Heiman G. Basic Statistics for the Behavioral Sciences

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308 CHAPTER 13 / The One-Way Analysis of Variance

Step 1 Find . Using the appropriate value of in the computations is what protects

the experiment-wise error for the number of means being compared. Find the value of

in Table 6 in Appendix C, entitled “Values of Studentized Range Statistic.” In the

table, locate the column labeled with the corresponding to the number of means in

your factor. Next, find the row labeled with the used to compute your . Then

find the value of for the appropriate .

For our study above, , , and , so .

Step 2 Compute the HSD.

5 3.77 5 .05df

5 12k 5 3

q

obt

is from the ANOVA, and is the number of scores in each level of the factor.

In the example, was 7 and was 5, so

Step 3 Determine the differences between each pair of means. Subtract each mean

from every other mean. Ignore whether differences are positive or negative (for each

pair, this is a two-tailed test of ).

The differences for the perceived difficulty study can be diagramed as shown

below:

Perceived Difficulty

Easy Medium Difficult

苶

 8 X

苶

 6 X

苶

 3

HSD  4.46

On the line connecting any two levels is the absolute difference between their means.

Step 4 Compare each difference to the HSD. If the absolute difference between two

means is greater than the HSD, then these means differ significantly. (It’s as if you per-

formed a -test on these means and was significant.) If the absolute difference

between two means is less than or equal to the HSD, then it is not a significant differ-

ence (and would not produce a significant ).

Above, the HSD was 4.46. The means from the easy level (8) and the difficult level

(3) differ by more than 4.46, so they differ significantly. The mean from the medium

level (6), however, differs from the other means by less than 4.46, so it does not differ

significantly from them.

obt

: 

2 

5 0

HSD 5 13.772a

b5 4.46

nMS

The formula for Tukey’s HSD test is

HSD 5 1q

Summary of Steps in Performing a One-Way ANOVA 309

■

Perform post hoc comparisons when is signifi-

cant to determine which levels differ significantly.

■

Perform Tukey’s HSD test when all are equal and

Fisther’s -test when are unequal.

MORE EXAMPLES

An is significant, with , , and

and , , and .

To compute Fisher’s -test on and ,

and These means do not

differ significantly.

To compute Tukey’s HSD, find For and

, . Then:

HSD 5 1q

b5 13.492a

20.61

b5 4.78

5 3.49df

5 30

k 5 3q

crit

5 ;2.042.t

obt

521.446

22.8

13.75

obt

2 X

4.0 2 6.8

20.61 a

5 30MS

5 20.61n 5 11X

5 6.8

5 1.5X

5 4.0F

obt

nst

obt

The differences are ;

;

Comparing each difference to ,

only and differ significantly.

For Practice

We have , , and ,

with in each condition, , and

1. Which post hoc test should we perform?

2. What is here?

3. What is the HSD?

4. Which means differ significantly?

Answers

1. Tukey’s HSD

2. For and ,

4. Only and differ significantly.X

HSD 5 13.4021163.44>2125 5.91

5 3.40.df

5 60k 5 3

5 60

5 63.44n 5 21

5 8.92X

5 11.50X

5 16.50

HSD 5 4.785 22.8.

2 X

5 4.0 2 6.8X

2 X

5 1.5 2 6.8 525.3

2 X

5 4.0 2 1.5 5 2.5

A QUICK REVIEW

SUMMARY OF STEPS IN PERFORMING A ONE-WAY ANOVA

It’s been a long haul, but here is everything we do when performing a one-way,

between-subjects ANOVA.

1. Create the hypotheses. The null hypothesis is , and the

alternative hypothesis is : not all are equal.

2. Compute . Compute the sums of squares and the degrees of freedom. Then

compute and Then compute .

3. Compare to Find , using and If is larger than ,

then is significant, indicating that the means in at least two conditions differ

significantly.

obt

crit

obt

.df

crit

obt

.MS

obt

␮sH

: ␮

5 ␮

5 . . . ␮

Thus, our final conclusion about this study is that we demonstrated a relationship

between scores and perceived difficulty, but only for the easy and difficult conditions. If

these two conditions were given to the population, we would expect to find one

population for easy with a around 8 and another population for difficult with a

around 3. We cannot say anything about the medium level, however, because it did not

produce a significant difference. Finally, as usual, we would now interpret the results in

terms of the behaviors being studied, explaining why this manipulation worked as it did.

