chapter 11 - simple analysis of variancestatdhtx/methods8/instructors manual... · 2011. 7. 18. ·...

107

Chapter 11 - Simple Analysis of Variance

11.1 Eysenck’s study:

Counting Rhyming Adjective Imagery Intentional Total

Mean 7.00 6.90 11.00 13.40 12.00 10.06

St. Dev. 1.83 2.13 2.49 4.50 3.74 4.01

Variance 3.33 4.54 6.22 20.27 14.00 16.058

2 2 2 2

..

2 2 2 2

..

9 10.06 8 10.06 ... 11 10.06

786.82

10 7 10.06 6.90 10.06 ... 12 10.06

10 35.152 351.52

786.82 351.52 435.30

total ij

treat j

error total treat

SS X X

SS n X X

SS SS SS

Summary Table

Source df SS MS F

Treatments 4 351.52 87.88 9.08

Error 45 435.30 9.67

Total 49 786.82

11.2 Recall in Eysenck (1974) for Intentional group:

a.

Descriptives a

RECALL

10 19.3000 2.6687 .8439

10 12.0000 3.7417 1.1832

20 15.6500 4.9019 1.0961

Younger

Older

Total

N Mean Std. Deviation Std. Error

PROCESS = Higha.

108

2 2 2 2

2 2 2

..

..

21 15.65 19 15.65 ... 11 15.65

456.55

10 19.3 15.65 12.0 15.65

10 26.642 266.45

456.55 266.45 190.1

total ij

treat j

error total treat

SS X X

SS n X X

SS SS SS

ANOVAa

RECALL

266.450 1 266.450 25.229 .000

190.100 18 10.561

456.550 19

Between Groups

Within Groups

Total

Sum of Squares df Mean Square F Sig.

PROCESS = Higha.

b. t test for two independent groups:

2

1 1

2

2 2

1 2

1 2

2 2

1 2

2 2

19.30 7.122

12.00 14.000

10 10

19.30 12.00 7.305.023

1.4537.12214.000

10

5.023 25.23

X s

X s

n n

X Xt

s s

n

t F

109

11.3 Recall in Eysenck (1974) for four Age/Levels of Processing groups:

Descriptives

RECALL

10 6.5000 1.4337 .4534

10 19.3000 2.6687 .8439

10 7.0000 1.8257 .5774

10 12.0000 3.7417 1.1832

40 11.2000 5.7699 .9123

1.00

2.00

3.00

4.00

Total


a.

2 2 2 2

2 2

2

2 2

..

..

8 11.2 6 11.2 ... 11 11.2

1298.4

6.5 11.2 19.3 11.210

7.0 11.2 12.0 11.2

10 105.98 1059.8

1298.4 1059.8 238.6

total ij

treat j

error total treat

SS X X

SS n X X

SS SS SS

ANOVA

RECALL

1059.800 3 353.267 53.301 .000

238.600 36 6.628

1298.400 39

Between Groups

Within Groups

Total


b. Groups 1 and 3 combined versus 2 and 4 combined:

Descriptives

RECALL

20 6.7500 1.6182 .3618

20 15.6500 4.9019 1.0961

40 11.2000 5.7699 .9123

Low

High

Total


110

2 2 2 2

2 2 2

..

..

8 11.20 6 11.20 ... 11 11.20

1298.40

20 6.75 11.20 15.65 11.20

20 39.605 792.1

1298.4 792.1 502.3

total ij

treat j

error total treat

SS X X

SS n X X

SS SS SS

ANOVA

RECALL

792.100 1 792.100 59.451 .000

506.300 38 13.324

1298.400 39

Between Groups

Within Groups

Total


c. The results are somewhat difficult to interpret because the error term now includes

variance between younger and older participants. Notice that this is roughly double

what it was in part a. In addition, we do not know whether the level of processing

effect is true for both age groups, or if it applies primarily to one group

11.4 Modification of Exercise 11.1 with groups 1 and 2 combined versus groups 3 and 4

combined.

Results as given by R:

Df Sum Sq Mean Sq F value Pr(>F)

group 1 105.62 105.62 3.4051 0.0728 .

Residuals 38 1178.75 31.02

---

Alternative results using a t test and pooling the variances:

t.test(younger, older, var.equal = T)

Two Sample t-test

data: younger and older

t = 1.8453, df = 38, p-value = 0.0728

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

111

-0.3154482 6.8154482

sample estimates:

mean of x mean of y

12.75 9.50

11.5 Rerun of Exercise 11.2 with additional subjects:

The following is abbreviated printout from SPSS

a.

Descriptives a

RECALL

12 18.4167 3.2039 .9249

10 12.0000 3.7417 1.1832

22 15.5000 4.6980 1.0016

Younger

Older

Total


PROCESS = Higha.

ANOVAa

RECALL

224.583 1 224.583 18.800 .000

238.917 20 11.946

463.500 21

Between Groups

Within Groups

Total


PROCESS = Higha.

b. & c. With and without pooling variances:

Independent Samples Test a

4.336 20 .000 6.4167 1.4799

4.273 17.893 .000 6.4167 1.5018

Equal variances assumed

Equal variances not assumed

RECALL

t df Sig. (2-tailed) Mean Difference

Std. Error

Difference

t-test for Equality of Means

PROCESS = Higha.

d. The squared t for the pooled case = 4.33592 = 18.80, which is the F in the analysis of

variance.

112

11.6 Magnitude of effect measures for Exercise 11.2:

2

2

266.45.58

456.55

1 266.45 2 1 10.56.55

456.55 10.56

group

total

group error

total error

SS

SS

SS k MS

SS MS

I would assume a fixed model because it is unlikely that Eysenck selected his age levels

at random.

11.7 Magnitude of effect measures for Exercise 11.3a:

2

2

1059.8.82

1298.4

1 1059.8 4 1 6.63.80

1298.4 6.63

group

total

group error

total error

SS

SS

SS k MS

SS MS

11.8 Foa et al.’s (1991) study of therapy:

Descriptives

SYMPTOMS

14 11.07 3.95 1.06

10 15.40 11.12 3.52

11 18.09 7.13 2.15

10 19.50 7.11 2.25

45 15.62 7.96 1.19

SIT

PE

SC

WL

Total


2

2 2 2

2

2 2 2 2

..

14 11.07 15.62 10 15.40 15.62 11 18.09 15.62

10 19.50 15.62

507.84

average variance (weighted)

13*3.95 9*11.12 10*7.13 9*7.1155.587

13 9 10 9

group j j

error

SS n X X

MS

113

ANOVA

SYMPTOMS

507.840 3 169.280 3.046 .039

2278.738 41 55.579

2786.578 44

Between Groups

Within Groups

Total


b. Welch’s procedure:

2

'

.

0.8973, 0.0809, 0.2164, 0.1978

1.3924

18.950713.6101

1.3924

kk

k

k

k k

k

nw

s

w

w XX

w

2

2

2

.'

1"2 2 1

1 11 1

17.25365.75123 5.3764

4 1 .06971 .261415

k k

k

k k

w X X

kFk w

k n w

2

2

1 15' 19.13

3 0.261413 1

1

k

k k

kdf

w

n w

The result would still be significant, although on a substantially reduced number of

degrees of freedom, but a substantially larger F. I feel more confident, because I

know that even when I take the differences in group variances into account, I still get

a significant result.

114

c.

d. It simply means that the groups do not come from populations with equal means.

11.9 Magnitude of effect for Foa et al. (1991) study:

2

2

507.840.18

2786.907

1 507.840 4 1 55.587.12

2786.907 55.587

group

total

group error

total error

SS

SS

SS k MS

SS MS

11.10 What happens if we double the sample size in Exercise 11.2?

The SSgroup would remain the same because the group means would remain the same..

MSerror would also remain the same because it is the average of the cell variances. You

would have more df, which would give you a slightly smaller critical value.

11.11 Giancola study with transformed data.

Because some of the values were negative, I added 3.0 to each observation. The results

below are still significant, but the F is smaller. The following boxplot shows the effect of

the transformation.

