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9 Sample Size for Power = .80 in Single-Sample Case

Chapter 12

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Robey and Barcikowski (1984) have given power tables for various alpha levels for the

single group repeated-measures design. Their tables assume a common correlation for the

repeated measures, which generally will not be tenable (especially in longitudinal studies);

however, a later paper by Green (1990) indicated that use of an estimated average correlation (from all the correlations among the repeated measures) is fine. Selected results from

their work are presented in TableÂ€12.9, which indicates sample size needed for powerÂ€=Â€.80

for small, medium, and large effect sizes at alphaÂ€=Â€.01, .05, .10, and .20 for two through

seven repeated measures. We give two examples to show how to use the table.

TableÂ€12.9:â•‡ Sample Sizes Needed for PowerÂ€=Â€.80 in Single-Group Repeated

Measures

Number of repeated measures

Effect sizea

2

.12

.30

.49

.14

.35

.57

.22

.56

.89

404

68

28

298

51

22

123

22

11

.12

.30

.49

.14

.35

.57

.22

.56

.89

268

45

19

199

34

14

82

15

8

.30

.12

.30

.49

.50

.14

.35

.57

.22

.56

.89

Average corr.

.30

.50

.80

.30

.50

.80

.80

3

4

5

6

7

273

49

22

202

38

18

86

19

11

238

44

21

177

35

18

76

18

12

214

41

21

159

33

18

69

18

12

195

39

21

146

31

18

65

18

13

192

35

16

142

27

13

60

13

8

170

32

16

126

25

13

54

13

9

154

30

16

114

24

13

50

14

10

141

29

16

106

23

14

47

14

10

209

35

14

αÂ€=Â€.01

324

56

24

239

43

19

100

20

11

αÂ€=Â€.05

223

39

17

165

30

14

69

14

8

αÂ€=Â€.10

178

31

14

154

28

13

137

26

13

125

25

13

116

24

13

154

26

11

64

12

6

131

24

11

55

11

7

114

22

11

49

11

7

102

20

11

44

11

8

93

20

11

41

12

9

87

19

12

39

12

9

(Continuedâ•›)

495

496

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Repeated-Measures Analysis

TableÂ€12.9:â•‡ (Continued)

Number of repeated measures

Average corr.

.30

.50

.80

Effect sizea

2

3

4

5

6

7

.12

.30

.49

.14

.35

.57

.22

.56

.89

149

25

10

110

19

8

45

8

4

αÂ€=Â€.20

130

23

10

96

17

8

40

8

5

114

21

10

85

16

8

36

9

6

103

20

10

76

16

9

33

9

7

94

19

11

70

15

9

31

10

8

87

19

11

65

15

10

30

10

8

a

These are small, medium, and large effect sizes, and are obtained from the corresponding effect size

measures for independent samples ANOVA (i.e., .10, .25, and .40) by dividing by 1- correl. Thus, for example,

.10

.40

, and .57 =

14 =

.

1- .50

1- .50

Example 12.1

An investigator has a three treatment design: That is, each of the subjects is exposed

to three treatments. He uses rÂ€=Â€.80 as his estimate of the average correlation of the

subjects’ responses to the three treatments. How many subjects will he need for

powerÂ€=Â€.80 at the .05 level, if he anticipates a medium effect size?

Reference to TableÂ€12.9 with correlÂ€=Â€.80, effect sizeÂ€=Â€.56, kÂ€=Â€3, and αÂ€=Â€.05, shows

that only 14 subjects are needed.

Example 12.2

An investigator will be carrying out a longitudinal study, measuring the subjects at five

points in time. She wishes to detect a large effect size at the .10 level of significance,

and estimates that the average correlation among the five measures will be about .50.

How many subjects will she need?

Reference to TableÂ€12.9 with correlÂ€=Â€.50, effect sizeÂ€=Â€.57, kÂ€=Â€5, and αÂ€=Â€.10, shows

that 11 subjects are needed.

12.10 MULTIVARIATE MATCHED-PAIRS ANALYSIS

It was mentioned in ChapterÂ€4 that often in comparing intact groups the subjects are

matched or paired on variables known or presumed to be related to performance on

Chapter 12

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the dependent variable(s). This is done so that if a significant difference is found, the

investigator can be more confident it was the treatment(s) that “caused” the difference.

