6 EXAMPLE: UNBALANCED ONE-WAY TREATMENT STRUCTURE
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Random Effects Models with Covariates
19
TABLE 12.8
Data for Example in Section 12.6 with One-Way Random Effects Treatment
Structure (Cities) Where the Response Variable is the Amount Spent on
Vocational Training and the Covariate is the Percent Unemployment
City 1
Spent
27.9
43.0
26.0
34.4
43.4
42.4
27.9
25.9
23.3
34.3
Unemp
4.9
9.4
4.2
6.6
9.5
8.8
4.6
4.6
4.2
6.8
City 6
Spent
28.0
18.2
29.2
25.0
22.6
34.9
39.3
37.3
30.5
Unemp
6.3
2.6
6.7
5.4
3.8
8.4
9.8
9.1
6.8
City 2
Spent
29.6
12.7
19.9
12.3
23.8
28.8
21.0
13.9
13.2
Unemp
8.1
0.9
4.6
1.9
5.9
8.0
4.4
2.1
1.3
City 7
Spent
37.6
34.7
38.6
33.8
10.1
29.8
15.1
35.0
Unemp
9.9
8.8
9.9
8.2
0.1
7.0
1.8
8.5
City 3
Spent
17.5
21.2
19.0
14.5
25.2
15.0
24.6
Unemp
3.1
6.4
4.0
2.0
8.4
2.4
8.7
City 8
Spent
10.6
21.4
32.1
28.3
38.7
19.8
40.0
36.3
29.0
Unemp
0.0
3.4
7.0
5.8
9.2
2.7
9.7
8.2
5.9
City 4
Spent
28.0
16.0
16.8
26.5
23.1
18.0
17.8
11.6
24.2
Unemp
9.6
3.3
4.2
8.6
6.2
3.1
3.9
0.5
7.4
City 9
Spent
10.9
12.0
14.1
15.4
11.2
17.9
26.2
14.2
21.4
Unemp
2.9
2.8
3.6
5.2
1.4
5.6
10.0
3.6
7.2
City 5
Spent
22.0
22.6
12.8
11.8
23.5
26.0
24.5
18.8
11.2
Unemp
6.7
6.9
2.2
1.4
8.4
9.3
8.0
5.7
1.8
City 10
Spent
8.6
9.1
13.6
13.7
13.4
15.3
23.1
22.2
Unemp
0.8
0.7
3.4
3.1
3.6
4.4
7.8
7.4
to be spent on vocational training (y) and the possible covariate is the level of
unemployment (x). The PROC MIXED code to fit Model 12.1 is in Table 12.9 where
ˆ a2 = 2.5001 from UN(1,1), σ
ˆ b2 = 0.4187 from
the estimates of the parameters are σ
ˆ ε2 = 0.6832. The mixed models estimates
UN(2,2), σˆ ab = 0.3368 from UN(2,1), and σ
ˆ = 9.6299, βˆ = 2.3956, so the estimate
of the mean of the intercepts and slopes are α
of the population mean spent income on vocational training at a given level of
ˆ y|EMP = 9.6299 + 2.3956 UNEMP. The solution for the random
unemployment is µ
effects that satisfy the sum-to-zero restriction within the intercepts and within the
slopes are in Table 12.10. The estimate statements in Table 12.11 provide predicted
values (estimated BLUPS) for each of the selected cities evaluated at 10% unemployment. Table 12.11 contains the predicted values for each of the cities evaluated
at 0, 8, and 10%. Tables 12.12, 12.13, and 12.14 contain the PROC MIXED code
and results for fitting Model 12.1 where the values of the covariate have been altered
by subtracting 2, then mean (5.4195402) and 8 from the percent unemployment.