ANOVA

lndv


Between Groups 4.280 4 1.070 4.117 .005

Within Groups 14.294 55 .260

Total 18.574 59

115

1 2 3 4 5

-20

24

6

dv

1 2 3 4 5

-1.0

0.0

1.0

2.0

Log dv

11.12 Foa et al. (1991) analysis

ANOVA

Score


Between Groups 507.840 3 169.280 3.046 .039

Within Groups 2278.738 41 55.579

Total 2786.578 44

Robust Tests of Equality of Means

Score

Statistica df1 df2 Sig.

Welch 5.375 3 19.128 .007

Brown-Forsythe 2.751 3 25.153 .064

a. Asymptotically F distributed.

Test of Homogeneity of Variances

Score

Levene Statistic df1 df2 Sig.

6.633 3 41 .001

116

The results are in agreement with those computed above. The variances are

heterogeneous by Levene’s test, but the difference is still significant when adjusted by

Welch or Brown and Forsythe.

11.13 Model for Exercise 11.1:

ij i ijX e

where

µ = grand mean

j = the effect of the jth treatment

eij = the unit of error for the ith subject in treatmentj


ij i ijX e

where

µ = grand mean

j = the effect of the jth treatment

eij = the unit of error for the ith subject in treatmentj


ij i ijX e

where

µ = grand mean

j = the effect of the jth treatment (where a ―treatment‖ is a particular combination of

Age and Task.

eij = the unit of error for the ith subject in treatment j

11.16 There is usually no practical meaning to an F appreciably less than 1. The second term

on the right of 2 2

treat eE MS n cannot be negative. The smallest that it can be is

zero. There is no reason to expect the MStreat estimate of 2

e to be appreciably less than

the MSerror estimate of 2

e.

117

11.17 Howell & Huessy (1981) study of ADD in elementary school vs. GPA in high school:

Group Group Means sj2

nj

Never ADD 2.6774 0.9450 201

2nd only 1.6123 1.0195 13

4th only 1.9975 0.5840 12

2nd & 4th 2.0287 0.2982 8

5th only 1.7000 0.7723 14

2nd & 5th 1.9000 1.0646 9

4th & 5th 1.8986 0.0927 7

all 3 yrs 1.4225 0.3462 8

Overall 2.4444 272

2

2 2 2201 2.6774 2.4444 13 1.6123 2.4444 ... 8 1.4225 2.4444

44.5570


200*0.9450 12*1.0195 ... 7*0.34620.8761

200 12 ... 7

..group j j

error

SS n X X

MS

ANOVA

SYMPTOMS

44.557 7 6.365 7.266 .000

231.282 264 .876

275.839 271

Between Groups

Within Groups

Total


11.18 Exercise 11.17 rerun, omitting the ―Never ADD‖ group:

2

2 2

..

13 1.6123 1.7849 ... 8 1.4225 1.7849

44.5570


12*1.0195 ... 7*0.34620.6607

12 ... 7

group j j

error

SS n X X

MS

GPA

118

This result, when compared with the previous one, indicates that the difference found in

Exercise 11.17 was at least mainly due to the fact that the first group differed from the

others.

11.19 Square Root Transformation of data in Table 11.6:

Original data:

Control 0.1 0.5 1 2

130 93 510 229 144

94 444 416 475 111

225 403 154 348 217

105 192 636 276 200

92 67 396 167 84

190 170 451 151 99

32 77 376 107 44

64 353 192 235 84

69 365 384 284

93 422 293

Means 109.4 258.6 390.56 248.5 156

S.D. 58.5 153.32 147.68 118.74 87.65

Var 3421.82 23506.04 21809.78 14098.86 7682.22

n 10 10 9 8 10

Square root transformed data:

Control 0.1 0.5 1 2

11.402 9.644 22.583 15.133 12.000

9.695 21.071 20.396 21.794 10.536

15.000 20.075 12.410 18.655 14.731

10.247 13.856 25.219 16.613 14.142

9.592 8.185 19.900 12.923 9.165

13.784 13.038 21.237 12.288 9.950

5.657 8.775 19.391 10.344 6.633

8.000 18.788 13.856 15.330 9.165

8.307 19.105 19.596 16.852

9.644 20.543 17.117

GPA

119

Means 10.13 15.31 19.40 15.39 12.03

S.D. 2.73 5.19 4.00 3.67 3.54

Var 7.48 26.96 16.03 13.49 12.55

n 10 10 9 8 10

11.20 Analysis of square root transformed data calculated in Exercise 11.19:

2

2 2

2 2 2 2

..

10 10.1327 14.3057 ... 10 12.0292 14.3057 478.773

( 1) 9 2.7343 9 5.1921 ... 9 3.5432 645.558

dosage j j

error j j

SS n X X

SS n s

(I could have calculated SStotal and then obtained SSerror by subtraction, but this was less

work.)

ANOVA

SQRTPOST

478.773 4 119.693 7.787 .000

645.558 42 15.370

1124.331 46

Between Groups

Within Groups

Total


11.21 Magnitude of effect for data in Exercise 11.17:

2

44.557.16

275.839

group

total

SS

SS

21

44.557 8 1 0.876.1389

275.839 0.876

group error

total error

SS k MS

SS MS

11.22 Darley and Latané (1968) -- reconstruction of ANOVA summary table:

Group n Mean

1 13 0.87

2 26 0.72

3 13 0.51

Total 52 0.705

120

2

2 2 2

..

13 0.87 0.705 26 0.72 0.705 13 0.51 0.705

0.854

treat j jSS n X X

MSerror was given in the problem.

Source df SS MS F

Bystanders 2 0.854 0.427 8.06*

Error 49 2.597 .053

Total 51 3.451

p .05 [ F

. 05( 2 , 49) 3 . 18]

11.23 Transforming Time to Speed in Exercise 11.22 involves a reciprocal transformation. The

effect of the transformation is to decrease the relative distance between large values.

11.24 Either a logarithmic or a square root transformation might be helpful in equalizing the

variances in Eysenck’s data in Table 11.2, especially since the variance seems to increase

with increasing values of the mean. But, as we saw in Exercise 11.21, applying the

logarithmic transformation made very little actual difference in the F value obtained.

11.25 The parts of speech (noun vs. verb) are fixed. But the individual items within those parts

of speech may well be random, representing a random sample of nouns and a random

sample of verbs.

11.26 We might wish to look at seasonal variation in mood (mood in Spring, Summer, Fall, and

Winter) across 20 different cities. The seasons would be a fixed variable, the cities would

be a random variable, and mood would be the dependent variable.

11.27 Analysis of Davey et al. data Report

dv

group Mean N Std. Deviation

1.00 12.6000 10 6.02218

2.00 7.0000 10 2.98142

3.00 8.7000 10 2.35938

Total 9.4333 30 4.62887

ANOVA

dv

Sum of

Squares df Mean Square F Sig.

Between Groups 164.867 2 82.433 4.876 .016

Within Groups 456.500 27 16.907

Total 621.367 29

121

11.28 Computer exercise. Reanalysis of data from Exercise 7.46.

The following is SPSS printout.

Descriptives

GSIT

135 62.474 9.553 .822

181 62.199 8.522 .633

59 60.593 10.577 1.377

375 62.045 9.242 .477

1

2

3

Total


ANOVA

GSIT

153.493 2 76.747 .898 .408

31790.736 372 85.459

31944.229 374

Between Groups

Within Groups

Total


11.29 Analysis of Epinuneq.dat, ignoring the effect of Interval. These results come from SPSS.

Descriptives

ERRORS

42 3.14 1.52 .24

42 4.81 1.25 .19

37 2.11 1.51 .25

121 3.40 1.80 .16

1

2

3

Total


ANOVA

ERRORS

147.970 2 73.985 36.197 .000

241.187 118 2.044

389.157 120

Between Groups

Within Groups

Total


122

11.30 Computer exercise. Analysis of Epinuneq.dat from Introini-Collison and McGaugh

(1986). These results come from SPSS.

INTERVAL = 1

Descriptives a

ERRORS

18 3.33 1.78 .42

18 5.33 .97 .23

18 1.83 1.47 .35

54 3.50 2.03 .28

1

2

3

Total


INT ERVAL = 1a.

