6 EXAMPLE: TWO-WAY MIXED EFFECTS TREATMENT STRUCTURE IN A CRD
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Mixed Models
9
design structure is a completely randomized design with four replications per fertilizer-variety combination. Weeds are a competitor of the wheat plants and since
there was not to be any weed control (or any other chemicals used) used on the
plots, a sample of the top one inch of the soil of each plot was obtained (100 g) and
the number of weed seeds were counted. The number of weed seeds per sample was
to be considered as a possible covariate. Table 13.2 contains the PROC MIXED code
to fit the independent slopes and intercepts model where the fixed effects are Fert,
Weed_s, and Weed_s*Fert and the random effects are Variety, Weed_s*Variety,
Variety*Fert and Weed_s*Varity*Fert. The significance levels associated with the
random effects involving the covariate are 0.0099 and 0.1275 for Weed_s*Variety
and Weed_s*Variety*Fert, respectively. This is an indication that possible
Weed_s*Variety*Fert is not needed in the model. To continue the investigation,
REML solutions were obtained with and without Weed_s*Variety*Fert and the AIC
were obtained to determine the need for the term. Table 13.3 contains the PROC
MIXED code to fit the full model and Table 13.4 contains the PROC MIXED code
to fit the reduced model. In both cases, the independent slopes and intercepts form
of the models were used, i.e., σab and σcd of Model 13.2 were set to zero. The AIC
for the full model is 417.83 and for the reduced model is 422.61, indicating that
Weed_s*Variety*Fert is a useful term in the model. The next step was to fit the
correlated slopes and intercepts model to the data. To specify the correlated slopes
and intercepts model or specify Σab and Σcd, the following random statements need
to be used:
Random int weed_s/type=un subject = variety; for ⌺ab and
Random int weed_s/type=un subject=variety*fert; for ⌺cd.
Unfortunately, that model with these two statements would not converge, so a simpler
covariance matrix was used. The model that did converge involved setting σab = 0,
which was accomplished by using the random statement:
Random int weed_s/type=un(1) subject = variety;
which sets the off diagonal elements of Σab equal to zero. Table 13.5 contains the
PROC MIXED code using the above random statements to fit Model 13.2 with σab =
0. The elements U(1,1), U(2,1), and U(2,2) corresponding to variety in the subject
column provide the elements of Σab and the elements U(1,1), U(2,1) and U(2,2)
corresponding to variety*fert in the subject column provide the elements of Σcd.
Element U(2,1) for variety is zero which is the result of using type = un(1) in the
random statement. The bottom part of Table 13.5 contains the tests for the fixed effects
in the model. The significance level associated with the weed_s*fert term in the model
is 0.0868 which is a test of the hypothesis that the fertility slopes are equal. One
could conclude there is not sufficient evidence (since the significance level is greater
than 0.05) to say the slopes are unequal and delete the weed_s*fert term from the
model. In which case an equal slopes model would be used to compare the levels of
fertilizer. However, on the other hand, one could say there is reasonable evidence