␮␮

310 CHAPTER 13 / The One-Way Analysis of Variance

4. Perform post hoc tests. If is significant and there are more than two levels of

the factor, determine which levels differ significantly by performing post hoc

comparisons. Perform Fisher’s -test if the ns are not equal or perform Tukey’s

HSD test if all are equal.

If you followed all of that, then congratulations, you’re getting good at this stuff. Of

course, all of this merely determines whether there is a relationship. Now you must

describe that relationship.

ADDITIONAL PROCEDURES IN THE ONE-WAY ANOVA

The following presents procedures for describing a significant relationship that we have

discussed in previous chapters, except that here they have been altered to accommodate

the computations in ANOVA.

The Confidence Interval for Each Population

As usual, we can compute a confidence interval for the represented by the mean of any

condition. This is the same confidence interval for that was discussed in Chapter 11,

but the formula is slightly different.



␮

obt

The is the two-tailed value found in the -tables using the appropriate and using

from the ANOVA as the is also from the ANOVA. The and are from

the level we are describing.

For example, in the easy condition, , , , and .

The two-tailed (at and ) is Thus,

This becomes

and finally,

Thus, if we were to test the entire population under our easy condition, we are 95%

confident that the population mean would fall between 5.42 and 10.58.

Follow the same procedure to describe the from any other significant level of the

factor.



5.42 #  # 10.58

122.57821 8.0 #  # 112.57821 8.0

7.0

b122.17921 8.0 #  # a

7.0

b112.17921 8.0

;2.179. 5 .05df 5 12t

crit

n 5 5df

5 12MS

5 7.0X 5 8.0

nXMS

df.df

tt

crit

The formula for the confidence interval for a single is

b12t

crit

21 X #  # a

b11t

crit

21 X

␮

Additional Procedures in the One-Way ANOVA 311

FIGURE 13.2

Mean number of

problems correctly

solved as a function of

perceived difficulty

Graphing the Results in ANOVA

As usual, graph your results by placing the dependent variable on the axis and the

independent variable on the axis. Then plot the mean for each condition. Figure 13.2

shows the line graph for the perceived difficulty study. Note that we include the

medium level of difficulty, even though it did not pro-

duce significant differences.

As usual, the line graph summarizes the relationship

that is present, and here it indicates a largely negative

linear relationship.

Describing Effect Size in

the ANOVA

Recall that in an experiment we describe the effect

size, which tells us how large of an impact the inde-

pendent variable had on dependent scores. The way to

do this is to compute the proportion of variance

accounted for, which tells us the proportional improve-

ment in predicting participants’ scores that we achieve

by predicting the mean of their condition. In ANOVA,

we compute this by computing a new correlation coefficient, called eta (pronounced

“ay-tah”), and then squaring it: eta squared indicates the proportion of variance in the

dependent variable that is accounted for by changing the levels of a factor in an

ANOVA. The symbol for eta squared is .

Both and are computed in the ANOVA. The reflects the differences

between the conditions. The reflects the total differences between all scores in the

experiment. Thus, reflects the proportion of all differences in scores that are associ-

ated with the different conditions.

For example, for our study, was 63.33 and was 147.33. So,

This is interpreted in the same way that we previously interpreted The larger

the , the more consistently the factor “caused” participants to have a particular

score in a particular condition, and thus the more scientifically important the factor is

for explaining and predicting differences in the underlying behavior. Thus, our

of .43 indicates that we are 43% more accurate at predicting participants’ scores

when we predict for them the mean of the difficulty level they were tested under. In

other words, 43% of all differences in these scores are accounted for (“caused”) by



tot

63.33

147.33

5 .43

tot



tot

Easy

Perceived difficulty

Medium Difficult

Mean problems correct

The formula for eta squared is



tot

When all is said and done, the -ratio is a convoluted way of measuring the differences

between the means of our conditions and then fitting those differences to a sampling

distribution. The larger the , the less likely that the means are representing the same

. A significant indicates that the means are unlikely to represent one population

mean. Then we determine which sample means actually differ significantly and

describe the relationship they form. That’s all there is to it.