ANOVAa

ERRORS

111.000 2 55.500 26.577 .000

106.500 51 2.088

217.500 53

Between Groups

Within Groups

Total


INTERVAL = 1a.

INTERVAL = 2

Descriptives a

ERRORS

12 2.83 1.27 .37

12 4.42 1.38 .40

12 2.17 1.40 .41

36 3.14 1.62 .27

1

2

3

Total


INT ERVAL = 2a.

ANOVAa

ERRORS

32.056 2 16.028 8.779 .001

60.250 33 1.826

92.306 35

Between Groups

Within Groups

Total


INTERVAL = 2a.

123

INTERVAL = 3

Descriptives a

ERRORS

12 3.17 1.40 .41

12 4.42 1.31 .38

6 2.33 1.63 .67

30 3.50 1.59 .29

1

2

3

Total


INT ERVAL = 3a.

ANOVAa

ERRORS

19.583 2 9.792 4.903 .015

53.917 27 1.997

73.500 29

Between Groups

Within Groups

Total


INTERVAL = 3a.

F = 0.56. There are no differences In the number of errors across the three Intervals.

11.31 Computer exercise. Repeating Exercise 11.29 using Epineq.dat. This output comes from

Minitab.

a. Analysis for Interval 1:

Analysis of Variance for Errors

Source DF SS MS F P

Dosage 2 71.72 35.86 14.93 0.000

Error 33 79.25 2.40

Total 35 150.97

Individual 95% CIs For Mean

Based on Pooled StDev

Level N Mean StDev ----+---------+---------+---------+--

1 12 3.167 1.801 (-----*-----)

2 12 5.333 1.073 (------*-----)

3 12 1.917 1.676 (-----*-----)

----+---------+---------+---------+--

Pooled StDev = 1.550 1.5 3.0 4.5 6.0

124

b. Analysis for Interval 2:


Source DF SS MS F P

Dosage 2 32.06 16.03 8.78 0.001

Error 33 60.25 1.83

Total 35 92.31



Level N Mean StDev ---------+---------+---------+-------

1 12 2.833 1.267 (------*-----)

2 12 4.417 1.379 (------*-----)

3 12 2.167 1.403 (------*------)

---------+---------+---------+-------

Pooled StDev = 1.351 2.4 3.6 4.8

c. Analysis for Interval 3:


Source DF SS MS F P

Dosage 2 35.06 17.53 7.76 0.002

Error 33 74.58 2.26

Total 35 109.64



Level N Mean StDev -+---------+---------+---------+-----

1 12 3.167 1.403 (------*-------)

2 12 4.417 1.311 (-------*------)

3 12 2.000 1.758 (-------*------)

-+---------+---------+---------+-----

Pooled StDev = 1.503 1.2 2.4 3.6 4.8

d. The average of the 9 variances :

2 2 2

2 1.801 1.073 ... 1.7582.162

9s

The average of the three error terms:

2.40 1.83 2.26average( ) 2.163

3errorMS

These two values agree within minor rounding error.

125

11.32 Analysis of data by Strayer et al. (2006)

11.33 Gouzoulis-Mayfrank et al. (2000) study:

126

b. The pairwise differences are 3.678, 3.464, and 0.214, and the square root of MSerror is

4.105. The gives d values of 0.896, 0.844, and 0.05. (c) it is reasonable to tentatively

conclude that Ecstacy produces lower scores than either the Control condition or the

Cannibis condition, which don’t differ.

11.34 I am trying to get students to commit themselves to the idea that transformations are not

outlandish things to do to data.

11.35 There should be no effect on the magnitude of the effect size measure because 2 is not

dependent on the underlying metric of the independent variable.

11.36 Students need to see that the pattern of differences among means is important in terms of

the overall F.

11.37 Teri et al. (1997) study:

Test of Homogeneity of Variances

Change

Levene Statistic df1 df2 Sig.

1.671 3 68 .181

127

Descriptives

Change


95% Confidence Interval for Mean

Lower Bound Upper Bound

1.00 23 4.2687 3.81590 .79567 2.6186 5.9188

2.00 19 5.0863 2.94113 .67474 3.6687 6.5039

3.00 10 -.7170 2.92656 .92546 -2.8105 1.3765

4.00 20 .3565 2.65011 .59258 -.8838 1.5968

Total 72 2.7053 3.89648 .45920 1.7897 3.6209

ANOVA

Change


Between Groups 391.391 3 130.464 12.922 .000

Within Groups 686.570 68 10.097

Total 1077.961 71

There was considerably, and significantly, more change in the two behavioral treatment groups.

128

Chapter 12 Multiple Comparisons Among Treatment Means

12.1 The effects of food and water deprivation on a learning task:

a. ANOVA with linear contrasts:

Groups: ad lib

(1)

2/day

(2)

food

(3)

water

(4)

f & w

(5)

Means: 18 24 8 12 11

aj: .5 .5 -.333 -.333 -.333 20.8333 ja

bj: 1 -1 0 0 0

2 b2

j

cj: 0 0 .5 .5 -1

21.5 jc

dj: 0 0 1 -1 0

2 d2

j

1 .5 18 .5 24 .333 8 .333 12 .333 11 10.667

2 1 18 1 24 0 8 0 12 0 11 6

3 0 18 0 24 .5 8 .5 12 1 11 1

4 0 18 0 24 1 8 1 12 0 11 4

1

2

1

2

5 10.667682.667

0.8333contrast

j

nSS

a

2

2

2

2

5 690

2contrast

j

nSS

b

3

2

3

2

5 13.333

1.5contrast

j

nSS

c

4

2

4

2

5 440.000

2contrast

j

nSS

d

129

Source df SS MS F

Deprivation 4 816.000 204.000 36.429*

1&2 vs 3,4,5 1 682.667 682.667 121.905*

1 vs 2 1 90.000 90.000 16.071*

3&4 vs 5 1 3.333 3.333 <1

3 vs 4 1 40.000 40.000 7.143*

Error 20 112.000 5.600

Total 24 928.000

p . 05 [ F . 05( 4 , 20) 2 . 87; F

. 05( 1 , 20) 4 . 35]

b. Orthogonality of contrasts:

Cross-products of coefficients:

.5 1 .5 1 .333 0 .333 0 .333 0 0

.5 0 .5 0 .333 .5 .333 .5 .333 1 0

.5 0 .5 0 .333 1 .333 1 .333 0 0

1 0 1 0 0 .5 0 .5 0 1 0

0 0 0 0 .5 1 .5 1 1 0 0

j j

j j

j j

j j

j j

a b

a c

a d

b c

c d

c.

SS treat SS contrast

816. 000 682. 667 90. 000 3 . 333 40. 000

12.2 Recall data from Exercise 11.1:

Group Counting Rhyming Adjective Imagery Intentional aj2

Means 7.00 6.90 11.00 13.40 12.00

aj -1/2 -1/2 1/2 1/2 0 1

bj 0 0 -1 1 0 2

1 .5 7.00 .5 6.90 .5 11.00 .5 13.40 5.25

2 1 11.00 1 13.40 2.40

1

22

2

10 5.25275.625

1contrast

j

nSS

a

2

22

2

10 2.4028.80

2contrast

j

nSS

b

130

Summary Table

Source df SS MS F

Treatments 4 351.52 87.88 9.08

1&2 vs 3&4

3 vs 4

1

1

275.625

28.80

275.625

28.80

28.50*

2.98

Error 45 435.30 9.67

Total 49 786.82

12.3 For = .05:

Per comparison error rate = = .05

Familywise error rate = 1 - (1 - )2 = .0975.

12.4 Linear contrast for data in Exercise 11.2:

1 1

2 2

19.3 10

12.0 10

X n

X n

1 1 19.3 1 12.0 7.3j ja X

22

2

10 7.3255.45

2

266.4525.23

10.56

contrast

i

contrast

error

nSS

a

SSF

MS

You can see that the F for this contrast is exactly the same as the F for the overall

analysis of variance in Exercise 11.2. With only two groups this will always be the case.

