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Chapter 8
One-way Analysis of Covariance
8
...

The one-way ANOCOVA model relates the response variable Y to a single factor and a
single explanatory variable X, (covariate)
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Uses of Analysis of covariance
1
...


To increase precision in randomised experiments
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Example:
A two-year experimental study was conducted to investigate whether vitamin C was
effective in the treatment of colds
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The students were monitored in the first and second years
of the study and the number of days that they had cold symptoms were recorded
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In the second year, they took pills
containing one of the following: no active ingredients, low doses of vitamin C, or high
doses of vitamin C
...
C
Placebo
Predays, X
Days, Y
Number of
Number of
days with cold
days with cold
symptoms in
symptoms in
the 1st year
the 2nd year
0
12
10
8
5
14
6
9
10
13
0
0
12
15
13
15
6
10
19
20

Predays, X
Number of
days with cold
symptoms in
the 1st year
14
16
5
12
0
8
12
5
19
14

Days, Y
Number of
days with cold
symptoms in
the 2nd year
12
13
8
10
0
4
9
10
10
8

High doses of Vit
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8
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There are four possible models that can be considered
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2
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To compare treatments ‘overall’ we need to assume parallel regression lines
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2
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2
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4
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The variances of the response variable for the conditional distributions described
above are equal
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The covariate is linearly related to the response variable within all levels of the
factor, and the slopes of the lines relating the covariate to the response variable
are equal across the levels of the factor
...
2
...
e
...
  k  0 , then
y ij    xij   ij
y

121

x

8
...
4 Model 4: One-way ANOVA model
If there is no linear relationship between X and Y but there are treatment effects, then
y ij     i   ij
y
treatment 1
treatment 3
treatment 2
x

8
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Tests of homogeneity-of-slopes (parallelism)
i
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Model 2 vs
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Tests of differences in adjusted treatment means
i
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Model 3 vs
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Tests of linear relationship between the response variable and the covariate
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e
...
Model 2

8
...


123

To fit non-parallel regression lines model
1
...
Click on days and then on  to move it to the Dependent Variable box
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Click on group and then on  to move it to the Fixed Factor(s) box
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Click on predays and then on  to move it to the Covariate(s) box
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Click on Model
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Click on Custom under Specify Model
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Click on group(F) in the Factors & Covariates box and then on  to move it to the
Model box
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Click on predays(C) in the Factors & Covariates box and then on  to move it to the
Model box
...
Check that the option Interaction is selected in the Build Terms box
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Click on group(F) and predays(C) in the Factors & Covariates box and then on  to
move it to the Model box
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11
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12
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You should then get the following ANOVA table from the SPSS output

125

Tests of Between-Subjects Effects
Dependent Variable: number of days with cold symptoms in the second year
Type III
Sum of
Squares a
df
351
...
218
1
10
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618
1
35
...
206 24
2960
...
800 29

Mean
Source
Square
F
Sig
...
319
5
...
001
Intercept
186
...
670
...
057

...
658
PREDAYS
156
...
179
...
519
1
...
249
Error
11
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R Squared =
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459)

Test for homogeneity of slopes
H 0 : 1   2   3  

F
Test statistic

MS ( Interaction)
MS ( Error )

F  Fk 1, N  2 k ,
Reject H 0 if observed
Observed F = 1
...
249 which is > 0
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The
results indicate the interaction between the covariate and the factor is not significant
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To fit ANOCOVA model
1
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Repeat steps 2 to 4 above
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Click on Model
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Click on Full Factorial
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6
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7
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8
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9
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10
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127

Click on Continue
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Mean
Deviation
11
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36
8
...
84
6
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47
8
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69

N

10
10
10
30

a
Levene's Test of Equality of Error Variances

Dependent Variable: number of days with cold symptoms in the
second year
F
df1

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...

a
...
555
3
172
...
955
1
158
...
245 26
2960
...
800 29

Mean
Source
Square
F
Sig
...
518 8
...
000
Intercept
172
...
973
...
955 14
...
001
GROUP
79
...
450

...
317
Total
Corrected Total
a
...
497 (Adjusted R Squared =
...
516
5
...
210

...
110
8
...
227
4
...


...
Error
t
Sig
...
563
1
...
711

...
457

...
812

...
337
1
...
400

...
041
1
...
655

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...

a
[GROUP=3]
0
a
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Estimated Marginal Means
Treatment group

Dependent Variable: number of days with cold symptoms in the second year
95% Confidence Interval
Lower
Upper
Mean a Std
...
011
1
...
719
14
...
715
1
...
404
10
...
674
1
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388
8
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Evaluated at covariates appeared in the model: number of days with
cold symptoms in the first year = 9
...


129

Dependent Variable: number of days with cold symptoms in the second year

Observed

Predicted

Std
...
Dev =
...
00
N = 30
...
50 -2
...
50 -1
...
50

0
...
50

1
...
50

2
...
45 with 2 and 24 degrees of freedom, p = 0
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The
results indicate that there are differences among the adjusted means for the three groups
This suggests that there is a relationship between treatment with vitamin C and the
number of post-treatment days with cold symptoms controlling for pre-treatment days
with cold symptoms
...
529 with 1 and 26 degrees of freedom, p = 0
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Adjusted Treatment Means
The mean number of days with cold symptoms adjusted for initial differences for the
three vitamin C groups are given in the Estimated Marginal Means table
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011), the low dose vitamin C group had a smaller adjusted mean
(7
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674)
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For a
description of how to conduct pairwise comparison, see Howell(1997); Chapter 16
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Hence the Normality assumption is satisfactory
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132

Practical 8: Analysis of Covariance
THC (tetrahydrocannibinol) is a major active ingredient in marijuana
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Different doses of
THC were injected into rats and their resulting activity measured in a vibration test
chamber
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The amount of THC could take 5 levels, the first level is zero acting as a control
and the amounts then increased (0, 0
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5 g, 1 g, 2 g)
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This was the activity
level observed under similar conditions before the THC was administered
...
34
3
...
33
2
...
62
5
...
95
1
...
42
1
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30
0
...
25
1
...
92
1
...
32
0
...
69
0
...
55
10
...
39
3
...
40
1
...
40
7
...
79
9
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1 g
Post

0
...
44
4
...
92
0
...
70
0
...
53
3
...
22

Pre

7
...
33
4
...
78
6
...
78
5
...
86
6
...
5 g
Post

5
...
16
1
...
36
3
...
51
3
...
92
3
...
94
6
...
90
3
...
76
4
...
32
7
...
29
4
...
48
2
...
67
1
...
07
2
...
00
4
...
62
3
...
90
2
...
82
4
...
69
5
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c)

Use the Analyze/Regression/Linear procedure to investigate whether the activity
level after treatment with THC is linearly related to the activity level before
treatment
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e)

Now use the General Factorial procedure to conduct an analysis of covariance
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133

Enter the data into SPSS from Shared (K):\sctms\som\ma2013\data\prac8
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2 g
Post

1
...
11
2
...
00
0
...
99
0
...
84
2
...
93

Questions for practical 8
General Questions
i)
ii)
iii)
iv)

What is meant by analysis of covariance?
What are the main purposes of using analysis of covariance?
Write down the analysis of covariance model and state its assumptions?
Explain how you would test the assumption of parallel regressions?

THC example
i)
ii)
iii)
iv)
v)
vi)

134

Comment on the scatter plot of post injection activity level against pre injection
activity level
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