STAT512
Analysis of Covariance
In Minitab
Example 1 An experiment has been set up to determine the effectiveness
of three new ergonomic designs for airplane control panels. Twentyfour pilots have been randomly selected for the experiment and
assigned to training in a flight simulator that contains one of the
control panels (eight planes per panel). After completion of training
on their respective control panels, the pilots are presented with
eight emergency situations in the flight simulator. The emergency
situations are presented in random order, and the total time (in
seconds) required to make all emergency responses is recorded for each
pilot. These data are found in the table below. The only factor of
interest in this experiment is panel configuration. The response
variable is reaction time. The table also gives the number of years of
experience each pilot has. The latter variable is not controlled by
the experimenter, but this uncontrolled variable (or covariate) may
influence reaction time. How can the effect of the covariate be
accounted for in the analysis of the data.
Inputting Data:
Reaction Time
6.7
6.1
6.0
5.9
7.3
7.7
6.0
6.4
5.8
6.5
6.8
6.3
6.0
5.4
5.7
5.4
6.0
6.5
7.0
7.0
7.2
6.8
6.6
7.4
Years Experience
7.7
17.4
10.7
22.1
6.1
4.1
16.5
8.8
11.2
2.6
4.1
5.3
4.7
13.1
14.6
8.1
13.8
8.2
7.0
6.0
10.9
11.2
8.9
3.0
Panel
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
2.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
3.0
Indicator 1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
Indicator 2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
Minitab Commands for Scatterplot: GRAPH > SCATTERPLOT > WITH GROUPS >
OK > Y-VARIABLE Reaction Time > X-VARIABLE Years of Experience >
CATEGORICAL VARIABLE Panel > OK
Reaction Time for the Three Panels with Covariate Years of Experience
8.0
Panel
1
2
3
Reaction Time
7.5
7.0
6.5
6.0
5.5
0
5
10
15
Years of Experience
20
25
Results from Performing One-Way Analysis of Variance
Note: These results can be used for comparison purposes later to learn
if any benefits are achieved from performing an analysis of
covariance.
One-way ANOVA: Reaction Time versus Panel
Source
Panel
Error
Total
DF
2
21
23
SS
2.790
6.346
9.136
MS
1.395
0.302
F
4.62
P
0.022
H0: 1   2   3
Ha: At least two of the treatment means are unequal
=.05
F=4.62
P=.022
Reject H0 in favor of Ha. The data provide sufficient evidence to
conclude that at least two of the treatment means are unequal.
Results from Performing an Analysis of Covariance
COMMANDS: STAT > ANOVA > GENERAL LINEAR MODEL > RESPONSE Reaction Time
MODEL ‘Years of Experience’ ‘Panel’ > COVARIATE Years of Experience >
OK > OK
Analysis of Variance for Reaction Time, using Adjusted SS for Tests
Source
Panel
Years of Experience
Error
Total
DF
2
1
20
23
Seq SS
2.7900
3.9175
2.4288
9.1363
Adj SS
3.8574
3.9175
2.4288
Adj MS
1.9287
3.9175
0.1214
F
15.88
32.26
P
0.000
0.000
H0:  Adj,1   Adj, 2   Adj,3
Ha: At least two of the adjusted treatment means are unequal
=.05
F=15.88
P=.000
Reject H0 in favor of Ha. The data provide sufficient evidence to
conclude that at least two of the adjusted treatment means are
unequal.
Note: Compared with the one-way analysis of variance, the analysis of
covariance has provided stronger evidence against the null hypothesis.
How Estimated Adjusted Treatment Means are Calculated
COMMANDS: STAT > REGRESSION > REGRESSION > RESPONSE VARIABLE Reaction
Time > PREDICTOR VARIABLES ‘Years of Experience’ ‘Indicator 1’
‘Indicator 2’ > OK
Regression Analysis: Reaction Tim versus Years of Exp, Indicator 1, ...