12.5 Studentized range statistic for data in Exercise 11.2:

1 1

2 2

19.3 10

12.0 10

X n

X n

1 2

2

19.3 12.0 7.37.101

1.02810.56

10

error

X Xq

MS

n

q2 = 7.10 = 5.023 2 = 7.10 = t 2

131

12.6 Linear contrasts for the data in Exercise 11.3:

1 1

1 1

1 1

1 1

6.5 10

19.3 10

7.0 10

12.0 10

X n

X n

X n

X n

Groups: Young/

Low

Young/

High

Old/

Low

Old/

High

Means: 6.5 19.3 7.0 12.0

aj: .5 .5 -.5 -.5 21 ja

bj: .5 -.5 .5 -.5 21 jb

cj: .5 -.5 -.5 .5 21 jb

1 .5 6.5 .5 19.3 .5 7.0 .5 12 3.4j ja X

2 .5 6.5 .5 19.3 .5 7.0 .5 12 8.9j jb X

3 .5 6.5 .5 19.3 .5 7.0 .5 12 3.9j jc X

1

22

2

10 3.4 115.6115.6 17.44

1 6.63contrast

j

nSS F

a

2

22

2

10 8.9 792.1792.1 119.47

1 6.63contrast

j

nSS F

a

3

22

2

10 3.9 152.1152.1 22.92

1 6.63contrast

j

nSS F

a

Notice that these sum to 1059.8, which is SSgroup in the overall analysis. These contrasts

are orthogonal, though they would not be deducible from the rules laid out earlier in the

chapter. Each of these Fs is significant on 1 and 36 df.

The first F tests the null hypothesis that Younger and Older subjects come from

populations with the same mean. The second tests the null hypothesis that there is no

difference in the recall of tasks requiring low versus high levels of processing. The third

tests for what we will later call an interaction—that the difference between low and high

levels of processing for younger subjects is the same as the comparable difference for

older subjects.

132

12.7 The Bonferroni test on contrasts in Exercise 12.2 (data from Exercise 11.1):

From Exercise 12.2: 1 = 5.25 2 = 2.40 n = 10

2 1ja

2 2jb MSerror = 9.67

2

j error

ta MS

n

'

1

5.255.34

1 9.67

10

t

'

1

2.401.72

2 9.67

10

t

.05[ 45;2comparisons) 2.32errort df Reject H0 for only the first comparison.

12.8 Tukey’s test on example in Table 11.2 :

recal l

Tukey HSDa

10 6.9000

10 7.0000

10 11.0000

10 12.0000

10 13.4000

1.000 .429

conditio

2.00

1.00

3.00

5.00

4.00

Sig.

N 1 2

Subset for alpha = .05

Means for groups in homogeneous subsets are display ed.

Uses Harmonic Mean Sample Size = 10.000.a.

The counting and imagery groups are homogeneous, but are different from the adjective,

intentional, and rhyming conditions, which are also homogeneous. This is the same

pattern of differences that we found with the REGWQ.

133

For the Tukey test SPSS also produces the following table:

Mult iple Comparisons

Dependent Variable: RECALL

Tukey HSD

1.00E-01 1.39 1.000 -3.85 4.05

-4.00* 1.39 .046 -7.95 -4.77E-02

-6.40* 1.39 .000 -10.35 -2.45

-5.00* 1.39 .007 -8.95 -1.05

-1.00E-01 1.39 1.000 -4.05 3.85

-4.10* 1.39 .039 -8.05 -.15

-6.50* 1.39 .000 -10.45 -2.55

-5.10* 1.39 .006 -9.05 -1.15

4.00* 1.39 .046 4.77E-02 7.95

4.10* 1.39 .039 .15 8.05

-2.40 1.39 .429 -6.35 1.55

-1.00 1.39 .951 -4.95 2.95

6.40* 1.39 .000 2.45 10.35

6.50* 1.39 .000 2.55 10.45

2.40 1.39 .429 -1.55 6.35

1.40 1.39 .851 -2.55 5.35

5.00* 1.39 .007 1.05 8.95

5.10* 1.39 .006 1.15 9.05

1.00 1.39 .951 -2.95 4.95

-1.40 1.39 .851 -5.35 2.55

(J) CONDTION

Rhyming

Adjective

Imagery

Intentional

Counting

Adjective

Imagery

Intentional

Counting

Rhyming

Imagery

Intentional

Counting

Rhyming

Adjective

Intentional

Counting

Rhyming

Adjective

Imagery

(I) CONDTION

Counting

Rhyming

Adjective

Imagery

Intentional

Mean

Difference (I-J) Std. Error Sig. Lower Bound Upper Bound

95% Confidence Interval

The mean dif ference is signif icant at the .05 level.*.

12.9 A post hoc test like the Tukey or the REGWQ often does not get at the specific questions

we have in mind, and, at the same time, often answers questions in which we have no

interest.

12.10 Games and Howell approach:

I have organized the solution into a set of tables to keep some sort of order to the

procedure.

(1) Matrix of mean differences

Group: 1 2 3 4 5

Mean: 10 18 19 20 29

1 10 -- 8* 9* 11* 19*

2 18 -- 1 3 11*

3 19 -- 2 10*

4 21 -- 8*

5 29 --

134

(2) Matrix of df’:

Group: 1 2 3 4 5

1 -- 8 13.92 12.78 15.00

2 -- 8.52 8.14 8.55

3 -- 12.96 14.89

4 -- 13.72

5 --

where

22

22 22

'

1 1

ji

i j

ji

ii

i j

ss

n ndf

ss

nn

n n

(3) Matrix of q(r, df)

Group: 1 2 3 4 5

1 -- 3.26 3.70 4.15 4.37

2 -- 3.20 4.04 4.42

3 -- 3.06 3.67

4 -- 3.03

5 --

(4) Matrix of

s 2

i

n i

s 2

j

n j

2

Group: 1 2 3 4 5

1 -- 1.16 1.00 0.99 0.99

2 -- 1.19 1.18 1.19

3 -- 1.03 1.03

4 -- 1.02

5 --

135

(5) Matrix of Wr = product of corresponding elements of (3) & (4):

Group: 1 2 3 4 5

1 -- 3.78 3.70 4.11 4.33

2 -- 3.81 4.77 5.26

3 -- 3.15 3.78

4 -- 3.09

5 --

The asterisks in (1) indicate the differences which are significant. Groups 1 and 5 differ

from each other and all other groups.

I imagine you're glad that you don’t have to do that one again—I know I am.

12.11 Scheffé’s test on the data in Exercise 12.10:

Group 1 2 3 4 5

X j 10 18 19 21 29

nj 8 5 8 7 9

sj2 7.4 8.9 8.6 7.2 9.3

aj -16 -16 -16 21 21

bj -20 8 8 8 0

218.2875

1

j j

error

j

n sMS

n

1

2

2 2

2

2 2

2

.05(4,32).05 1,

3416113.26

12432 8.2875

151261.57

4480 8.2875

1 4 4 2.69 10.76error

contrast

j j error

contrast

j j error

crit k df

LF

n a MS

LF

n b MS

F k F F

Thus both contrasts are significant.

12.12 Tukey’s HSD test applied to the THC data in Table 11.6

Group: 1 2 3 4 5

µg THC 0 0.1 0.5 1 2

nj 10 10 9 8 10

n h k

( 1

n j )

5

1

10

1

10

1

9

1

8

1

10

9 . 326

136

Group 1 5 2 4 3 r qHSD Wr

Means 1.981 2.124 2.318 2.353 2.557

1 -- 0.143 0.337 0.372* 0.576* 5 4.04 0.3373

5 -- 0.194 0.229 0.433* 4 4.04 0.3373

2 -- 0.035 0.239 3 4.04 0.3373

4 -- 0.204 2 4.04 0.3373

3 --

.05

0.065, 4.04 0.337

9.326

errorr

h

MSw q r df

n

The 0.5µg group and the 1.0 group are different from the control group and the .5µg is

different from the 2.0 g group. No other differences are significant. The maximum

familywise error rate is .05.

12.13 Dunnett’s test on data in Table 11.6:

critical value

2 0.0652

2.58 0.3059.326

errorc j d

h

MSX X t

n

The control group is significantly different from the 0.1 µg, the 0.5 µg, and the 1.0 µg

groups.