The estimate of the variance component for the intercepts depends on the amount
subtracted from the covariate as described in Section 12.4. When X0 is subtracted
© 2002 by CRC Press LLC
20
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 12.9
PROC MIXED Code to Fit the Random Coefficients Model to the Vocational
Training Spending Data
PROC MIXED CL COVTEST DATA=CITY IC;
CLASS CITY;
MODEL Y=X/SOLUTION DDFM=KR;
RANDOM INT X/TYPE=UN SUB=CITY SOLUTION;
CovParm
UN(1,1)
UN(2,1)
UN(2,2)
Residual
Neg2LogLike
284.40
Effect
Intercept
X
Subject
CITY
CITY
CITY
Estimate
2.5001
0.3368
0.4187
0.6832
StdErr
1.3572
0.3910
0.2035
0.1180
ZValue
1.84
0.86
2.06
5.79
ProbZ
0.0327
0.3890
0.0198
0.0000
Alpha
0.05
0.05
0.05
0.05
Parameters
4
AIC
292.40
AICC
292.90
HQIC
291.07
BIC
293.61
CAIC
297.61
Estimate
9.6299
2.3956
StdErr
0.5401
0.2074
DF
9.2
8.9
tValue
17.83
11.55
Probt
0.0000
0.0000
Lower
1.0819
–0.4295
0.1944
0.5003
TABLE 12.10
Predicted Intercepts and Slopes for Each of the Cities
in the Study
Effect
Intercept
X
Intercept
X
Intercept
X
Intercept
X
Intercept
X
Intercept
X
Intercept
X
Intercept
X
Intercept
X
Intercept
X
© 2002 by CRC Press LLC
CITY
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
Estimate
0.9717
1.1022
–0.2659
0.0511
1.7366
–0.7753
0.8222
–0.5249
–1.0013
–0.4962
0.7862
0.5100
0.3338
0.4642
1.3563
0.6307
–2.3763
–0.5371
–2.3633
–0.4249
StdErrPred
0.8266
0.2280
0.6818
0.2249
0.7532
0.2282
0.7136
0.2225
0.7233
0.2215
0.8307
0.2271
0.7395
0.2180
0.7176
0.2201
0.7147
0.2260
0.6952
0.2290
df
23
13
19
12
22
13
21
12
21
11
24
13
22
11
21
11
21
12
20
13
tValue
1.18
4.83
–0.39
0.23
2.31
–3.40
1.15
–2.36
–1.38
–2.24
0.95
2.25
0.45
2.13
1.89
2.87
–3.33
–2.38
–3.40
–1.85
Probt
0.2516
0.0003
0.7009
0.8239
0.0308
0.0049
0.2624
0.0366
0.1807
0.0457
0.3535
0.0435
0.6561
0.0570
0.0727
0.0151
0.0033
0.0344
0.0029
0.0865
Upper
10.6709
1.1031
1.4648
0.9892
Random Effects Models with Covariates
21
TABLE 12.11
Estimate Statements Used to Provide Predicted Values for Each City at 10%
Unemployment with Predicted Values at 0, 8, and 10% Unemployment
estimate
estimate
estimate
estimate
estimate
estimate
estimate
estimate
estimate
estimate
‘city 1
‘city 2
‘city 3
‘city 4
‘city 5
‘city 6
‘city 7
‘city 8
‘city 9
‘city 10
at
at
at
at
at
at
at
at
at
at
10’
10’
10’
10’
10’
10’
10’
10’
10’
10’
intercept
intercept
intercept
intercept
intercept
intercept
intercept
intercept
intercept
intercept
1
1
1
1
1
1
1
1
1
1
x
x
x
x
x
x
x
x
x
x
10
10
10
10
10
10
10
10
10
10
|int
|int
|int
|int
|int
|int
|int
|int
|int
|int
At 0%
City
1
2
3
4
5
6
7
8
9
10
Estimate
10.6016
9.3641
11.3666
10.4522
8.6286
10.4162
9.9638
10.9862
7.2536
7.2667
1
1
1
1
1
1
1
1
1
1
x
x
x
x
x
x
x
x
x
x
10/subject
10/subject
10/subject
10/subject
10/subject
10/subject
10/subject
10/subject
10/subject
10/subject
1;
0 1;
0 0 1;
0 0 0 1;
0 0 0 0 1;
0 0 0 0 0 1;
0 0 0 0 0 0 1;
0 0 0 0 0 0 0 1;
0 0 0 0 0 0 0 0 1;
0 0 0 0 0 0 0 0 0 1;
At 10%
StdErr
0.7186
0.4799
0.6041
0.5370
0.5538
0.7246
0.5807
0.5438
0.5392
0.5048
Estimate
45.5793
33.8311
27.5696
29.1593
27.6221
39.4721
38.5619
41.2490
25.8383
26.9738