There is, however, one other type of procedure that you should be aware of. All of

the research designs in this book involve one dependent variable, and the statistics we

perform are called univariate statistics. We can, however, measure participants on two

or more dependent variables in one experiment. Statistics for multiple dependent vari-

ables are called multivariate statistics. These include the multivariate -test and the

multivariate analysis of variance (MANOVA). Even though these are very complex

procedures, the basic logic still holds: The larger the or , the less likely it is that

the samples represent no relationship in the population. To discuss multivariates further

would require another book.

Using the SPSS Appendix As discussed in Appendix B.7, SPSS will perform the

one-way between-subjects ANOVA. This includes reporting the significance level of

obt



obt

PUTTING IT

ALL TOGETHER

312 CHAPTER 13 / The One-Way Analysis of Variance

changing the levels of perceived difficulty. Because 43% is a very substantial

amount, this factor is important in determining participants’ performance, so it is

important for scientific study.

Recall that our other measure of effect size is Cohen’s , which describes the magni-

tude of the differences between our means. However, with three or more levels, the pro-

cedure is extremely complicated. Therefore, instead, is the preferred measure of

effect size in ANOVA.

STATISTICS IN PUBLISHED RESEARCH: REPORTING ANOVA

In research reports, the results of an ANOVA are reported in the same ways as with pre-

vious procedures. However, now we are getting to more complicated designs, so there

is an order and logic to the report. Typically, we report the means and standard devia-

tion from each condition first. (Rather than use an incredibly long sentence for this,

often a table is presented.) Then we describe the characteristics of the primary ANOVA

and report the results. Then we report any secondary procedures. Thus, for our per-

ceived difficulty study, you might see:

“A one-way, between-subjects ANOVA was performed on the scores from

the three levels of perceived difficulty. The results were significant,

, p< A Tukey HSD test revealed that only the means for

the easy and difficult conditions differed significantly (p< ). This

manipulation accounted for .43 of the variance in scores (using ). The

95% confidence interval for the easy condition is . . .”

Notice that for , we report and then We also indicate that the Tukey

test was performed, although usually the HSD value is not reported. The alpha

level we used is reported (as ), as is a summary of the levels that differ sig-

nificantly. Then we report other secondary analyses, such as and confidence

intervals.



p 6 .05

.df

obt

␩

.05

.05F121225 4.52



Chapter Summary 313

, performing the HSD test, and graphing the means. The program also computes,

the , , and 95% confidence interval for for each level. It does not compute .

(The “partial eta squared” is not what we’ve discussed.

Appendix B.11 describes using SPSS to perform the one-way, within-subjects

ANOVA (discussed in Appendix A.3). The program provides the same information as

above, except that it does not perform the HSD test.

CHAPTER SUMMARY

1. The general terms used previously and their corresponding ANOVA terms are

shown in this table:

General Term ⴝ ANOVA Term

independent variable ⫽ factor

condition ⫽ level of treatment

sum of squared deviations ⫽ sum of squares (SS)

variance (s

) ⫽ mean square (MS)

effect of independent variable ⫽ treatment effect

2. A one-way analysis of variance tests for significant differences between the

means from two or more levels of a factor. In a between-subjects factor, each

condition involves an independent sample. In a within-subjects factor, the

conditions involve related samples.

3. The experiment-wise error rate is the probability that a Type I error will occur in

an experiment. ANOVA keeps the experiment-wise error rate equal to

4. ANOVA tests the that all being represented are equal; is that not all

are equal.

5. The mean square within groups ( ), measures the differences among the

scores within the conditions. The mean square between groups , measures

the differences among the level means.

6. When is true, and estimate the error variance. When is false,

also reflects added treatment variance.

7. is computed using the -ratio, which equals the mean square between groups

divided by the mean square within groups.

8. may be greater than 1 because either (a) there is no treatment effect, but the

sample data are not perfectly representative of this, or (b) two or more sample

means represent different population means.

9. If is significant with more than two levels, perform post hoc comparisons to

determine which means differ significantly. When the are not equal, perform

Fisher’s protected -test on each pair of means. If all are equal, perform Tukey’s

HSD test.

10. Eta squared describes the effect size—the proportion of variance in depen-

dent scores accounted for by the levels of the independent variable.

1␩

nst

obt

1MS

␮sH

␣.