12.14 If you are willing to sacrifice using a common error term, you simply run the relevant t

tests but evaluate them at ' = /c.

12.15 They are sequentially modified because you change the critical value each time you reject

another null hypothesis.

12.16 Linear and quadratic trend in Conti and Musty (1984).

The results given below assume that you have added the three observations mentioned in

the exercise.

Group: Control 0.1 0.5 1 2

Means 1.981 2.318 2.557 2.353 2.124

Linear -0.72 -0.62 -0.22 0.28 1.28 aj2 = 2.668

Quadratic 0.389 0.199 -0.362 -0.612 0.387 bj2 = 0.846

137

2

2

2

2

0.72 1.981 0.62 2.318 0.22 2.557

0.28 2.353 1.28 2.124 0.04846

0.389 1.981 0.199 2.318 0.362 2.557

0.612 2.353 0.387 2.124 0.31179

10 0.048460.0088

2.668

Linear j j

Quad j j

Linear

j

Quad

a X

b X

nSS

a

nSS

b

2

2

10 0.311791.1487

0.846j

Source df SS MS F

Treatments 4 1.857 0.491 8.103*

Linear 1 0.0088 0.0088 0.145

Quadratic 1 1.1490 1.1487 18.987*

Error 45 2.726 0.0605

Total 49 4.689

There is a significant quadratic trend, but no significant linear trend. This quadratic trend

is clearly visible in the means. SPSS will come very close to these results if you code

group membership as 0, .1, .5, 1, & 2.

12.17 Conti and Musty (1984) recorded locomotive behavior in rats in response to injection of

THC in the an active brain region. The raw data showed a clear linear relationship

between group means and standard deviations, but a logarithmic transformation of the

data largely removed this relationship. Mean locomotive behavior increased with dosage

up to 0.5 g, but further dose increases resulting in decreased behavior. Polynomial trend

analysis revealed no linear trend but a significant quadratic trend.

12.18 Computer example.

12.19 If there were significant differences due to Interval and we combined across intervals,

those differences would be incorporated into the error term, decreasing power.

138

12.20 Trend analysis for Epineq.dat separately at each interval.

One Day: FLinear = 9.44 (p = .0042); FQuad = 20.43 (p = .0001)

One Week: FLinear = 4.33 (p = .0453); FQuad = 13.23 (p = .0009)

One Month: FLinear = 6.91 (p = .0129); FQuad = 8.60 (p = .0061)

12.21 At all three intervals there was a significant linear and quadratic trend, indicating that the

effect of epinephrine on memory increases with a moderate dose but then declines with a

greater dose. The linear trend reflects the fact that in the high dose condition the animals

do even worse than with no epinephrine.

12.22 Stone et al. (1992): Glucose and memory:

a.

Using group number on X

Group

76543210

en

cy

800

700

600

500

400

300

200

100

0

Using dose on X

DOSE

6005004003002001000-100

LA

TE

NC

Y

800

700

600

500

400

300

200

100

0

b. Trend analysis using actual dose:

ANOVA

LATENCY

772106.472 5 154421.294 11.193 .000

7561.039 1 7561.039 .548 .465

764545.433 4 191136.358 13.855 .000

413869.833 30 13795.661

1185976.306 35

(Combined)

Contrast

Deviation

Linear

Term

Between

Groups

Within Groups

Total


139

c. Trend analysis using 1, 2, …6 as coding:

ANOVA

LATENCY

772106.472 5 154421.294 11.193 .000

152114.402 1 152114.402 11.026 .002

619992.070 4 154998.017 11.235 .000

413869.833 30 13795.661

1185976.306 35

(Combined)

Contrast

Deviation

Linear

Term

Between

Groups

Within Groups

Total


d. When we use the group number coding in our trend analysis we find a significant

linear trend. As the dose of sucrose increases, memory increases accordingly.

e. The choice of coding system in not always obvious. Using 1, 2, … ,6 actually ranks

the dose levels and ignores the fact that dose increases in an extreme way. (In other

words, the difference between the first 2 doses is 1 mg/kg, whereas the difference

between the last two doses is 250 mg/kg. Using 1, 2, …, 6 deliberately ignores this

relationship. Apparently the human body responds in a nonlinear way to the increase

in actual dose levels.

12.23 The first comparison calls for comparing the two control groups with the experimental

groups. The solution from SPSS follows for the contrast itself. (SPSS only allows me to

specify 1/3 as .33, rather than using more decimal places, which is why it complains that

the coefficients don’t sum to 0 and gives the contrast as 10.77 rather than 10.6

The square root of MSerror = 2.366, which I will use to compute the confidence interval. I

will use 10.67 as the (correct) contrast, even though that is not what SPSS reported. Then

140

.95 .05

1 2

10.67 2.086 2.366

10.67 4.935

5.735 15.605

errorCI t s

12.24 Effect sizes for contrasts in Exercise 12.1

i j

e e

a Xd

s s

From the answers to Exercise 12.1:

1 = 10.667 2 = -6 3 = -1 4 = -4 MSerror = 5.60

1

2

3

4

10.667 10.6674.51

2.3665.60

6 62.54

2.3665.60

1 10.42

2.3665.60

4 41.69

2.3665.60

e

e

e

e

ds

ds

ds

ds

All contrasts show large differences, with the smallest difference being about 4/10 of a

standard deviation.

These are called standardized effect sizes because we divide the mean difference (the

contrast) by the size of the standard deviation. This represents the difference in standard

deviation units. An unstandardized effect would just be the difference between the means

(or the means of sets of groups).

12.25 The study by Davey et al. (2003):

The group means are Negative mood =12.6, Positive mood = 7.0, No induction = 8.7

The SPSS ONEWAY solution with one contrast comparing the Negative and Positive

141

mood groups is shown below.

ANOVA

Things listed to check

164.867 2 82.433 4.876 .016

456.500 27 16.907

621.367 29

Between Groups

Within Groups

Total

Sum of

Squares df Mean Square F Sig.

Contrast Coefficients

1 -1 0

Contrast

1

Negative Positive None

Group

Contrast Tests

5.6000 1.83888 3.045 27 .005

5.6000 2.12498 2.635 13.162 .020

Contrast

1

1

Assume equal v ariances

Does not assume equal

variances

Things listed to check

Value of

Contrast Std. Error t df Sig. (2-tailed)

The contrast between the Positive and Negative mood conditions was significant (t(27) =

3.045, p < .05). This leads to an effect size of / 5.6 / 16.907errord MS

5.6 / 4.11 1.36 . The two groups differ by over 1 1/3 standard deviations. It is evident

that inducing a negative mood leads to more checking behavior than introducing a

positive mood. (If we had compared the Positive and No mood conditions, the difference

would not have been significant. However I had not planned to make that comparison.

12.26 Benjamini & Hochberg test applied to Foa’s data

I start with running Fisher’s LSD test comparing all means because I need to know the t

values and probabilities.

142

We need to arrange the mean differences in decreasing order of p

Comparison Mean Diff p i pcrit Sig.

SC-WL -1.409 .668 6 .05 No

PE-SC -2.691 .414 5 .042 No

PS-WL -4.100 .226 4 .033 No

SIT-PE -4.329 .168 3 .025 No

SIT-SC -7.019 .024 2 .016 Yes

SIT-WL -8.429 .009 1 .008 Yes

12.27 This requires students to make up their own example.

12.28 This requires students to find an example in the literature.

143

Chapter 13 - Factorial Analysis of Variance

Note: Because of severe rounding in reporting and using means, there will be visible rounding

error in the following answers, when compared to standard computer solutions. I have made the

final answer equal the correct answer, even if that meant that it is not exactly the answer to the

calculations shown. (e.g. 3(3.3) would be shown as 10.0, not 9.9)

13.1 Mother/infant interaction for primiparous/multiparous mothers under or over 18 years of

age with LBW or full-term infants:

Table of cell means

Size/Age

LBW

< 18

LBW

> 18

NBW

Mother’s

Parity

Primi- 4.5 5.3 6.4 5.40

Multi- 3.9 6.9 8.2 6.33

4.2 6.1 7.3 5.87

2

22 352

2404 338.9360

total

XSS X

N

2

2 2

. ..

