9 EXAMPLE: TEACHING METHODS WITH ONE COVARIATE MEASURED ON THE LARGE SIZE EXPERIMENTAL UNIT AND ONE COVARIATE MEASURED ON THE SMALL SIZE EXPERIMENTAL UNIT
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Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
37
experimental design being a one-way treatment structure in a completely randomized
design structure. The levels of sex were conceptually randomly assigned to the
students within each class; thus the student is the experimental unit for levels of
sex. The student experimental design is a one-way treatment structure in a randomized complete block design structure where there are multiple students of each sex
within each class or block. The data in Table 15.18 shows that there were 8 to
13 students of each sex within each classroom. Table 15.19 contains an analysis of
variance table for this split-plot design without using any of the covariate information. The teacher error term is based on 6 degrees of freedom and the student error
term is based on 181 degrees of freedom. Thus the approximate number of degrees
of freedom associated with the denominator for the various tests from the mixed
model can be between 2 and 183, depending on the number of parameters in the
covariate part of the model. Pre-tests were given to each student to measure the
initial knowledge of mathematics and the number of years of teaching experience
for each teacher were used as possible covariates. A model with factorial effects for
the intercepts, the slopes for years of experience and the slopes for pre-test scores
is in Equation 15.8. The results of fitting Model 15.8 to the teaching method and the
PROC MIXED code to do so are in Table 15.20. There are several terms in the covariate
part of the model with coefficients that are not significantly different from zero; thus
some model building needed to be carried out. The first term deleted was
pre*method*sex as it has the largest significance level — 0.9131. The additional terms
deleted using a backward or stepwise deletion process were pre*sex, yrs*method*sex,
and yrs*method. The resulting reduced model is
}
}
Sc ijkm = µ + ρi + θ Yrs ij + δ Pr e ijkm + κ i Pr e ijkm + a ij
teacher
+ τ k + (ρτ)ik + λ k Yrs ij + ε ijkm
student
(15.39)
The PROC MIXED code in Table 15.21 enables PROC MIXED to fit Model 15.39
to the teaching method data. The significance levels corresponding to pre*method
and yrs*sex are 0.0098 and 0.0494, respectively, indicating there is sufficient evidence to conclude that the slopes for years of experience are different for each sex
and the slopes for pre-test scores are different for the teaching methods. The full
rank model for the intercepts, the slopes for years of experience, and the slopes for
pre-test scores is
Scijkm = µ ik + κ i Pr e ijkm + λ k Yrsij + a ij + ε ijkm
(15.40)
The PROC MIXED code in Table 15.22 is used to fit the full rank model,
Equation 15.40. The REML estimates of the variance components and combined
estimates of the intercepts and slopes are in Table 15.22. The significance levels
corresponding to method*sex, pre*method, and yrs*sex are very small, indicating
there is sufficient evidence to conclude that all of the parameters are not equal to
zero. The adjusted means evaluated at four values of pre-test scores, 77.67 (mean
© 2002 by CRC Press LLC
38
TABLE 15.18
Test Scores for Male and Female Students Taught by One of Three Teaching Methods Where a Pre-Test
Score and Years of Teaching Experience are Possible Covariates
Teacher 1 with 7 years of exp
Method
I
I
I
I
I
I
I
I
I
I
I
I
ID
1
2
3
4
5
6
7
8
9
10
11
12
Score
82
80
75
77
80
82
77
75
79
77
Pre Sc
94
92
81
72
90
86
65
65
65
80
Female
Score
76
81
74
81
76
76
78
77
77
78
79
76
Pre Sc
69
79
79
79
70
77
80
63
84
70
83
87
Teacher 4 with 7 years of exp