At 8%
StdErr
0.4894
0.6548
0.6448
0.5329
0.4907
0.4746
0.3932
0.4620
0.6225
0.7496
Estimate
38.5837
28.9377
24.3290
25.4178
23.8234
33.6609
32.8423
35.1964
22.1214
23.0324
StdErr
0.3255
0.4814
0.4650
0.3858
0.3561
0.3241
0.3091
0.3401
0.4469
0.5517
TABLE 12.12
PROC MIXED Code to Fit the Random Coefficients Model Using X2 = X – 2
as the Covariate
PROC MIXED CL COVTEST DATA=CITY IC;
CLASS CITY;
MODEL Y=X2 /SOLUTION DDFM=KR;
RANDOM INT X2/TYPE=UN SUB=CITY SOLUTION;
CovParm
UN(1,1)
UN(2,1)
UN(2,2)
Residual
Neg2LogLike
284.40
Effect
Intercept
X2
Subject
CITY
CITY
CITY
Estimate
5.5225
1.1742
0.4187
0.6832
StdErr
2.7060
0.6471
0.2034
0.1180
ZValue
2.04
1.81
2.06
5.79
ProbZ
0.0206
0.0696
0.0198
0.0000
Alpha
0.05
0.05
0.05
0.05
Parameters
4
AIC
292.40
AICC
292.90
HQIC
291.07
BIC
293.61
CAIC
297.61
Estimate
14.4211
2.3956
StdErr
0.7576
0.2074
DF
9.0
8.9
tValue
19.04
11.55
Probt
0.0000
0.0000
© 2002 by CRC Press LLC
Lower
2.5516
–0.0942
0.1944
0.5003
Upper
19.5882
2.4425
1.4648
0.9892
22
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 12.13
PROC MIXED Code to Fit the Random Coefficients Model Using XMN = X –
5.4195402, the Mean of the X Values or Percent Unemployment Data
PROC MIXED CL COVTEST DATA=CITY IC;
CLASS CITY;
MODEL Y=XMN /SOLUTION DDFM=KR;
RANDOM INT XMN/TYPE=UN SUB=CITY SOLUTION;
CovParm
UN(1,1)
UN(2,1)
UN(2,2)
Residual
Neg2LogLike
284.40
Effect
Intercept
XMN
Subject
CITY
CITY
CITY
Estimate
18.4486
2.6059
0.4187
0.6832
StdErr
8.7381
1.2808
0.2034
0.1180
ZValue
2.11
2.03
2.06
5.79
ProbZ
0.0174
0.0419
0.0198
0.0000
Alpha
0.05
0.05
0.05
0.05
Parameters
4
AIC
292.40
AICC
292.90
HQIC
291.07
BIC
293.61
CAIC
297.61
Estimate
22.6128
2.3956
StdErr
1.3615
0.2074
DF
9.0
8.9
tValue
16.61
11.55
Probt
0.0000
0.0000
Lower
8.7033
0.0956
0.1944
0.5003
Upper
61.9455
5.1162
1.4648
0.9892
TABLE 12.14
PROC MIXED Code to Fit the Random Coefficients Model Using X8 = X – 8
as the Covariate
PROC MIXED CL COVTEST DATA=CITY IC;
CLASS CITY;
MODEL Y=X8 /SOLUTION DDFM=KR;
RANDOM INT X8/TYPE=UN SUB=CITY SOLUTION;
CovParm
UN(1,1)
UN(2,1)
UN(2,2)
Residual
Neg2LogLike
284.40
Effect
Intercept
X8
Subject
CITY
CITY
CITY
Estimate
34.6853
3.6863
0.4187
0.6832
StdErr
16.4247
1.7897
0.2034
0.1180
ZValue
2.11
2.06
2.06
5.79
ProbZ
0.0174
0.0394
0.0198
0.0000
Alpha
0.05
0.05
0.05
0.05
Parameters
4
AIC
292.40
AICC
292.90
HQIC
291.07
BIC
293.61
CAIC
297.61
Estimate
28.7945
2.3956
StdErr
1.8670
0.2074
DF
9.0
8.9
tValue
15.42
11.55
Probt
0.0000
0.0000
Lower
16.3654
0.1785
0.1944
0.5003
Upper
116.4205
7.1941
1.4648
0.9892
from X, the resulting variance component is related to the variance of the models
evaluated at X0. Table 12.15 contains the estimates of the variance components for
the intercepts evaluated at X = 0, 2, 5.4195402, and 8 obtained from Tables 12.9,
12.12, 12.13, and 12.14 as well as the sample variance of the predicted values at
each of those values of X. The variance at X = 0 is the variance of the intercept
© 2002 by CRC Press LLC
Random Effects Models with Covariates
23
TABLE 12.15
Estimates of the Intercept Variance Components and the