10(3)[ 5.40 5.87 6.33 5.87 ]

30 0.4356 13.067

Parity iSS ns X X

2

2 22

. ..

10 2 [(4.200 5.87) 6.10 5.87 7.30 5.87 ]

20 2.79 0.05 2.04 20 4.89

97.733

size jSS np X X

2

2 2

..

10[ 4.5 5.87 ... 8.2 5.87 ]

10 12.853 128.53

cells ijSS n X X

128.53 13.067 97.733

17.733

PS cells P SSS SS SS SS

338.93 128.53

210.40

error total cellsSS SS SS

144

Source df SS MS F

Parity 1 13.067 13.067 3.354

Size/Age 2 97.733 48.867 12.541*

P x S 2 17.733 8.867 2.276

Error 54 210.400 3.896

Total 59 338.933

*p < .05 F.05(2,54) = 3.17

13.2 It is hard to believe that mothers who are less than 18 years old and have had at least their

second child don't differ in many respects from the rest of the mothers.

13.3 The mean for these primiparous mothers would not be expected to be a good estimate of

the mean for the population of all primiparous mothers because 50% of the population of

primiparous mothers do not give birth to LBW infants. This would be important if we

wished to take means from this sample as somehow representing the population means

for primiparous and multiparous mothers.

13.4 Simple effect of size/age in multiparous mothers in Exercise 13.1:

2

at 2 2

2 22

.

10[(3.90 6.33) 6.90 6.33 8.20 6.33 ]

10 5.90 0.32 3.50 10 9.727

97.267

S M jSS n X X

at

at

97.26748.633

2

S MS at M

S M

SSMS

df

at at

48.63312.483*

3.896

S MS M

error

MSF

MS

p . 05 [ F . 05( 1 , 54) 4 . 03]

13.5 Memory of avoidance of a fear-producing stimulus:

Area of Stimulation

Neutral Area A Area B Mean

50 28.6 16.8 24.4 23.27

Delay 100 28.0 23.0 16.0 22.33

150 28.0 26.8 26.4 27.07

Mean 28.2 22.2 22.27 24.22

145

2

22

2

1090 28374 45 5 3 3

109028374 1971.778

45

ij

total

X X N n a b

XSS X

N

2

2 22

. ..

5(3)[(23.27 24.22) 22.33 24.22 27.07 24.22 ]

5 3 0.90 3.57 8.12 30 12.60

188.578

Delay iSS na X X

2

2 22

. ..

5(3)[(28.20 24.22) 22.20 24.22 22.27 24.22 ]

356.044

Area jSS nd X X

2

2 22

..

5[(28.60 24.22) 16.80 24.22 ... 26.4 24.22 ]

916.578

Cells ijSS n X X

SS DASS cellsSS DSS A 916.578 188.578 356.044 371.956

SS errorSS totalSS cells 1971.778 916.578 1055.200

Source df SS MS F

Delay 2 188.578 94.289 3.22

Area 2 356.044 178.022 6.07*

D x A 4 371.956 92.989 3.17*

Error 36 1055.200 29.311

Total 44 1971.778

p . 05 [ F

. 05( 2 , 36) 3 . 27; F . 05( 4 , 36) 2 . 64]

13.6 Plot of cell means from Exercise 13.5:

Means:

Area of Stimulation

Neutral Area A Area B

50 28.6 16.8 24.4

Delay 100 28.0 23.0 16.0

150 28.0 26.8 26.4

146

13.7 In Exercise 13.5, if A refers to Area:

1 = the treatment effect for the Neutral site

= X .1 – X ..

= 28.2 – 24.22 = 3.978

13.8 Simple effects to clarify the results for Area in Exercise 13.5:

2

A at 50 at 50 1

2 22

.

5[(28.60 23.27) 16.80 23.27 24.40 23.27 ]

357.733

ASS n X X

A at 50 at 50

A at 50

357.733178.867

2A

SSMS

df

at 50

at 50

178.8676.10*

29.311

A

A

error

MSF

MS

2

A at 100 at 100 2

2 22

.

5[(28.00 22.33) 23.00 22.33 16.00 22.33 ]

363.333

ASS n X X

147

A at 100 at 100

A at 100

363.333181.666

2A

SSMS

df

at 100

at 100

181.6666.20*

29.311

A

A

error

MSF

MS

2

A at 150 at 150 3

2 22

.

5[(28.00 27.07) 26.80 27.07 26.40 27.07 ]

6.933

ASS n X X

A at 150 at 150

A at 150

6.9333.466

2A

SSMS

df

at 150

at 150

3.4661

29.311

A

A

error

MSF

MS

p . 05 [ F . 05( 2 , 36) 3 . 27]

SSA at 50 + SSA at 100 + SSA at 140 = 375.733 + 363.333 + 6.933 = 728.000

SSA + SSDA = 356.004 + 371.956 = 728.000

13.9 The Bonferroni test to compare Site means.

0 0

28.20 22.00 28.20 22.27

29.311 29.311 29.311 29.311

15 15 15 15

3.03 (Reject ) 3.03 (Reject )

error error error error

N A N B

N A N Bt t

MS MS MS MS

n n n n

H H

[t’.025(2,36) = ± 2.34]

We can conclude that both the difference between Groups N and A and between Groups

N and B are significant, and our familywise error rate will not exceed = .05.

148

13.10 Simple effects of Delay in Area A in Exercise 13.5:

2

at 2 2

2 2 2

.

5[ 16.8 22.2 23.0 22.2 26.8 22.2 ]

254.8

D A iSS n X X

D at AD at A

D at A

254.8127.4

2

SSMS

df

D at A

at A

127.44.35

29.311D

error

MSF

MS

p . 05 [ F

. 05( 2 , 36) 3 . 27]

13.11 Rerunning Exercise 11.3 as a factorial design:

The following printout is from SPSS

Tests of Between-Subjects Effects

Dependent Variable: Recall

1059.800a 3 353.267 53.301 .000

5017.600 1 5017.600 757.056 .000

115.600 1 115.600 17.442 .000

792.100 1 792.100 119.512 .000

152.100 1 152.100 22.949 .000

238.600 36 6.628

6316.000 40

1298.400 39

Source

Corrected Model

Intercept

Age

LevelProc

Age * LevelProc

Error

Total

Corrected Total

Type II I Sum

of Squares df Mean Square F Sig.

R Squared = .816 (Adjusted R Squared = .801)a.

[ The Corrected Model is the sum of the main effects and interaction. The Intercept is the correction factor,

which is (X) 2. The Total (as opposed to Corrected Total) is X

2. The Corrected Total is what we have

called Total.]

149

Estimated Marginal Means

3. Age * LevelProc

Dependent Variable: Recall

6.500 .814 4.849 8.151

19.300 .814 17.649 20.951

7.000 .814 5.349 8.651

12.000 .814 10.349 13.651

LevelProc

1.00

2.00

1.00

2.00

Age

1.00

2.00

Mean Std. Error Lower Bound Upper Bound

95% Conf idence Interval

The results show that there is a significance difference between younger and older

subjects, that there is better recall in tasks which require more processing, and that there

is an interaction between age and level of processing (LevelProc). The difference

between the two levels of processing is greater for the younger subjects than it is for the

older ones, primarily because the older ones do not do much better with greater amounts

of processing.

13.12 Difference between Groups 1 and 3 combined and 2 and 4 combined:

The level effect in the factorial design has the same df, SS, and MS as it did in Exercise

11.3. But the F is more than twice as large because the error term here does not include

variation due to Age, which was included in the error term in Exercise 11.3.

13.13 Made-up data with main effects but no interaction:

Cell means: 8 12

4 6

13.14 Made-up data with interaction and no main effects:

Cell means: 8 4

4 12

150

13.15 The interaction was of primary interest in an experiment by Nisbett in which he showed

that obese people varied the amount of food they consumed depending on whether a lot

or a little food was visible, while normal weight subjects ate approximately the same

amount under the two conditions.