␩

␮s

obt

314 CHAPTER 13 / The One-Way Analysis of Variance

KEY TERMS

HSD

analysis of variance 292

ANOVA 290

between-subjects ANOVA 291

between-subjects factor 291

error variance 298

eta squared 311

experiment-wise error rate 293

factor 290

-distribution 304

Fisher’s protected -test 307

-ratio 297

level 290



treat

error

crit

obt

mean square 295

mean square between groups 295

mean square within groups 295

multivariate statistics 312

one-way ANOVA 291

post hoc comparisons 294

sum of squares 295

treatment 290

treatment effect 290

treatment variance 298

Tukey’s HSD multiple comparisons

test 307

univariate statistics 312

within-subjects ANOVA 291

within-subjects factor 291

REVIEW QUESTIONS

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

1. What does each of the following terms mean? (a) ANOVA (b) one-way design

2. (a) What is the difference between and ? (b) What does stand for?

3. What are two reasons for conducting a study with more than two levels of a

factor?

4. (a) What are error variance and treatment variance? (b) What are the two types of

mean squares, and what does each estimate?

5. (a) What is the experiment-wise error rate? (b) Why does ANOVA solve the prob-

lem with the experiment-wise error rate created by multiple -tests?

6. Summarize the steps involved in analyzing an experiment when

7. (a) When is it necessary to perform post hoc comparisons? Why? (b) When is it

unnecessary to perform post hoc comparisons?

8. When do you use each of the two post hoc tests discussed in this chapter?

9. What does indicate?

10. (a) Why should equal 1 if the data represent the situation? (b) Why is

greater than 1 when the data represent the situation? (c) What does a

significant indicate about differences between the levels of a factor?

11. A research article reports the results of a “multivariate” analysis. What does this

term communicate about the study?

APPLICATION QUESTIONS

12. A researcher conducts an experiment with three levels of the independent

variable. (a) Which two versions of a parametric procedure are available to her?

(b) How does she choose between them?

13. (a) In a study comparing four conditions, what is for the ANOVA? (b) What is

in the same study? (c) Describe in words what and say for this study.H

obt



k 7 2.

kNn

Application Questions 315

14. Foofy obtained a significant from an experiment with five levels. She

therefore concludes that changing each condition of the independent variable

results in a significant change in the dependent variable. (a) Is she correct? Why

or why not? (b) What must she now do?

15. (a) Poindexter computes an of .63. How should this be interpreted? (b) He

computes another of How should this be interpreted?

16. A report says that the between-subjects factor of participants’ salary produced sig-

nificant differences in self-esteem. (a) What does this tell you about the design?

(b) What does it tell you about the results?

17. A report says that a new diet led to a significant decrease in weight for a group of

participants. (a) What does this tell you about the design? (b) What do we call this

design?

18. A researcher investigated the number of viral infections people contract as a

function of the amount of stress they experienced during a 6-month period.

She obtained the following data:

Amount of Stress

Negligible Minimal Moderate Severe

Stress Stress Stress Stress

2465

1357

4278

1354

(a) What are and ? (b) Compute and complete the ANOVA summary

table. (c) With what is ? (d) Report your statistical results. (e) Perform

the appropriate post hoc comparisons. (f) What do you conclude about this study?

(g) Describe the effect size and interpret it. (h) Estimate the value of that is likely

to be found in the severe stress condition.

19. Here are data from an experiment studying the effect of age on creativity scores:

Age 4 Age 6 Age 8 Age 10

3997

51112 7

714 9 6

410 8 4

310 9 5

(a) Compute and create an ANOVA summary table. (b) With what

do you conclude about ? (c) Perform the appropriate post hoc comparisons.

What should you conclude about this relationship? (d) Statistically, how important

is the relationship in this study? (e) Describe how you would graph these results.

20. In a study in which , , , , and , you

compute the following sums of squares:

Source Sum of Squares df Mean Square F

Between 147.32

㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Within 862.99 㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Total 1010.31 㛬㛬㛬㛬㛬㛬

5 8.2X

5 16.9X

5 45.3n 5 21k 5 3

obt

 5 .05,F

obt



crit

 5 .05,

obt

21.7.F

obt

316 CHAPTER 13 / The One-Way Analysis of Variance

(a) Complete the ANOVA summary table. (b) With , what do you conclude

about ? (c) Report your results in the correct format. (d) Perform the appropri-

ate post hoc comparisons. What do you conclude about this relationship? (e) What

is the effect size in this study, and what does this tell you about the influence of the

independent variable?

