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15
Examples
277
Given the large difference (p < 0.001) between tmt means we may wish to allow
for heterogeneity of variance associated with tmt. Thus we fit a separate variety
variance for each level of tmt so that instead of assuming var (u2 ) = σ22 I 88 we
assume
"
#
2
σ2c
0
var (u2 ) =
⊗ I 44
2
0 σ2t
2 and σ 2 are the tmt.variety interaction variances for control and
where σ2c
2t
treated respectively. This model can be achieved using a diagonal variance structure for the treatment part of the interaction. We also fit a separate run variance
for each level of tmt and heterogeneity at the residual level, by including the
uni(tmt,2) term. We have chosen level 2 of tmt as we expect more variation
for the exposed treatment and thus the extra variance component for this term
should be positive. Had we mistakenly specified level 1 then ASReml would have
estimated a negative component by setting the !GU option for this term. The
portion of the ASReml output for this analysis is
6
7
8
9
LogL=-343.428
LogL=-343.234
LogL=-343.228
LogL=-343.228
Source
variety
run
pair
uni(tmt,2)
Variance
tmt.variety
tmt.variety
tmt.run
tmt.run
S2=
S2=
S2=
S2=
1.1498
1.1531
1.1572
1.1563
262
262
262
262
Model terms
Gamma
44
44
2.01903
66
66 0.276045
132
132 0.853941
264
264 0.176158
264
262
1.00000
DIAGonal
1
1.30142
DIAGonal
2 -0.321901
DIAGonal
1
1.20098
DIAGonal
2
1.92457
Analysis of Variance
7 mu
4 tmt
NumDF
1
1
df
df
df
df
:
1 components constrained
Component
2.33451
0.319178
0.987372
0.203684
1.15625
1.50477
-0.372199
1.38864
2.22530
DenDF
56.5
60.6
F_inc
1276.73
448.83
Comp/SE
3.01
0.59
2.59
0.32
2.77
2.26
-0.82
2.18
3.07
%
0
0
0
0
0
0
0
0
0
C
P
P
P
P
P
U
U
U
U
Prob
<.001
<.001
The estimated variance components from this analysis are given in column (b)
of table 15.8. There is no significant variance heterogeneity at the residual or
tmt.run level. This indicates that the square root transformation of the data has
successfully stabilised the error variance. There is, however, significant variance
heterogeneity for tmt.variety interactions with the variance being much greater
for the control group. This reflects the fact that in the absence of bloodworms the
potential maximum root area is greater. Note that the tmt.variety interaction
variance for the treated group is negative. The negative component is meaningful