(since the significance level is close to 0.05) to conclude the slopes are likely to be
unequal and use the unequal slopes model to compare the levels of fertilizer. The
LSMEAN statements in Table 13.6 provide estimates of the regression models at
© 2002 by CRC Press LLC
10
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 13.2
PROC MIXED Code to Fit the Independent Intecepts and Slopes Model with
Type III Sums of Squares to Test Hypotheses about the Variance Components
proc MIXED cl covtest cl method=type3 data=ex_13_5;
class variety fert;
MODEL yield=fert weed_s weed_s*fert/ddfm=kr;
RANDOM variety weed_s*variety variety*fert weed_s*variety*fert;
Source
fert
df
3
weed_s
1
weed_s*fert
3
10.77039
3.59013
variety
5
44.87971
8.97594
weed_s*variety
5
34.91409
6.98282
variety*fert
15
29.52654
1.96844
weed_s*variety*fert
15
29.45335
1.96356
Residual
48
61.07080
1.27231
Source
fert
ErrorDF
22.0
FValue
10.12
ProbF
0.0002
weed_s
5.5
856.05
0.0000
weed_s*fert
24.6
2.06
0.1321
variety
20.7
4.94
0.0039
weed_s*variety
23.1
3.95
0.0099
variety*fert
weed_s*variety*fert
48
48
1.55
1.54
0.1262
0.1275
Source
Estimate
variety
17.09539
weed_s*variety
0.01056
variety*fert
5.18887
weed_s*variety*fert 0.00403
Residual
1.27231
© 2002 by CRC Press LLC
SS
54.40389
MS
18.13463
EMS
Var(Residual) + 0.1001
Var(variety*fert) + Q(fert)
Var(Residual) + 95.402
Var(weed_s*variety*fert) +
381.61 Var(weed_s*variety) +
Q(weed_s,weed_s*fert)
Var(Residual) + 117.5
Var(weed_s*variety*fert) +
Q(weed_s*fert)
Var(Residual) + 0.1047
Var(variety*fert) + 0.4188
Var(variety)
Var(Residual) + 123.46
Var(weed_s*variety*fert) +
493.85 Var(weed_s*variety)
Var(Residual) + 0.1342
Var(variety*fert)
Var(Residual) + 171.48
Var(weed_s*variety*fert)
Var(Residual)
4866.61369 4866.61369
StdErr
10.37430
0.00724
2.95962
0.00223
0.23428
ZValue
1.65
1.46
1.75
1.81
5.43
ErrorTerm
0.7458 MS(variety*fert) +
0.2542 MS(Residual)
0.7727 MS(weed_s*variety) +
0.2273 MS(Residual)
0.6852 MS(weed_s*variety*fert) +
0.3148 MS(Residual)
0.7805 MS(variety*fert) +
0.2195 MS(Residual)
0.72 MS(weed_s*variety*fert) +
0.28 MS(Residual)
MS(Residual)
MS(Residual)
ProbZ
0.0994
0.1446
0.0796
0.0708
0.0000
Mixed Models
11
TABLE 13.3
PROC MIXED Code to Fit the Independent Slopes and Intercepts Model
proc mixed cl covtest ic data=ex_13_5;
class variety fert;
model yield=fert weed_s weed_s*fert/ddfm=kr;
random variety weed_s*variety variety*fert weed_s*variety*fert;
CovParm
variety
weed_s*variety
variety*fert
weed_s*variety*fert
Residual
Neg2LogLike
407.83
Effect
fert
weed_s
weed_s*fert
Estimate
22.31045
0.01170
6.50305
0.00496
1.19896
StdErr
16.53183
0.00914
3.94944
0.00293
0.22239
ZValue
1.35
1.28
1.65
1.69
5.39
ProbZ
0.0886
0.1003
0.0498
0.0455
0.0000
Alpha
0.05
0.05
0.05
0.05
0.05
Lower
7.72770
0.00389
2.60757
0.00202
0.85950
Parameters
5
AIC
417.83
AICC
418.56
HQIC
413.66
BIC
416.79
CAIC
421.79
NumDF
3
1
3
DenDF
26.0
5.3
26.5
FValue
9.76
845.95
2.08
ProbF
0.0002
0.0000
0.1264
Upper
214.86710
0.13646
35.39015
0.02538
1.78936
TABLE 13.4
PROC MIXED Code to Fit the Reduced Independent Slopes
and Intercepts Model
proc mixed cl covtest ic data=ex_13_5;
class variety fert;
model yield=fert weed_s weed_s*fert/ddfm=kr;
random variety weed_s*variety variety*fert;
CovParm
variety
weed_s*variety
variety*fert
Residual
Estimate