13.16 Unequal sample sizes Klemchuk, Bond, & Howell (1990):

Cell ns: Age

Younger Older

Daycare No 14 12 26

Yes 10 4 14

24 16 40 = N

1 1 1 1

14 12 10 4

1

47.925

i

h

n

kn

Cell Means: Age

Younger Older

Daycare No -1.2089 0.0750 -0.5669

Yes -0.5631 0.5835 0.0102

-0.8860 0.3292 13.895

2 2 2

. ..

2 2 2

. ..

2*7.925 0.5669 ( .2784) 0.0102 ( .2784)

2*7.925*.0.1665 2.639

2*7.925 0.8860 ( .2784) 0.3292 ( .2784)

2*7.925*0.7384 11.704

Daycare h i

Age h j

SS an X X

SS dn X X

151

22

2

.. 2 2

1.2089 ( .2784) 0.0750 .27847.925

0.5631 .2784 0.5835 .2784

7.925*1.8146*14.3811

14.381 2.639 11.704 0.038

cells h ij

AD cells D A

SS n X X

SS SS SS SS

Cell variances: Age

Younger Older

Daycare No 0.7407 0.1898

Yes 0.9551 0.2456

21 13(0.7407) 11(0.1898) 9(0.9551) 3(0.2456) 21.050error ij ijSS n s

Source df SS MS F

Age 1 11.704 11.704 20.02*

Daycare 1 2.639 2.639 4.51*

A×D 1 0.038 0.038 <1

Error 36 21.050 0.585

Total 39

.05,(1,36)* .05 [ 4.11]p F

We can conclude that there are effects due to both Age and Daycare, but there is no

interaction between the two main effect variables. (One variance is nearly 4 times

another, but with such small sample sizes we can’t tell if there is heterogeneity of

variance.)?

13.17 Magnitude of effect for mother-infant interaction data in Exercise 13.1:

152

2

2

2

2

2

13.067.04

338.933

97.733.29

338.933

17.733.05

338.933

1 13.067 1 3.896.03

338.933 3.896

1

parity

P

total

size

s

total

ps

Ps

total

parity error

p

total errpr

size error

s

to

SS

SS

SS

SS

SS

SS

SS p MS

SS MS

SS s MS

SS

2

97.733 2 3.896.26

338.933 3.896

1 1 17.733 1 2 3.896.03

338.933 3.896

tal errpr

ps error

ps

total errpr

MS

SS p s MS

SS MS

13.18 d̂ for the data in Exercise 13.1

Both of these independent variables normally vary in the general population, so I will

correct the error term in both cases. For the Birthweight I will use the contrast of LBW

infants versus full term infants, which will involve coefficients (.5, .5, -1).

LBW versus Full-term

= (.5)(4.2) + (.5)(6.1) + (-1)(7.3) = -2.15

210.400 13.067 17.333 240.804.225 2.055

54 1 2 57

error P pxs

e

error P pxs

SS SS SSs

df df df

2.15ˆ 1.052.055e

ds

The full term mothers exceed the LBW mothers by 9/10 of a standard deviation in terms

of mean mother-infant interaction.

Parity

y = (1)(5.40) = (-1)(6.33) = -0.93

153

210.400 97.733 17.333 325.4665.611 2.369

54 2 2 58

error s pxs

e

error s pxs

SS SS SSs

df df df

0.93ˆ 0.392.369e

ds

Multiparous mothers outperform first-time mothers by about a 1/3 of a standard

deviation.

13.19 Magnitude of effect for avoidance learning data in Exercise 13.5:

2

2

2

2

2

188.578.10

1971.778

356.044.18

1971.778

17.733.19

1971.778

1 188.578 2 29.311.06

1971.778 29.311

1

delay

D

total

area

A

total

DA

DA

total

delay error

D

total errpr

area er

A

SS

SS

SS

SS

SS

SS

SS d MS

SS MS

SS a MS

2

356.044 2 29.311.15

1971.778 29.311

1 1 371.956 2 2 29.311.13

1971.778 29.311

ror

total errpr

DA error

DA

total errpr

SS MS

SS d a MS

SS MS

13.20 d̂ for the data in Exercise 13.5

In this case Area of Stimulation does not vary normally in the population, nor does the

delay of stimulation, because this is an artificial situation. Therefore the denominator for

d in each case will be the square root of MSerror, which is 5.414.

For Area I will take the contrast of the difference between the means of the Neutral

location and the mean of the other two areas, which is 5.96. then

154

5.414ˆ .9085.965e

ds

For the Delay variable, the overall difference was not significant and it doesn’t look as if

there is much profit in creating a contrast. This is fortunate, because I cannot think of a

logic contrast to run on an interval variable like this. I guess it would be perfectly

possible to generate a polynomial contrast on Delay and then use that y in calculating an

estimate of d. However I would not really know how I would interpret that statistic.

13.21 Three-way ANOVA on Early Experience x Intensity of UCS x Conditioned Stimulus

(Tone or Vibration):

n = 5 in all cells SStotal = 41,151.00

E×I×C

Cells

CS = Tone CS = Vibration

Exper: Hi Med Low Hi Med Low

Control 11 16 21 12.0 19 24 29 24.00 20.00

Tone 25 28 34 29.0 21 26 31 26.00 27.50

Vib 6 13 20 13.0 40 41 52 44.33 28.67

Both 22 30 30 27.33 35 38 48 40.33 33.83

16 21.75 105 21.33 28.75 32.25 40.00 33.66 27.50

E×I Cells Intensity

Experience: High Med Low

Control 15 20 25 20.00

Tone 23 27 32.5 27.50

Vib 23 27 36 28.67

Both 28.5 34 39 33.83

22.38 27.00 33.12 27.50

E×C Cells Conditioned Stimulus

Experience: Tone Vib

Control 16.00 24.00 20.00

Tone 29.00 26.00 27.50

Vib 13.00 44.33 28.67

Both 27.33 40.33 33.83

21.33 33.66 27.50

I×C Cells Conditioned Stim

Intensity: Tone Vib

High 16.00 28.75 22.38

Med 21.75 32.25 27.00

Low 26.25 40.00 33.12

21.33 33.67 27.50

155

2 2

2

2 2

2 2 2 2

2 2

.

.. ...

. . ...

...

20 27.5 27.5 27.55 3 2

28.67 27.5 33.83 27.5

2931.667

5 4 2 22.38 27.5 27.00 27.5 33.12 27.5

2326.250

5 2 15.00 27.50 ... 39.00

E i

I j

cellsEI ij

SS nic X X

SS nec X X

SS nc X X

2

2 2 2

.. ...

27.50

5325.000

5325.000 2931.667 2326.250 67.083

5 4 3 21.33 27.5 33.66 27.5

4563.333

E I cellsEI E I

C k

SS SS SS SS

SS nei X X

2 2 2

. ... 5 3 16.00 27.50 ... 40.33 27.50

12,110.000

12,110.000 2931.667 4563.333 4615.000

cellsEC i k

E C cellsEC E C

SS ni X X

SS SS SS SS

2 2 2

. ... 5 4 15.00 27.50 ... 39.00 27.50

6945.000

6945.000 2326.250 4563.333 55.417

cellsIC ij

I C cellsIC I C

SS ne X X

SS SS SS SS

2 2 2

... 5 11.00 27.50 ... 48.00 27.50

14,680.000

14,680.000 2931.667 2326.250 4563.333 67.083 4615.000 55.417

121.25

cellsEIC ijk

E I C cellsEIC E I C EI EC IC

SS n X X

SS SS SS SS SS SS SS SS

41,151.000 14,680.000 26.471.000error total CellsC E ISS SS SS

156

Source df SS MS F

Experience 3 2931.667 977.222 3.544*

Intensity 2 2326.250 1163.125 4.218*

Cond Stim 1 4563.333 4563.333 16.550*

E x I 6 67.083 11.181 <1

E x C 3 4615.000 1538.333 5.579*

I x C 2 55.417 27.708 <1

E x I x C 6 121.250 20.208 <1

Error 96 26,471.000 275.740

Total 119 41,151.000

p . 05 [ F

. 05( 1 , 96) 3 . 94; F . 05( 2 , 96) 3 . 09; F

. 05( 3 , 96) 2 . 70; F . 05( 6 , 96)

2 . 19]

There are significant main effects for all variables with a significant Experience ×

Conditioned Stimulus interaction.