Variance of the Predicted Values at Four Values of X
Amount Subtracted
from X
0
2
5.4195402
8
Estimate of the Intercept
Variance Component
2.5001
5.5225
18.4486
34.6853
Variance of the
Predicted Values
2.1678
5.3460
18.3628
34.5209
predictions from Table 12.10. The variance at X = 8 is the variance of the predicted
values from Table 12.11. The sample variances of the predicted values are not quite
equal to the REML estimates of the variance components since the sample variances
do not take into account the number of other parameters being estimated in the
models. However, the estimates of the variance components of the models evaluated
at a given value of X are predictable using the results from Section 12.4. In this
case, using the estimates of the variance components from Table 12.9, the estimate
of the variance component for the intercepts at X = 8 is computed as
σˆ a2* = σˆ a2 + 2 × 8 × σˆ ab + 82 × σˆ b2
= 2.5001 + 2 × 8 × 0.3368 + 82 × 0.4187
= 34.6853
where a* denotes the intercepts of the models with the covariate taking on the values
of X – 8. The REML estimates of variance components for the slopes [UN(2,2)] and
residual are the same in all four of the analyses: Tables 12.9, 12.12, 12.13, and 12.14.
Using the results in Section 12.2, a prediction band can be constructed about the
population of regression models. Table 12.16 contains the estimates of the amount to
be spent on vocational training from the estimated population regression model at 0,
1, …,10% unemployment. The estimated standard error of prediction at each of the
values of x and the simultaneous lower and upper 95% prediction intervals are also
included in Table 12.16. The Scheffé percentage point used was 2 F 0.05, 2, 70 = 2.50.
Figure 12.4 is a graph of the ten predicted regression lines, the mean and the upper
and lower prediction limits over the range of 0 to 10% unemployment. The graph
clearly shows that as the percent of unemployment increases, the width of the
prediction interval also increases. The next section applies the analysis to a two-way
random effects treatment structure.
12.7 EXAMPLE: TWO-WAY TREATMENT STRUCTURE
Three genetic lines of Hereford cattle were randomly selected from a population of
Hereford cattle producers, and three genetic lines of Black Angus cattle were randomly
© 2002 by CRC Press LLC
24
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 12.16
The Mean, Estimated Standard Error
of Prediction, and Lower and Upper
Prediction Intervals for Selected
Values of X (% Unemployment)
x
0
1
2
3
4
5
6
7
8
9
10
mean
9.63
12.03
14.42
16.82
19.21
21.61
24.00
26.40
28.79
31.19
33.59
se
1.78
2.07
2.49
3.00
3.55
4.13
4.72
5.33
5.95
6.57
7.20
low
5.17
6.85
8.19
9.32
10.34
11.29
12.19
13.07
13.92
14.76
15.59
up
14.09
17.20
20.65
24.31
28.08
31.93
35.81
39.73
43.67
47.62
51.58
Random Coefficient Models for Cities
60
Income ($1000)
50
40
30
20
10
0
0
1
2
3
4
5
6
7
8
9
10
Unemployment %
7
1
8
2
9
3
10
4
L
5
M
6
U
FIGURE 12.4 Graph of the predicted models for each city with the lower (L) and upper (U)
prediction band and the mean (M) of the regression model.