Male
II
II
II
1
2
3
Score
80
75
78
© 2002 by CRC Press LLC
Pre Sc
90
76
84
Female
Score
79
85
85
Pre Sc
66
67
84
Teacher 2 with 4 years of exp
Male
Score
79
80
78
79
81
73
77
79
76
80
Pre Sc
67
74
92
72
75
60
85
76
81
85
Female
Score
77
78
78
77
77
76
79
79
77
Pre Sc
66
74
81
64
75
64
88
62
75
Teacher 5 with 8 years of exp
Male
Score
75
81
76
Pre Sc
81
80
74
Female
Score
79
82
86
Pre Sc
65
63
94
Teacher 3 with 4 years of exp
Male
Score
77
76
81
79
77
79
74
79
76
83
78
76
Pre Sc
67
93
79
72
62
82
62
88
63
91
62
93
Female
Score
79
76
71
76
72
79
77
74
Pre Sc
90
92
66
85
61
95
91
95
Teacher 6 with 3 years of exp
Male
Score
72
75
70
Pre Sc
89
63
80
Female
Score
76
76
74
Pre Sc
82
85
64
Analysis of Messy Data, Volume III: Analysis of Covariance
Male
4
5
6
7
8
9
10
11
12
76
78
71
70
78
78
77
74
76
76
94
85
87
89
91
90
63
86
85
80
79
81
80
81
79
76
68
66
91
84
90
90
Teacher 7 with 8 years of exp
Male
III
III
III
III
III
III
III
III
III
III
III
III
III
1
2
3
4
5
6
7
8
9
10
11
12
13
Score
85
81
84
91
88
86
82
81
82
81
80
83
Female
Score
89
94
85
84
85
84
91
88
88
89
88
84
83
Pre Sc
93
75
75
72
63
65
78
73
69
68
62
66
66
71
74
84
86
86
83
70
64
83
83
84
86
85
82
81
75
65
86
87
68
72
69
Teacher 8 with 6 years of exp
Male
Score
83
80
78
77
82
88
83
82
88
84
77
Pre Sc
63
64
61
65
74
89
83
87
77
75
65
Female
Score
84
84
88
84
85
84
88
89
90
85
Pre Sc
79
66
94
90
67
74
75
78
92
75
73
74
74
76
67
73
65
84
69
83
91
70
72
62
77
77
75
78
76
80
76
83
74
90
78
73
75
80
Teacher 9 with 6 years of exp
Male
Score
86
88
89
87
90
90
86
91
86
90
Pre Sc
66
85
86
86
94
91
83
90
74
89
Female
Score
99
99
89
95
90
91
92
92
92
90
94
90
92
Pre Sc
88
89
78
75
76
82
90
74
75
70
92
78
89
39
© 2002 by CRC Press LLC
Pre Sc
89
66
76
92
86
93
82
66
76
62
68
65
77
78
76
78
80
78
74
78
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
II
II
II
II
II
II
II
II
II
40
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 15.19
Analysis of Variance Table without Covariate
Information for the Teaching Method Study
Source
df
EMS
Method
2
σ + 21.293 σ a2 + φ 2 (method )
Error(teacher) = Class(method)
6
σ ε2 + 21.333 σ a2
Sex
1
σ ε2 + φ 2 (sex)
Method*sex
2
σ ε2 + φ 2 (method ∗ sex)
Error(student)
181
2
ε
σ ε2
TABLE15.20
PROC MIXED Code and Results for Fitting a Model with Two-Way Factorial
Effects for the Intercepts and Slopes for the Teaching Method Data Set
proc mixed data=score method=reml cl covtest;
class method teacher sex;
model score=method|sex pre pre*method pre*sex Pre*sex*method;
yrs yrs*method yrs*sex yrs*sex*method/ddfm=kr;
random teacher(method);
CovParm
Teacher(method)
Residual
Estimate
0.2385
5.5977
StdErr
0.4422
0.6039
ZValue
0.54
9.27
ProbZ
0.2948
0.0000
Effect
method
sex
method*sex
pre
pre*method
pre*sex
pre*method*sex
yrs
yrs*method
yrs*sex
yrs*method*sex
NumDF
2
1
2
1
2
1
2
1
2
1
2
DenDF
22.4
174.7
174.1
172.0
172.0
174.8
174.4
2.7
2.8
172.3
172.2
FValue
2.28
1.84
0.58
44.65
4.69
1.84
0.09
32.84
4.16
3.65
0.12
ProbF
0.1258
0.1763
0.5631
0.0000
0.0104
0.1766
0.9131
0.0134
0.1471
0.0577
0.8852
Alpha
0.05
0.05
Lower
0.0371
4.5798
Upper
32153.4762
6.9990
pre-test score), 70, 80, and 90, and the mean number of years of experience
(6.59 years) are in Table 15.23. The adjusted means evaluated at three values of
years of experience, 0, 5, and 10 years, and the mean pre-test score (77.67) are in
Table 15.24. The adjusted means evaluated at 0 years of experience correspond to
the expected mean test scores for new teachers. Since the slopes for years of
experience are a function of sex, the female and male students need to be compared
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
41
TABLE 15.21
PROC MIXED Code and Results for Fitting a Model with Two-Way
Factorial Effects for the Intercepts and One-Way Effects for the Slopes