The regression equation is
Reaction Time = 7.55 - 0.0885 Years of Experience - 0.853 Indicator 1
+ 0.030 Indicator 2
Predictor
Constant
Years of Experience
Indicator 1
Indicator 2
S = 0.348480
Coef
7.5453
-0.08846
-0.8534
0.0302
R-Sq = 73.4%
SE Coef
0.2197
0.01558
0.1836
0.1806
T
34.35
-5.68
-4.65
0.17
P
0.000
0.000
0.000
0.869
R-Sq(adj) = 69.4%
From the output above,
ŷ = 7.545 - .088YrsExp - .853Indicator1 + .03Indicator2
The estimated adjusted treatment means can be obtained using the
following regression equations:
Control Panel 1: ŷ =7.545 - .088YrsExp - .853(0) + .03(0) = 7.545 - .088YrsExp
Control Panel 2: ŷ =7.545 - .088YrsExp - .853(1) + .03(0) = 6.692 - .088YrsExp
Control Panel 3: ŷ =7.545 - .088YrsExp - .853(0) + .03(1) = 7.575 - .088YrsExp
Estimated adjusted treatment means are calculated at the overall mean
for YrsExp which is 9.42. Thus the estimated adjusted treatment mean
for control panel 1 is
Control Panel 1: ŷ = 7.545 - .088YrsExp = 7.545-.088(9.42) = 6.7
Pairwise Comparisons Using Bonferroni Procedure
COMMANDS: STAT > ANOVA > GENERAL LINEAR MODEL > RESPONSE Reaction Time
MODEL ‘Years of Experience’ ‘Panel’ > COVARIATE Years of Experience >
OK > COMPARISONS > TERMS Panel > Bonferroni > OK > OK
Grouping Information Using Bonferroni Method and 95.0% Confidence
Panel
3
1
2
N
8
8
8
Mean
6.7
6.7
5.9
Grouping
A
A
B
Means that do not share a letter are significantly different.
Bonferroni 95.0% Simultaneous Confidence Intervals
Response Variable Reaction Time
All Pairwise Comparisons among Levels of Panel
Panel = 1 subtracted from:
Panel
2
3
Lower
-1.333
-0.442
Panel = 2
Panel
3
Center
-0.8534
0.0302
Upper
-0.3738
0.5020
-------+---------+---------+--------(-----*-----)
(-----*-----)
-------+---------+---------+---------0.80
0.00
0.80
subtracted from:
Lower
0.4276
Center
0.8836
Upper
1.340
-------+---------+---------+--------(-----*-----)
-------+---------+---------+---------0.80
0.00
0.80
Assessing Reasonableness of Normality and Equal Variance Assumption
COMMANDS: STAT > ANOVA > GENERAL LINEAR MODEL > RESPONSE Reaction Time
MODEL ‘Years of Experience’ ‘Panel’ > COVARIATE Years of Experience >
GRAPHS > ‘Histogram of Residuals’ ‘Normal Plot of Residuals’
‘Residuals versus fits’ > OK > OK
Assessing Reasonableness of Normality Assumption
Histogram
Normal Probability Plot
(response is Reaction Time)
(response is Reaction Time)
99
5
95
4
90
Frequency
Percent
80
70
60
50
40
30
3
2
20
10
1
5
1
-0.8
-0.6
-0.4
-0.2
0.0
Residual
0.2
0.4
0.6
0
0.8
-0.6
-0.3
0.0
Residual
0.3
0.6
From the above plots, the normality assumption is reasonable.
Assessing Reasonableness of Equal Variance Assumption
Versus Fits
(response is Reaction Time)
0.50
Residual
0.25
0.00
-0.25
-0.50
-0.75
5.5
6.0
6.5
Fitted Value
7.0
7.5
From the above plots, the equal variance assumption is reasonable.
Assessing Reasonableness of Equal Slopes Assumption
COMMANDS: STAT > ANOVA > GENERAL LINEAR MODEL > RESPONSE Reaction Time
MODEL ‘Years of Experience’ ‘Panel’ ‘Years of Experience*Panel’ >
COVARIATE Years of Experience > OK > OK
Source
Panel
Years of Experience
Panel*Years of Experience
Error
Total
DF
2
1
2
18
23
Seq SS
2.7900
3.9175
0.0015
2.4272
9.1363
Adj SS
0.8226
3.0918
0.0015
2.4272
Adj MS
0.4113
3.0918
0.0008
0.1348
F
3.05
22.93
0.01
P
0.072
0.000
0.994
H0: Panel and Years of Experience Do Not Interact (The slopes of the
different treatment regression lines are equal)
Ha: Panel and Years of Experience Do Interact
=.05
F = .01
p-value=.994
Fail to reject H0. The assumption of equal slopes for the different
treatment regression lines is reasonable.