In ChapterÂ€4 we gave a univariate example, where kindergarteners were compared

against nonkindergarteners on first-grade readiness, after they were matched on IQ,

SES, and number of children in the family.

Now consider a multivariate example, that is, where there are several dependent

variables. Kvet (1982) was interested in determining whether excusing elementary

school children from regular classroom instruction for the study of instrumental

music affected sixth-grade reading, language, and mathematics achievement. These

were the three dependent variables. Instrumental and noninstrumental students from

four public school districts were used in the study. We consider the analysis from just

one of the districts. The instrumental and noninstrumental students were matched

on the following variables: sex, race, IQ, cumulative achievement in fifth grade,

elementary school attended, sixth-grade classroom teacher, and instrumental music

outside the school.

TableÂ€12.10 shows the control lines for running the analysis on SAS and SPSS. Note

that we compute three difference variables, on which the multivariate analysis is done,

and that it is these difference variables that are used in the MODEL (SAS) and GLM

(SPSS) statements. We are testing whether these three difference variables (considered

jointly) differ significantly from the 0 vector, that is, whether the group mean differences on all three variables are jointly 0.

Again we obtain a Tâ•›2 value, as for the single sample multivariate repeated-measures

analysis; however, the exact F transformation is somewhat different:

F=

N-p 2

T , with p and ( N - p ) df ,

( N - 1) p

where N is the number of matched pairs and p is the number of difference variables.

The multivariate test results shown in TableÂ€12.11 indicate that the instrumental

group does not differ from the noninstrumental group on the set of three difference

variables (FÂ€=Â€.9115, p < .46). Thus, the classroom time taken by the instrumental

group does not appear to adversely affect their achievement in these three basic academic areas.

12.11 ONE-BETWEEN AND ONE-WITHIN DESIGN

We now add a grouping (between) variable to the one-way repeated measures design.

This design, having one-between and one-within subjects factor, is often called a

split plot design. For this design, we consider hypothetical data from a study comparing the relative efficacy of a behavior modification approach to dieting versus a

497

82 83 69 99 63 66â•… 69 60 87 80 69

55 61 52 74 55 67â•… 87 87 88 99 95

91 99 99 99 99 87â•… 78 72 66 76 52

78 62 79 69 54 65â•… 72 58 74 69 59

85 99 99 75 66 61

END DATA.

COMPUTE ReaddiffÂ€=Â€read1-read2.

COMPUTE LangdiffÂ€=Â€lang1-lang2.

COMPUTE MathdiffÂ€=Â€math1-math2.

LIST.

GLM Readdiff Langdiff Mathdiff

/INTERCEPT=INCLUDE

/EMMEANS=TABLES(OVERALL)

/PRINT=DESCRIPTIVE.

71

82

74

58

DATA LIST FREE/read1 read2 lang1 lang2 math1 math2.

BEGIN DATA.

62 67 72 66 67 35â•… 95 87 99 96 82 82

66 66 96 87 74 63â•… 87 91 87 82 98 85

70 74 69 73 85 63â•… 96 99 96 76 74 61

85 99 99 71 91 60â•… 54 60 69 80 66 71

DATA MatchedPairs;

INPUT read1 read2 lang1 lang2 math1 math2;

LINES;

62 67 72 66 67 35

66 66 96 87 74 63

70 74 69 73 85 63

85 99 99 71 91 60

82 83 69 99 63 66

55 61 52 74 55 67

91 99 99 99 99 87

78 62 79 69 54 65

85 99 99 75 66 61

95 87 99 96 82 82

87 91 87 82 98 85

96 99 96 76 74 61

54 60 69 80 66 71

69 60 87 80 69 71

87 87 88 99 95 82

78 72 66 76 52 74

72 58 74 69 59 58

PROC PRINT DATAÂ€=Â€MatchedPairs;

RUN;

DATA MatchedPairs; SET MatchedPairs;

ReaddiffÂ€=Â€read1-read2;

LangdiffÂ€=Â€lang1-lang2;

MathdiffÂ€=Â€math1-math2;

RUN;

PROC GLM;

MODEL Readdiff Langdiff MathdiffÂ€=Â€/;

MANOVA H =INTERCEPT;

RUN;