Corresponding to Pre-Test Scores and for Years of Teaching Experience
proc mixed data=score method=reml cl covtest;
class method teacher sex;
model score=method|sex pre pre*method
yrs yrs*sex/ddfm=kr;
random teacher(method);
CovParm
Teacher(method)
Residual
Estimate
0.9198
5.5054
StdErr
0.7554
0.5853
ZValue
1.22
9.41
ProbZ
0.1117
0.0000
Effect
method
sex
method*sex
pre
pre*method
yrs
yrs*sex
NumDF
2
1
2
1
2
1
1
DenDF
150.4
177.5
177.3
181.2
181.1
5.0
177.6
FValue
0.40
0.13
28.64
49.17
4.75
23.67
3.92
ProbF
0.6685
0.7212
0.0000
0.0000
0.0098
0.0047
0.0494
Alpha
0.05
0.05
Lower
0.2938
4.5169
Upper
13.1064
6.8600
at three or more values. Figures 15.19 and 15.20 contain plots of the teaching method
by sex means in Tables 15.23 and 15.24. The female and male regression lines within
a teaching method are parallel when graphed over the levels of pre-test scores. The
regression models for the three teaching methods are parallel within each level of
sex when graphed over the years of experience. Table 15.25 contains the pairwise
comparisons of the female and male mean test scores within each level of teaching
method for the for 0, 5, 10, and 6.59 years of experience and a pre-test score of
77.67. The means of the females and the means of the males are not significantly
different (p = 0.05) for teaching method I with 6.59 and 10 years of experience and
for teaching method III with 0 years of experience. The slopes for pre-test scores
are a function of teaching method, the teaching methods need to be compared at
three or more values of pre-test scores. Tables 15.26 and 15.27 are comparisons of
the means of the three teaching methods within females and within males evaluated
at 6.59 years of experience and pre-test scores of 70, 80, 90, and 77.67. The means
of teaching method III are significantly larger than the means of teaching method I
which are larger than the means of teaching method II for all comparisons. The
above are LSD type comparisons, but other multiple comparison techniques could
be used in making this set of comparisons.
This example involves two covariates where one is measured on the whole plot
or large experimental unit and one is measured on the small experimental unit, a
combination of the examples in Sections 15.7 and 15.8. As long as the computations
are carried out using a mixed models approach, there is really no difference in the
analysis of this split-plot design than carrying out an analysis of covariance with a
© 2002 by CRC Press LLC
42
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 15.22
PROC MIXED Code to Fit a Full Rank Means Model for the Intercepts
and Slopes
proc mixed data=score method=reml cl covtest;
class method teacher sex;
model score=method*sex pre*method yrs*sex/noint ddfm=kr solution;
random teacher(method);
CovParm
Teacher(method)
Residual
Estimate
0.9198
5.5054
StdErr
0.7554
0.5853
ZValue
1.22
9.41
ProbZ
0.1117
0.0000
Alpha
0.05
0.05
Lower
0.2938
4.5169
Upper
13.1064
6.8600
Effect
method*sex
method*sex
method*sex
method*sex
method*sex
method*sex
pre*method
pre*method
pre*method
yrs*sex
yrs*sex
method
I
I
II
II
III
III
I
II
III
sex
F
M
F
M
F
M
F
M
Estimate
64.607
67.829
66.854
63.521
63.809
62.608
0.086
0.089
0.202
1.108
0.750
StdErr
2.569
2.520
2.956
2.938
2.972
2.872
0.029
0.033
0.031
0.210
0.213
df
82.7
79.7
81.6
80.7
36.2
32.7
179.5
180.8
182.0
7.3
7.7
tValue
25.15
26.91
22.62
21.62
21.47
21.80
3.00
2.69
6.58
5.28
3.52
Probt
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0031
0.0077
0.0000
0.0010
0.0084
Effect
method*sex
pre*method
yrs*sex
NumDF
6
3
2
DenDF
109.7
181.0
12.2
FValue
227.84
19.63
13.26
ProbF
0.0000
0.0000
0.0009
two-way treatment structure in a completely randomized or randomized complete
block design structure. The next example involves a strip-plot design with three sizes
of experimental units.