selected from a population of Black Angus cattle producers. The set of three genetic
lines from each population is not a large enough sample to provide reliable estimates
of the variance components in a genetic study, but the smaller number was used for
© 2002 by CRC Press LLC
Random Effects Models with Covariates
25
demonstration purposes. Cows (20) from each line of the Hereford cattle were
crossed with bulls from each line of the Black Angus, and 20 cows from each line
of Black Angus cattle were crossed with bulls from each line of Hereford. Thus,
there were 180 cows in the study where each cow gave birth to 1 calf. Five pastures
(denoted as Rep) were used for the study where 36 cows with calves were put into
each pasture where there were 4 cows and calves from each of the 9 crosses. The
response variable was the weight of the calf at weaning time and the birth weight
was considered as a possible covariate. Table 12.17 contains the data where Rep
represents the pasture and (yij, xij) represent the weaning weight and the birth weight,
respectively, for the ijth cross.
The two-way random coefficient model of Equation 12.5 was to be fit to the
data set. The PROC MIXED code in Table 12.18 fits Model 12.5 where all of the
covariances between the slopes and intercepts for each set are zero, “a” denotes
Hereford, “b” denotes Black Angus, y denotes the weaning weight and x denotes
the birth weight. In this analysis, method = type3 was used to provide method of
moments estimates of the variance components using type III sums of squares. This
option causes PROC MIXED to compute sums of squares, mean squares, expected
mean squares, appropriate error terms (Milliken and Johnson, 1992), approximate
error degrees of freedom, and F-statistics to test a hypothesis about each of the
effects in the model. Using the parameters in the description of Model 12.5, source
“a” tests H0: σ a2 = 0 vs Ha: σ2a > 0, source “x*a” tests H0: σ 2b = 0 vs. Ha: σb2 > 0,
source “b” tests H0: σc2 = 0 vs. Ha: σc2 > 0, source “x*b” tests H0: σd2 = 0 vs. Ha:
σd2 > 0, source “a*b” tests H0: σf2 = 0 vs. Ha: σf2 > 0, and source “x*a*b” tests H0:
σg2 = 0 vs. Ha: σg2 > 0. The results indicate that all of the variance components are
important except σa2 (source a or Hereford) and σ c2 (source b or Black Angus) with
significance levels of 0.1565 and 0.2140, respectively. The method of moments
estimates of the variance components are in Table 12.19 as well as the estimates of
the population intercept (230.2420) and slope (0.6281).
Based on the results of Section 12.4, the interpretations of the variance components σa2, σc2, and σf2 are that they represent the variation of the intercepts of the
regression models or the variation of the predicted models evaluated at X = 0 or at
a birth weight of 0, a place that is not meaningful to compare the regression models.