SPSS

SAS

TableÂ€12.10:â•‡ SAS and SPSS Control Lines for Multivariate Matched-Pairs Analysis

Chapter 12

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TableÂ€12.11:â•‡ Multivariate Test Results for Matched Pairs Example

SAS Output

MANOVA Test Criteria and Exact F Statistics for the Hypothesis of No Overall Intercept Effect

HÂ€=Â€Type III SSCP Matrix for Intercept

EÂ€=Â€Error SSCP Matrix

S=1Â€M=0.5 N=6

Statistic

Value

F Value

NumÂ€DF

DenÂ€DF

PrÂ€>Â€F

Wilks’ Lambda

Pillai’s Trace

Hotelling-Lawley

Trace

Roy’s Greatest Root

0.83658794

0.16341206

0.19533160

0.91

0.91

0.91

3

3

3

14

14

14

0.4604

0.4604

0.4604

0.19533160

0.91

3

14

0.4604

SPSS Output

Multivariate Testsa

Effect

Intercept

a

b

Pillai’s Trace

Wilks’ Lambda

Hotelling’s Trace

Roy’s Largest Root

Value

F

Hypothesis df

Error df

Sig.

.163

.837

.195

.195

.912b

.912b

.912b

.912b

3.000

3.000

3.000

3.000

14.000

14.000

14.000

14.000

.460

.460

.460

.460

Design: Intercept

Exact statistic

behavior modification plus exercise approach (combination treatment) on weight

loss for a group of overweight women. There is also a control group in this study.

In this experimental design, 12 women are randomly assigned to one of the three

treatment conditions, and weight loss is measured 2 months, 4 months, and 6 months

after the program begins. Note that weight loss is relative to the weight measured at

the previous occasion.

When a between-subjects variable is included in this design, there are two additional

assumptions. One new assumption is the homogeneity of the covariance matrices on

the repeated measures for the groups. That is, the population variances and covariances

for the repeated measures are assumed to be the same for all groups. In our example,

the group sizes are equal, and in this case a violation of the equal covariance matrices

assumption is not serious. That is, the within-subjects tests (for the within-subject

main effect and the interaction) are robust (with respect to type IÂ€error) against a

violation of this assumption (see Stevens, 1986, chap. 6). However, if the group sizes

are substantially unequal, then a violation is serious, and Stevens (1986) indicated

in TableÂ€6.5 what should be added to test this assumption. AÂ€key assumption for the

499

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Repeated-Measures Analysis

validity of the within-subjects tests that was also in place for the single-group repeated

measures is the assumption of sphericity that now applies to the repeated measures

within each of the groups. It is still the case here that the unadjusted univariate F tests

for the within-subjects effects are not robust to a violation of sphericity. Note that the

combination of the sphericity and homogeneity of the covariance matrices assumption

has been called multisample sphericity. The second new assumption is homogeneity

of variance for the between-subjects main effect test. This assumption applies not to

the raw scores but to the average of the outcome scores across the repeated measures

for each subject. As with the typical between-subjects homogeneity assumption, the

procedure is robust when the between-subjects group sizes are similar, but a liberal or

conservative F test may result if group sizes are quite discrepant and these variances

are not the same.

TableÂ€12.12 provides the SAS and SPSS commands for the overall tests associated

with this analysis. TableÂ€12.13 provides selected SAS and SPSS results. Note that

this analysis can be considered as a two-way ANOVA. As such, we will test main

effects for diet and time, as well as the interaction between these two factors. The

time main effect and the time-by-diet interaction are within-subjects effects because

they involve change in means or change in treatment effects across time. The univariate tests for these effects appear in the first output selections for SAS and SPSS

in TableÂ€12.13. Using the Greenhouse–Geisser procedure, the main effect of time

is statistically significant (p < .001) as is the time-by-diet interaction (p = .003).

(Note that these effects are also significant using the multivariate approach, which

is not shown to conserve space.) The last output selections for SAS and SPSS in

TableÂ€12.13 indicate that the main effect of diet is also statistically significant, F(2,

33)Â€=Â€4.69, p = .016.

