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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
Sum of Squares df Mean Square F Sig.
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
Sum of Squares df Mean Square F Sig.
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
11.14 Model for Exercise 11.2:
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
11.15 Model for Exercise 11.3:
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
average variance (weighted)
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
Sum of Squares df Mean Square F Sig.
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
average variance (weighted)
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
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
ANOVA
GSIT
153.493 2 76.747 .898 .408
31790.736 372 85.459
31944.229 374
Between Groups
Within Groups
Total
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
ANOVA
ERRORS
147.970 2 73.985 36.197 .000
241.187 118 2.044
389.157 120
Between Groups
Within Groups
Total
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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:
Analysis of Variance for Errors
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
Individual 95% CIs For Mean
Based on Pooled StDev
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:
Analysis of Variance for Errors
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
Individual 95% CIs For Mean
Based on Pooled StDev
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
N Mean Std. Deviation Std. Error
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
Sum of Squares df Mean Square F Sig.
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
Sum of Squares df Mean Square F Sig.
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
Sum of Squares df Mean Square F Sig.
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
of Squares df Mean Square F Sig.
R Squared = .234 (Adjusted R Squared = .221)a.
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
of Squares df Mean Square F Sig.
R Squared = .397 (Adjusted R Squared = .349)a.
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.