The researcher wanted to evaluate the variance components of the intercepts at a
birth weight of 75 pounds so a new covariate was computed as x75 = x – 75. The
PROC MIXED code in Table 12.20 fits the independent slopes and intercepts model
using x75 as the covariate. The sums of squares and tests for all terms involving
x75 provides identical values as those same terms in Table 12.18. The sums of
squares and tests for a, b, and a*b have changed as they are evaluating the variance
components of the new intercepts or the models evaluated at x = 75. At x = 75, the
significance levels corresponding to sources a , b, and a*b are 0.0569, 0.1039, and
0.000, respectively. The method of moments estimates of the variance components
are in Table 12.21. The estimates of the three variance components corresponding
ˆ a2 = 217.3864, σ
ˆ c2 = 129.4336, and σ
ˆ f2 = 121.1095. These
to the models at x = 75 are σ
variance components are much larger than those obtained for the models evaluated
at x = 0. The next step is to fit the correlated slopes and intercepts model to the
data. Table 12.22 contains the PROC MIXED code to fit the correlated slopes and
© 2002 by CRC Press LLC
26
TABLE 12.17
Weights of Calves at Weaning (yij ) and Birth Weights (xij ) of Crosses of the ith Hereford Genetic Line with
the jth Black Angus Genetic Line
y11
283
258
285
270
273
285
279
295
243
244
258
247
283
260
270
259
252
270
269
280
x11
94
53
96
71
63
83
71
98
63
65
90
70
88
53
69
50
51
82
80
99
© 2002 by CRC Press LLC
y12
262
248
252
261
264
270
273
268
237
241
241
240
263
252
264
261
254
257
247
249
x12
92
53
64
88
71
87
97
83
83
92
94
91
88
58
92
84
87
91
64
72
y13
261
258
272
263
285
286
267
278
244
255
236
239
273
267
266
274
259
264
269
257
x13
58
52
85
63
95
98
52
78
71
98
53
60
83
69
64
85
68
79
91
64
y21
304
293
311
303
293
287
286
288
261
283
281
283
281
287
310
290
309
306
287
299
x21
86
72
96
88
61
55
54
56
61
88
85
88
56
64
91
67
100
96
73
87
y22
280
259
271
279
285
287
279
273
241
250
254
237
266
274
261
277
254
270
260
262
x22
100
53
80
96
89
96
77
63
61
80
89
55
63
81
53
87
57
91
67
72
y23
283
266
270
264
294
277
286
273
264
245
247
254
282
283
277
284
266
275
272
278
x23
94
59
66
55
97
64
83
57
99
60
64
79
86
89
77
91
71
89
82
94
y31
334
336
344
302
345
321
331
319
320
300
303
312
329
321
337
337
318
324
343
313
x31
86
88
94
57
88
67
76
66
93
76
78
86
80
73
86
87
78
83
99
73
y32
282
281
288
297
308
287
315
319
266
283
285
288
294
292
291
291
298
301
294
286
x32
58
57
65
77
80
54
88
94
65
86
89
92
71
67
67
68
86
90
80
71
y33
278
280
278
275
295
276
272
273
252
266
263
246
269
278
288
291
268
258
282
269
x33
77
82
79
73
94
58
50
53
70
96
90
59
56
74
92
98
71
54
97
72
Analysis of Messy Data, Volume III: Analysis of Covariance
Rep
1
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1
1
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
Random Effects Models with Covariates
27
TABLE 12.18
PROC MIXED Code to Fit the Independent Slope and Intercept Model Using
Type III Sums of Squares with Expected Mean Squares and Error Terms
proc mixed scoring=2 data=twoway method=type3;
class rep a b;
model y=x/solution ddfm=kr;
random rep a x*a b x*b a*b x*a*b;
Source
x
df
1
SS
12837.5138
MS
12837.5138
rep
a
x*a
b
x*b
a*b
x*a*b
Residual
4
2
2
2
2
4
4
158
18310.2743
25.8699
686.0917
19.7484
811.8033
17.2849
212.1297
41.8099
4577.5686
12.9349
343.0459
9.8742
405.9017
4.3212
53.0324
0.2646
Source
x
ErrorDF
3.4
FValue
18.55
ProbF
0.0176
rep
a
x*a
b
x*b
a*b
x*a*b
158.0
4.0
4.0
4.0
4.0
158.0
158.0
17298.68
3.07
6.58
2.40
7.99
16.33
200.41
0.0000
0.1552
0.0543
0.2063
0.0400
0.0000
0.0000
EMS
Var(Residual) + 3616.3 Var(x*a*b) + 10849 Var(x*b) +
10849 Var(x*a) + Q(x)
Var(Residual) + 34.044 Var(rep)
Var(Residual) + 0.6016 Var(a*b) + 1.8047 Var(a)
Var(Residual) + 3696.7 Var(x*a*b) + 11090 Var(x*a)
Var(Residual) + 0.5877 Var(a*b) + 1.763 Var(b)
Var(Residual) + 3601.7 Var(x*a*b) + 10805 Var(x*b)
Var(Residual) + 0.6183 Var(a*b)