19.96009
0.01321
13.15558
1.41400
StdErr
16.22121
0.00937
4.94595
0.24846
ZValue
1.23
1.41
2.66
5.69
ProbZ
0.1093
0.0794
0.0039
0.0000
Alpha
0.05
0.05
0.05
0.05
Lower
6.42950
0.00473
7.07120
1.03013
Neg2LogLike
414.61
Parameters
4
AIC
422.61
AICC
423.09
HQIC
419.27
BIC
421.77
CAIC
425.77
NumDF
3
1
3
DenDF
37.3
5.3
67.5
FValue
6.26
834.73
3.78
ProbF
0.0015
0.0000
0.0144
Effect
fert
weed_s
weed_s*fert
Upper
272.16194
0.11026
32.52760
2.06208
weed_s = 0, 25, 34.79 (the mean of weed_s), and 45 seeds per plot using the unequal
slopes model. For this study it is reasonable to estimate and compare the levels of
fert at weed_s equal to zero since with the use of chemical the researcher could
eliminate the weeds from the plots. The regression lines for the levels of fert are
© 2002 by CRC Press LLC
12
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 13.5
PROC MIXED Code to Fit the Final Model to the Data in Section 13.5
proc mixed cl covtest data=ex_13_5;
class variety fert;
model yield=fert weed_s weed_s*fert/solution;
random int weed_s/type=un(1) subject=variety solution;
random int weed_s/type=un subject=variety*fert solution;
CovParm
UN(1,1)
UN(2,1)
UN(2,2)
UN(1,1)
UN(2,1)
UN(2,2)
Residual
Subject
Estimate StdErr ZValue ProbZ Alpha Lower
Upper
variety
21.62078 15.68449 1.38 0.0840 0.05
7.61169 193.86792
variety
0.00000
variety
0.01156 0.00875 1.32 0.0931 0.05
0.00394
0.11970
variety*fert 3.83690 4.95720 0.77 0.2195 0.05
0.82909 1308.83111
variety*fert 0.07919 0.10718 0.74 0.4600 0.05 –0.13088
0.28926
variety*fert 0.00311 0.00355 0.88 0.1907 0.05
0.00075
0.32554
1.25670 0.25640 4.90 0.0000 0.05
0.87408
1.96094
Effect
fert
weed_s
weed_s*fert
NumDF
3
1
3
DenDF
15.0
5.0
15.0
FValue
12.59
893.71
2.65
ProbF
0.0002
0.0000
0.0868
TABLE 13.6
Least Squares Means for the Levels of Fertilizer at Four Levels
of Weed Infestation
LSMEANS
LSMEANS
LSMEANS
LSMEANS
fert/diff
fert/diff
fert/diff
fert/diff
Weeds = 34.79
fert
1
2
3
4
Estimate
65.749
71.224
75.371
80.903
StdErr
2.861
2.863
2.860
2.860
at
at
at
at
means adjust=simulate;
weed_s=45 adjust=simulate;
weed_s=25 adjust=simulate;
weed_s=0 adjust=simulate;
Weeds = 45
Estimate
50.050
56.790
60.301
66.623
StdErr
3.242
3.268
3.261
3.244
Weeds = 25
Estimate
80.808
85.070
89.826
94.600
StdErr
2.580
2.557
2.575
2.562
Weeds = 0
Estimate
119.254
120.420
126.732
129.571
StdErr
2.414
2.353
2.500
2.297
displayed in Figure 13.1. Table 13.7 contains the pairwise comparisons between the
levels of fert at each value of weed_s where the “adjust=simulate” option was
selected to control the error rates for multiple comparisons (Westfall et al., 1999).
The adjusted significance levels are in the column titled “Adjp.” At the 0.05 level,
the means for Fert levels 1 and 2 are never significantly different, the mean of Fert 1
is always significantly less than the means of Fert 3 and Fert 4, the mean of Fert 2
is significantly less than the mean of Fert 3 only at weed_s = 0, the mean of Fert 2
is always less than the mean of Fert 4, and the means of Fert 3 and 4 are never different.
© 2002 by CRC Press LLC
Mixed Models
13
Fertility Level Models with Random Varieties
150
Yield lbs
130
110
90
70
50
0.00
10.00
20.00
40.00
30.00
50.00
Number of Weeds
1
2
3
4
FIGURE 13.1 Graph of the fertility level models with unequal slopes.