15.10 EXAMPLE: COMFORT STUDY IN A STRIP-PLOT
DESIGN WITH THREE SIZES OF EXPERIMENTAL
UNITS AND THREE COVARIATES
An environmental engineer designed an experiment to evaluate the affect of three
environmental conditions on the comfort of female and male human subjects. On a
given day, one female and one male were subjected to one of the three environments
where both persons were put into an environmental chamber at the same time. On
two other days, the same two persons were subjected to the other two environmental
conditions; thus each person is subjected to all three environmental conditions on
different days. After 1 hour in the environmental chamber, each person was given
a set of questions which were used to compute a comfort score where the larger
score indicates the person is warmer and the smaller score indicates the person is
© 2002 by CRC Press LLC
Analysis of Covariance for Split-Plot and Strip-Plot Design Structures
43
TABLE 15.23
LSMEANS Code to Provide Adjusted method*sex Means Evaluated
at the Mean Number of Years of Teaching Experience and Four
Values of Pre-Test Scores
lsmeans
lsmeans
lsmeans
lsmeans
Effect
method*sex
method*sex
method*sex
method*sex
method*sex
method*sex
method*sex/at
method*sex/at
method*sex/at
method*sex/at
method
I
I
II
II
III
III
sex
F
M
F
M
F
M
means diff;
(pre)=(70) diff;
(pre)=(80) diff;
(pre)=(90) diff;
Years
6.59
6.59
6.59
6.59
6.59
6.59
Pre=77.76
Estimate
78.57
79.43
81.04
75.35
86.82
83.26
Pre=70
Estimate
77.91
78.77
80.36
74.67
85.26
81.70
Pre=80
Estimate
78.77
79.63
81.25
75.56
87.29
83.73
Pre=90
Estimate
79.63
80.49
82.14
76.44
89.31
85.75
TABLE 15.24
LSMEANS Code to Provide Adjusted method*sex Means
Evaluated at the Mean Number of Pre-Test Score
and for Three Values of Years of Teaching Experience
lsmeans method*sex/at (yrs)=(0) diff;
lsmeans method*sex/at (yrs)=(5) diff;
lsmeans method*sex/at (yrs)=(10) diff;
Effect
method*sex
method*sex
method*sex
method*sex
method*sex
method*sex
method
I
I
II
II
III
III
sex
F
M
F
M
F
M
pre
77.67
77.67
77.67
77.67
77.67
77.67
yr=0
Estimate
71.27
74.49
73.74
70.41
79.52
78.32
yr=5
Estimate
76.81
78.24
79.28
74.16
85.06
82.07
yr=10
Estimate
82.35
81.99
84.83
77.91
90.60
85.82
cooler (Table 15.28). Figure 15.21 is a graphical representation of the randomization
process for this experiment. The picture depicts just 3 of the 12 blocks where 1 block
consists of 3 horizontal rectangles and 2 vertical rectangles. The two vertical rectangles represent two persons and the three horizontal rectangles represent the days
on which the two persons were observed within an environmental chamber set to
one of the environmental conditions. The randomization process is to randomly
assign sex of person to the two persons within each block (conceptually at least).
Thus the person is the experimental unit for sex of person. If day is ignored, the
person design is a one-way treatment structure (2 sexes) in a randomized complete
block design structure (12 blocks). The error term for persons is computed as the
© 2002 by CRC Press LLC