21. Performing ANOVA protects our experimentwise error rate. (a) Name and explain

the error that this avoids. (b) Why is this an important error to avoid?

22. A researcher investigated the effect of volume of background noise on partici-

pants’ accuracy rates while performing a boring task. He tested three groups

of randomly selected students and obtained the following means and sums of

squares:

Low Volume Moderate Volume High Volume

苶

61.5 65.5 48.25

n 457

Source Sum of Squares df Mean Square F

Between groups 652.16

㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Within groups 612.75 㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬㛬

Total 1264.92 㛬㛬㛬㛬㛬㛬

(a) Complete the ANOVA (b) At , what is ? (c) Report the statistical results

in the proper format. (d) Perform the appropriate post hoc tests. (e) What do you con-

clude about this study? (f) Compute the effect size and interpret it.

INTEGRATION QUESTIONS

23. (a) How do we create related samples? (b) In part (a) what two inferential proce-

dures are appropriate? (c) How do we create independent samples? (d) In part

24. (a) In this chapter we tested the relationship between performance scores and

perceived difficulty. Describe this relationship using “as a function of.”

(b) An experimenter computes the mean anxiety level for samples of freshmen,

sophomores, juniors, and seniors. To graph these results, how would you label

the axes? (c) Would this be a bar or line graph and why? (Chs. 2, 4)

25. (a) How is h similar to ? (b) How do we interpret this proportion? (Chs. 12, 13)

26. (a) Name and explain the error that power prevents. (b) Why is it important to

maximize the power of any experiment? (c) How is this done in a design using

ANOVA? (d) How does influencing the differences between and within conditions

influence the size of and its likelihood of being significant? (Chs. 10, 11, 13)

27. (a) How do you identify which variable in a study is the factor? (b) How do you

identify the levels of a factor? (c) How do you identify the variable that is the

dependent variable? (Chs. 2, 13)

28. For the following, identify the inferential procedure to perform and the key infor-

mation for answering the research question. Note: If no inferential procedure is

appropriate, indicate why. (a) Doing well in statistics should reduce students’

obt

crit

 5 .05

obt

 5 .05

Integration Questions 317

math phobia. We measure their math phobia after selecting groups who received

either an A, B, C, or D in statistics. (b) To determine if recall is better or worse

than recognition, participants study a list of words, and then half of them recall

the words and the other half performs a recognition test. (c) We repeatedly test the

aggressiveness of rats after 1, 3, 5 and 7 weeks, to see if they become more

aggressive as they grow older. (d) Using the grades of 100 students, we want to

use students’ score on the first exam to predict their final exam grade. (e) We test

for gender differences in voting history by counting the number of males and

females who voted in the last election. (f) We ask if pilots are quicker than naviga-

tors, comparing the reaction time of a group of pilots to that of their navigators.

(Chs. 7, 8, 10, 12, 13)

29. In question 28, identify the levels of the factor and the dependent variable in

experiments, and the predictor/criterion variables in correlational studies.

(Chs. 8, 12, 13)

■ ■ ■ SUMMARY OF

FORMULAS

The format for the summary table for a one-way

ANOVA is

Summary Table of One-Way ANOVA

Source Sum of Squares df Mean Square F

Between SS

obt

Within SS

Total SS

tot

1. To perform the one-way, between-subjects

ANOVA

a. Compute the sum of squares,

b. Compute the mean squares,

5 N 2 kMS

5 k 2 1MS

5 SS

tot

2 SS

2 a

1©X

tot

5 © a

1Sum of scores in the column2

n of scores in the column

tot

5 ©X

tot

2 a

1©X

tot

c. Compute the -ratio

2. The formula for the protected -test is

Use the two-tailed for

3. The formula for the HSD is

Values of are found in Appendix C, Table 6.

4. The formula for the confidence interval for is

Use the two-tailed for

5. The formula for eta squared is



tot

df 5 df

crit

b12t

crit

21 X #  # a

b11t

crit

21 X



HSD 5 1q

df 5 df

crit

obt

2 X

obt