over the long run that we expect to see precisely 80% typos.

People who fail to understand that probability implies over the long run fall victim

to the “gambler’s fallacy.” For example, after observing my errorless typing for a while,

the fallacy is thinking that errors “must” occur now, essentially concluding that errors

have become more likely. Or, say we are flipping a coin and get seven heads in a row.

The fallacy is thinking that a head is now less likely to occur because it’s already

occurred too often (as if the coin says, “Hold it. That’s enough heads for a while!”).

The mistake of the gambler’s fallacy is failing to recognize that whether an event

occurs or not in a sample does not alter its probability because probability is deter-

mined by what happens “over the long run.”

.80

p 5 .20.

1 2 .80 5 .20