To interpret the significant effects, we display in TableÂ€12.14 the means involved

in the main effects and interaction as well as a plot of the cell means for the two

factors. Recall that graphically an interaction is evidenced by nonparallel lines. In

this graph you can see that the profiles for diets 1 and 2 are essentially parallel;

however, the profile for diet 3 is definitely not parallel with the profiles for diets

1 and 2. And, in particular, it is the relatively greater weight loss at time 2 for

diet 3 (i.e., 5.9 pounds) that is making the profile distinctly nonparallel. The main

effect of diet, evident in TableÂ€12.14, indicates that the population row means are

not equal. The sample means suggest that, weight loss averaging across time, is

greatest for diet 3. The main effect of time indicates that the population column

means differ. The sample column means suggest that weight loss is greater after

month 2 and 4, than after month 6. In addition to the graph, the cell means in

TableÂ€12.14 can also be used to describe the interaction. Note that weight loss for

each treatment was relatively large at 2 months, but only those in the diet 3 condition experienced essentially the same weight loss at 2 and 4 months, whereas the

weight loss for the other two treatments tapered off at the 4-month period. This

created much larger differences between the diet groups at 4 months relative to

the other months.

DATA LIST FREE/diet wgtloss1 wgtloss2 wgtloss3.

BEGIN DATA.

1 4 3 3 1 4 4 3 1 4 3 1

1 3 2 1 1 5 3 2 1 6 5 4

1 6 5 4 1 5 4 1 1 3 3 2

1 5 4 1 1 4 2 2 1 5 2 1

2 6 3 2 2 5 4 1 2 7 6 3

2 6 4 2 2 3 2 1 2 5 5 4

2 4 3 1 2 4 2 1 2 6 5 3

2 7 6 4 2 4 3 2 2 7 4 3

3 8 4 2 3 3 6 3 3 7 7 4

3 4 7 1 3 9 7 3 3 2 4 1

3 3 5 1 3 6 5 2 3 6 6 3

3 9 5 2 3 7 9 4 3 8 6 1

END DATA.

(2) GLM wgtloss1 wgtloss2 wgtloss3 BY diet

â•…â•… /WSFACTOR=time 3

(3) /PLOT=PROFILE(time*diet)

(4) /EMMEANS=TABLES(time) COMPARE ADJ(BONFERRONI)

â•…â•… /PRINT=DESCRIPTIVE

(5) /WSDESIGN=time

â•…â•… /DESIGN=diet.

DATA weight;

INPUT diet wgtloss1 wgtloss2 wgtloss3;

LINES;

1 4 3 3

1 4 4 3

1 4 3 1

1 3 2 1

1 5 3 2

1 6 5 4

1 6 5 4

1 5 4 1

1 3 3 2

1 5 4 1

1 4 2 2

1 5 2 1

2 6 3 2

2 5 4 1

2 7 6 3

2 6 4 2

2 3 2 1

2 5 5 4

2 4 3 1

2 4 2 1

2 6 5 3

2 7 6 4

(Continuedâ•›)

SPSS

SAS

TableÂ€12.12:â•‡ SAS and SPSS Control Lines for One-Between and One-Within Repeated Measures Analysis

SPSS

(1) CLASS indicates diet is a grouping (or classification) variable.

(2) MODEL (in SAS) and GLM (in SPSS) indicates that the weight scores are function of diet.

(3) PLOT requests a profile plot.

(4) The EMMEANS statement requests the marginal means (pooling over diet) and Bonferroni-adjusted multiple comparisons associated with the within-subjects factor (time).

(5) WSDESIGN requests statistical testing associated with the within-subjects time factor, and the DESIGN command requests testing results for the between-subjects diet

factor.

2 4 3 2

2 7 4 3

3 8 4 2

3 3 6 3

3 7 7 4

3 4 7 1

3 9 7 3

3 2 4 1

3 3 5 1

3 6 5 2

3 6 6 3

3 9 5 2

3 7 9 4

3 8 6 1

PROC GLM;

(1) CLASS diet;

(2) MODEL wgtloss1 wgtloss2 wgtloss3Â€=Â€diet/

NOUNI;

REPEATED time 3 /SUMMARY MEAN;

RUN;

SAS

TableÂ€12.12:â•‡ (Continued)