13.22 Sternberg study:

Tests of Between-Subjects Ef fects

Dependent Variable: RXTIME

11825.360a 5 2365.072 17.926 .000

1089380.280 1 1089380.280 8257.060 .000

9257.220 2 4628.610 35.083 .000

2165.453 1 2165.453 16.413 .000

402.687 2 201.343 1.526 .219

38788.360 294 131.933

1139994.000 300

50613.720 299

Source

Corrected Model

Intercept

NSTIM

YESNO

NSTIM * YESNO

Error

Total

Corrected Total

Type III Sum



Estimated Marginal Means

NSTIM * Present of Absent

Dependent Variable: RXTIME

49.000 1.624 45.803 52.197

57.540 1.624 54.343 60.737

59.120 1.624 55.923 62.317

62.180 1.624 58.983 65.377

64.600 1.624 61.403 67.797

69.120 1.624 65.923 72.317

Present of Absent

Present

Absent

Present

Absent

Present

Absent

NSTIM

1.00

3.00

5.00

Mean Std. Error Lower Bound Upper Bound

95% Confidence Interval

157

As the number of stimuli increase, so does the reaction time. The reaction time is also

longer when the test stimulus is not one of the comparison stimuli.

13.23 Analysis of Epineq.dat:

Tests of Between-Subjects Ef fects

Dependent Variable: Trials to reversal

141.130a 8 17.641 8.158 .000

1153.787 1 1153.787 533.554 .000

133.130 2 66.565 30.782 .000

2.296 2 1.148 .531 .590

5.704 4 1.426 .659 .622

214.083 99 2.162

1509.000 108

355.213 107

Source

Corrected Model

Intercept

DOSE

DELAY

DOSE * DELAY

Error

Total

Corrected Total

Type III Sum



13.24 The average of the nine within-cell variances will equal MSerror (2.162) in Exercise 13.23.

13.25 Tukey on Dosage data from Exercise 13.25

Multiple Comparisons

Dependent Variable: Trials to reversal

Tukey HSD

-1.67* .35 .000

1.03* .35 .010

1.67* .35 .000

2.69* .35 .000

-1.03* .35 .010

-2.69* .35 .000

(J) dosage of epinephrine

0.3 mg/kg

1.0 mg/kg

0.0 mg/kg

1.0 mg/kg

0.0 mg/kg

0.3 mg/kg

(I) dosage of epinephrine

0.0 mg/kg

0.3 mg/kg

1.0 mg/kg

Mean

Difference (I-J) Std. Error Sig.

Based on observed means.

The mean difference is significant at the .05 level.*.

.

All of these groups differed from each other at p < .05.

13.26 Three-way analysis of variance for Tab13-14.dat.

158

Source df SS MS F

Experience 1 1302.083 1302.083 48.78*

Road 2 1016.667 508.333 19.04*

Exper*Road 2 116.667 58.333 2.19

Condition 1 918.750 918.750 34.42*

Exper*Cond 1 216.750 216.750 8.12*

Road*Cond 2 50.000 25.000 0.94

Exper*Road*Cond 2 146.000 73.000 2.73

Error 36 961.000 26.694

Total 47 4727.917

13.27 Simple effects on data in Exercise 13.26.

Source df SS MS F

Condition 1 918.750 918.75 34.42*

Cond @ Inexp. 1 1014.00 1014.00 37.99*

Cond @ Exp. 1 121.50 121.50 4.55*

Cond*Exper 1 216.750 216.75 8.12*

Other Effects 9 2631.417

Error 36 961.000 26.694

Total 47 4727.917

*P < .05 [F.05(1.36) = 4.12]

13.28 Comparison of composers from different periods

2

..

2 2 2

2

.

2 2

2

( ) .

2 2

(12 12.325) (14 12.325) ... (8 12.325)

550.775

..

5*4[(14.750 12.325) (9.900 12.325) ]

235.225

..

5[(13.6 14.750) (10.8 14.750) ... (17.8 14.750

total

P i

C Classical j

SS X X

SS nc X X

SS n X X

2

2

( ) .

2 2 2

( ) ( ) ( )

( )

) ]

5 30.430 152.15

..

5[(10.2 9.900) (10.2 9.900) ... (10.0 9.900) ]

5 0.68 3.40

152.15 3.40 155.55

550.7

C Romantic j

Composer Period C Classical C Romantic

error total P C P

SS n X X

SS SS SS

SS SS SS SS

75 235.225 155.550 160.000

159

Source df SS MS F

Period

Error1

Composer(Period)

Error2

1

6

6

32

235.225

155.550

155.550

160.000

235.225

25.925

25.925

5.000

9.073*

5.185*

Total 39 550.775

* p < .05

There is a significant effect due to both Period ((Classical is rated higher) and Composer.

13.29 Dress codes and Performance:

2

..

2 2 2

2

.

2 2

(91 72.050) (78 72.050) ... (56 72.050)

13554.65

..

10*7[(73.929 72.050) (70.171 72.050) ]

494.290

total

Code i

SS X X

SS nc X X

2

( ) .

2 2 2

2

( ) .

2 2 2

(

..

10[(79.7 73.929) (71.5 73.929) ... (73.5 73.929) ]

10 147.414 1474.14

..

10[(68.5 70.171) (73.7 70.171) ... (71.1 70.171) ]

10 126.314 1263.14

School Yes j

School No j

School Co

SS n X X

SS n X X

SS

) ( ) ( )

( )

1474.14 1263.14 2737.28

13554.65 494.29 2737.28 10323.08

de School Yes School No

error total C S C

SS SS

SS SS SS SS

Source df SS MS F

Code

Error1

School(Code)

Error2

1

12

12

126

494.290

2737.280

2737.280

10323.08

494.290

228.107

288.107

81.931

2.166

2.784*

Total 139 13554.65

* p < .05

160

The F for Code is not significant but the F for the nested effect is. But notice that the two

F values are not all that far apart but their p values are very different. The reason for this

is that we only have 12 df for error to test Code, but 126 df for error to test School(Code).

161

13.30 Treating both variables in Ex13-31 as crossed

SPSS printout

* ( )

161.1 2576.186 2737.286

school school code school codeSS SS SS

13.31 Gartlett & Bos (2010) Same versus opposite sex parents. Cell means with variances in

parentheses.

Males Females

Same-

Sex

25.80

(12.96)

26.30

(25.00)

26.05

Opposite-

Sex

23.00

(16.00)

20.30

(20.25)

21.65

24.40

23.3

22

2 2

2 2 2 2

*

2 43 [(26.05 23.85) 21.65 23.85 ] 832.48

2 43 [ 24.40 23.85 23.3 23.85 ] 52.03

43 [ 25.8 23.85 26.3 23.85 23.00 23.85 20.3 23.85 ]

994.59

944.59 832.48 52.03 110.08

Parents

Gender

Cells

P G

error

SS

SS

SS

SS

MS

(12.96 25.00 16.00 20.25) / 4 18.55

162

Source df SS MS F

Parents

Gender

P*G

Error2

1

1

1

168

832.38

52.03

110.08

832.38

52.03

110.08

18.55

44.87*

2.80

5.93*

Total 171

* p < .05

There is a significant effect due to Same-Sex versus Opposite-Sex parents, with those

children raised by Same-Sex couples showing higher levels of competence. There is no

effect due to the gender of the child, but there is an interaction, with the male versus

female difference being greater in the Opposite-sex condition.

13.32 Analysis of Seligman et al. (1990)

If we think that males are generally more optimistic than females, then the sample sizes

themselves are part of the ―treatment‖ effect. We probably would not want to ignore that

if we are looking at sex as an independent variable. In fact, the lack of independence

between sample size and the effect under study is an important problem when it occurs.

13.33 This question does not have a specific answer.

chapter 11 - simple analysis of variancestatdhtx/methods8/instructors manual... · 2011. 7. 18. ·...

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