Var(Residual) + 3762.7 Var(x*a*b)
Var(Residual)
ErrorTerm
0.9783 MS(x*a) + 1.0041 MS(x*b) – 0.9611
MS(x*a*b) – 0.0212 MS(Residual)
MS(Residual)
0.9729 MS(a*b) + 0.0271 MS(Residual)
0.9825 MS(x*a*b) + 0.0175 MS(Residual)
0.9505 MS(a*b) + 0.0495 MS(Residual)
0.9572 MS(x*a*b) + 0.0428 MS(Residual)
MS(Residual)
MS(Residual)
intercepts model where four random statements are needed to complete the model
specification. The statement “random rep;” specifies that the replications or
pastures are a random effect. The statement “random int x75/type = un
subject = a solution;” specifies covariance structure for the intercepts and
slopes for the main effect of a or Hereford genetic lines. The statement “random
int x75/type = un subject = b solution;” specifies covariance structure
for the intercepts and slopes for the main effect of b or Black Angus genetic lines.
The statement “random int x75/type = un subject = a*b solution;”
specifies covariance structure for the intercepts and slopes for the interaction between
the levels of a and the levels of b or a cross between a Hereford line and a Black
Angus line. PROC MIXED would not fit this model without including some initial
values for the variance components. The estimates of the variance components
obtained from the uncorrelated model in Table 12.21 were used as starting values
where “0” was inserted in place of the covariances of the slopes and intercepts for
the main effect of a, for the main effect of b, and for the interaction of a and b.
© 2002 by CRC Press LLC
28
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 12.19
Method of Moments Estimates of the Variance Components
and Resulting Estimates and Tests of the Fixed Effects
CovParm
rep
a
x*a
b
x*b
a*b
x*a*b
Residual
Estimate
134.4528
4.8338
0.0262
3.2637
0.0329
6.5611
0.0140
0.2646
Effect
Intercept
x
Estimate
230.2420
0.6281
StdErr
5.5108
0.1458
df
5.0
3.9
tValue
41.78
4.31
Effect
x
NumDF
1
DenDF
3.9
FValue
18.55
ProbF
0.0131
Probt
0.0000
0.0131
These starting values are included in the “Parameters” statement in Table 12.22.
Table 12.22 contains the REML estimates of the variance components, which are
similar to those obtained from the method of moments, but not identical since covariances of the slopes and intercepts were also estimated. The REML estimates of the
three variance components from the correlated slopes and intercepts structure corresponding to the models evaluated at x = 75 are σˆ a2 = 225.0213 (from U(1,1) for subject =
a), σˆ c2 = 132.7903 (from UN(1,1) for subject = b), and σˆ f2 = 119.1411 (from UN(1,1)
for subject = a*b). These are the important variance components to be used in any
genetic analysis of this data set (no genetic analysis is included here). Table 12.23
contains the predicted values for the slopes (x75) and the models at X = 75 (Intercept)
where each set satisfies the sum to zero restriction within each type of effect. These
predicted values can be used to obtain the predicted slope and intercept for each level
of a (Hereford), for each level of b (Black Angus), and for each combination of levels
of a and b (crosses of Hereford and Black Angus). The predicted intercept for the first
Hereford genetic line is âp a1 = αˆ + âa1 = 277.3511 – 11.9311 = 265.4200 and the
predicted slope is bˆ p a1 = βˆ + bˆ a1 = 0.6275 – 0.1312 = 0.4963. The predicted model
ˆ p a = 265.42 + .4963 (birth wt – 75). Figure 12.5
for the first Hereford genetic is wt
1
is a graph of the three predicted models for the three Hereford genetic lines. Similar
models can be constructed for the three Black Angus genetic lines, and those
predicted models are displayed in Figure 12.6. The predicted slope and intercept for
a cross of the first Hereford genetic line and the first Black Angus genetic line is
bˆ p a11 = βˆ + bˆ a1 + bˆ b1 + bˆ ab1 = 0.6275 – 0.1312 + 0.1862 – 0.0793 = 0.6032 and
ˆ + âa1 + âb1 + âab1 = 277.3511 – 11.9311 + 11.3392 – 8.2703 = 268.4889.