2

33

diet

Error

181.352

181.352

181.352

181.352

SPSS Results

18.4537037

3.9351852

Mean Square

2

1.556

1.717

1.000

Df

<.0001

0.0012

Â€

Pr > F

90.676

116.574

105.593

181.352

Mean Square

Tests of Within-Subjects Effects

36.9074074

129.8611111

Type III SS

Time

Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

88.37

5.10

Â€

F Value

The GLM Procedure

Repeated Measures Analysis of Variance

Tests of Hypotheses for Between Subjects Effects

90.6759259

5.2314815

1.0260943

Mean Square

Type III Sum of Squares

181.3518519

20.9259259

67.7222222

Type III SS

Source

Measure: MEASURE_1

DF

2

4

66

Time

time*diet

Error(time)

Source

DF

Source

SAS Results

The GLM Procedure

Repeated Measures Analysis of Variance

Univariate Tests of Hypotheses for Within Subject Effects

TableÂ€12.13:â•‡ Selected Output for One-Between One-Within Design

4.69

Â€

88.370

88.370

88.370

88.370

F

F Value

<.0001

0.0033

Â€

G–G

0.0161

Â€

PrÂ€>Â€F

Â€

<.0001

0.0029

H-F-L

(Continuedâ•›)

.000

.000

.000

.000

Sig.

Adj Pr > F

Type III Sum of Squares

1688.231

36.907

129.861

Source

Intercept

Diet

Error

1

2

33

Df

20.926

20.926

20.926

20.926

67.722

67.722

67.722

67.722

time * diet Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Error(time) Sphericity Assumed

Greenhouse-Geisser

Huynh-Feldt

Lower-bound

Measure: MEASURE_1

Transformed Variable: Average

Type III Sum of Squares

Source

TableÂ€12.13:â•‡ (Continued)

5.231

6.726

6.092

10.463

1.026

1.319

1.195

2.052

Mean Square

1688.231

18.454

3.935

Mean Square

Tests of Between-Subjects Effects

4

3.111

3.435

2.000

66

51.337

56.676

33.000

Df

429.009

4.689

F

5.098

5.098

5.098

5.098

F

.000

.016

Sig.

.001

.003

.002

.012

Sig.

Chapter 12

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TableÂ€12.14:â•‡ Cell and Marginal Means for the One-Between One-Within Design

TIME

1

2

3

DIETS

COLUMN MEANS

1

2

3

ROW MEANS

4.50

5.33

6.00

5.278

3.33

3.917

5.917

4.389

2.083

2.250

2.250

2.194

3.304

3.832

4.722

Diet 3

6

Weight loss

Diet 2

4

Diet 1

2

1

2

Time

3

12.12 POST HOC PROCEDURES FOR THE ONE-BETWEEN

AND ONE-WITHIN DESIGN

In the previous section, we presented and discussed statistical test results for the main

effects and interaction. We also used cell and marginal means and a graph to describe

results. When three or more levels of a factor are present in a design, researchers may also

wish to conduct follow-up tests for specific effects of interest. In our example, an investigator would likely focus on simple effects given the interaction between diet and time. We

will provide testing procedures for such simple effects, but for completeness, we briefly

discuss pairwise comparisons associated with the diet and time main effects. Note that for

the follow-up procedures discussed in this section, there is more than one way to obtain

results via SAS and SPSS. In this section, we use procedures, while not always the most

efficient, are intended to help you better understand the comparisons you are making.

12.12.1 Comparisons Involving Main Effects

As an example of this, to conduct pairwise comparisons for the means involved in a

statistically significant main effect of the between-subjects factor (here, diet), you can

simply compute the average of each participant’s scores across the time points of the

505

## 2016 keenan a pituch, james p stevens applied multivariate statistics for the social sciences analyses with SAS and IBMs SPSS routledge (2015)

## 2 Type I Error, Type II Error, and Power

## 2 Addition, Subtraction, and Multiplication of a Matrix by a Scalar

## K-Group MANOVA: A Priori and Post Hoc Procedures

## 14 Power Analysis—A Priori Determination of Sample Size

## 9 MANCOVA—Several Dependent Variables and Several Covariates

## 10 McFadden’s Pseudo R-Square for Strength of Association

## 5 Example 1: Examining School Differences in Mathematics

## 8 Example 2: Evaluating the Efficacy of a Treatment

## 6 Example 1: Using SAS and SPSS to Conduct Two-Level Multivariate Analysis

## 7 Example 2: Using SAS and SPSS to Conduct Three-Level Multivariate Analysis

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