The final step taken in this analysis was to provide predicted values for the
slopes and intercepts for each variety grown at each level fertilizer. The predicted
values were constructed by using the estimates of the slopes and intercepts for each
fertilizer level using the fixed effects solution and the predicted values of the slopes
and intercepts for each variety and for each variety*fert combination from the
ˆ i + ãj + c˜ ij and
random effects part of the solution. The predicted values are ãpij = α
b˜ pij = βˆ i + b˜ j + d˜ ij, where ãj are the predicted intercepts correspond to the jth variety, c˜ ij
are the predicted intercepts corresponding to the ijth combination of fert and variety, b˜ j are the predicted slopes correspond to the jth variety, and d˜ ij are the predicted
slopes corresponding to the ijth combination of fert and variety. The predicted
intercepts are in Table 13.8 and the predicted slopes are in Table 13.9. Figures 13.2
through 13.5 are plots of the predicted regression models for each of the fertilizer
levels. Prediction bands could be constructed for the population of variety models
at each level of fertilizer using the method in Chapter 12. The construction of the
prediction bands is left as an exercise.
13.7 EXAMPLE: TREATMENTS ARE FIXED AND LOCATIONS
ARE RANDOM WITH A RCB AT EACH LOCATION
The data in Table 13.10 is from an experiment to evaluate the effect of three levels
of a compound (denoted by Drug) designed to promote faster growth of calves. A
diet with none of the drug was included as a control and used as a fourth level of
drug (0 mg/kg). The experiment involved six blocks of four pens of eight calves at
each of seven locations where the levels of the drug were randomly assigned to the
four pens within each block. The response variable was the average daily gain (ADG)
© 2002 by CRC Press LLC
14
Analysis of Messy Data, Volume III: Analysis of Covariance
TABLE 13.7
Pairwise Comparisons of the Fertilizer Means Using the Simulate
Multiple Comparison Adjustment Method at Weed Seed Densities
of 0, 25, 34.79, and 45 per Sample for the Data in Section 13.5
Effect
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
fert
1
1
1
2
2
3
1
1
1
2
2
3
1
1
1
2
2
3
1
1
1
2
2
3
_fert
2
3
4
3
4
4
2
3
4
3
4
4
2
3
4
3
4
4
2
3
4
3
4
4
weed_s
34.79
34.79
34.79
34.79
34.79
34.79
45
45
45
45
45
45
25
25
25
25
25
25
0
0
0
0
0
0
Estimate
–5.475
–9.621
–15.153
–4.147
–9.679
–5.532
–6.739
–10.250
–16.573
–3.511
–9.833
–6.322
–4.262
–9.018
–13.792
–4.756
–9.530
–4.774
–1.166
–7.478
–10.317
–6.312
–9.151
–2.839
StdErr
2.125
2.120
2.121
2.121
2.121
2.117
2.485
2.478
2.455
2.511
2.489
2.474
1.891
1.909
1.899
1.884
1.868
1.892
2.022
2.168
1.963
2.143
1.911
2.052
df
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
15
tValue
–2.58
–4.54
–7.14
–1.95
–4.56
–2.61
–2.71
–4.14
–6.75
–1.40
–3.95
–2.56
–2.25
–4.72
–7.26
–2.52
–5.10
–2.52
–0.58
–3.45
–5.25
–2.94
–4.79
–1.38
Probt
0.0210
0.0004
0.0000
0.0695
0.0004
0.0196
0.0161
0.0009
0.0000
0.1823
0.0013
0.0219
0.0396
0.0003
0.0000
0.0234
0.0001
0.0234
0.5728
0.0036
0.0001
0.0100
0.0002
0.1867
Adjp
0.0893
0.0023
0.0000
0.2512
0.0022
0.0833
0.0673
0.0047
0.0001
0.5244
0.0062
0.0908
0.1534
0.0018
0.0000
0.0973
0.0009
0.0973
0.9378
0.0187
0.0009
0.0463
0.0017
0.5319
TABLE 13.8
Predicted Intercepts for Each Variety at Each Level
of Fert for the Data in Section 13.5
variety
Fertilizer
1
2
3
4
1
116.84
116.53
120.81
121.99
2
116.38
115.73
121.76
122.96
3
118.17
119.28
124.56
127.68
4
121.08
121.68
128.55
131.93
5
119.32
122.58
130.03
133.64
6
123.75
126.72
134.69
139.23
of the calves within each pen (grams/day). All of the calves did not weigh the same
nor were they of the same age at the start of the experiment. Eight calves were
randomly assigned to each pen. The average age in days and the average weight in
© 2002 by CRC Press LLC