âp ab11 = α
ˆ p ab11 = 268.4889 + 0.6032 (birth wt –
Thus, the model for the (1,1) combination is wt
75). The nine predicted regression lines for the Hereford and Black Angus crosses
are displayed in Figure 12.7.
© 2002 by CRC Press LLC
Random Effects Models with Covariates
29
TABLE 12.20
PROC MIXED Code to Fit the Independent Slope and Intercept Model with
x75 = x – 75 as the Covariate Using Type III Sums of Squares with Expected
Mean Squares and Error Terms
proc mixed scoring=2 data=twoway method=type3;
class rep a b;
model y=x75/solution ddfm=kr;
random rep a x75*a b x75*b a*b x75*a*b;
Source
x75
df
1
SS
12837.5138
MS
12837.5138
rep
a
x75*a
4
2
2
18310.2743
28862.0375
686.0917
4577.5686
14431.0187
343.0459
b
x75*b
2
2
19003.6174
811.8033
9501.8087
405.9017
a*b
x75*a*b
Residual
4
4
158
9067.4377
212.1297
41.8099
2266.8594
53.0324
0.2646
Source
x75
ErrorDF
3.4
FValue
18.55
ProbF
0.0176
rep
a
x75*a
b
x75*b
a*b
x75*a*b
158.0
4.0
4.0
4.0
4.0
158.0
158.0
17298.68
6.38
6.58
4.21
7.99
8566.49
200.41
0.0000
0.0569
0.0543
0.1039
0.0400
0.0000
0.0000
EMS
Var(Residual) + 3616.3 Var(x75*a*b) + 10849
Var(x75*b) + 10849 Var(x75*a) + Q(x75)
Var(Residual) + 34.044 Var(rep)
Var(Residual) + 18.662 Var(a*b) + 55.986 Var(a)
Var(Residual) + 3696.7 Var(x75*a*b) + 11090
Var(x75*a)
Var(Residual) + 18.652 Var(a*b) + 55.956 Var(b)
Var(Residual) + 3601.7 Var(x75*a*b) + 10805
Var(x75*b)
Var(Residual) + 18.715 Var(a*b)
Var(Residual) + 3762.7 Var(x75*a*b)
Var(Residual)
ErrorTerm
0.9783 MS(x75*a) + 1.0041 MS(x75*b) – 0.9611
MS(x75*a*b) – 0.0212 MS(Residual)
MS(Residual)
0.9972 MS(a*b) + 0.0028 MS(Residual)
0.9825 MS(x75*a*b) + 0.0175 MS(Residual)
0.9966 MS(a*b) + 0.0034 MS(Residual)
0.9572 MS(x75*a*b) + 0.0428 MS(Residual)
MS(Residual)
MS(Residual)
12.8 SUMMARY
Random effects treatment structures with covariates provides the usual random
coefficient regression model. The strategy of determining the simplest covariate part
of the model is applied here as one would for a fixed effects treatment structure,
except we are looking for variance and covariance components which help describe
the data instead of regression models. Predictions of the coefficients for a family of
regression models can be obtained and for some cases it might be important to obtain
the BLUP of the model for a specific level of the random effect and to compare
BLUPs for two or more specific levels of the random effect. It is also important to
evaluate the intercept variance components at values of the covariates that are
important to the study. The default is to provide estimates of the variance components
for the intercepts of the models, which may not be meaningful.
© 2002 by CRC Press LLC