D . Muldowney et al. Livestock Production Science 67 2001 241 –251
245
concentration diet, ad is the interaction term for
joint distribution of y and y
and using the delta
ijk 1
2
the jth breed and the kth diet, b is the regression method based on Taylor series expansion of the
i
coefficient for the covariate logw, the natural back-transform Kendall and Stuart, 1977
logarithm of side weight and e
is the residual error
ijkl
term. This was taken as the most appropriate model for the data because there was no significant inter-
3. Results
action between the covariate and either of the factors, breed and dietary energy concentration, for
3.1. Compositional data analysis y
or y . The analysis is somewhat simplified for
1 2
expository purposes by the omission of year effects. The assumption of multivariate normality of y
1
This model was estimated using analysis of co- and y
was not rejected by the Anderson–Darling
2
variance or multiple regression in the statistical test Payne et al., 1993.
package Genstat Payne et al., 1993. Predictions of Analysis of covariance Eq. 8 with the factors,
treatment means for factors and interactions between breed, diet, their interaction and the covariate,
factors can be made from the estimated model on the logarithm of side weight logside was performed for
y scale and back-transformed to the x or proportional response variables y and y . Breed and diet affected
1 2
scale using the following back-transformations: logmuscle bone and logfat bone, with P values
of ,0.001, ,0.001, ,0.001 and 0.002, respectively.
ˆ ˆ
y y
1 2
e e
There was a significant P ,0.05 breed by diet ˆ
]]]] ˆ
]]]] ˆ
x 5 ,
x 5 and x
m ˆ
ˆ f
ˆ ˆ
b y
y y
y
1 2
1 2
interaction for logmuscle bone, but not for logfat e 1 e 1 1
e 1 e 1 1 bone. The covariate, logside had a significant effect
ˆ ˆ
5 1 2 x 1 x
9
m f
on both response variables, with P values of 0.001 ˆ
ˆ where y and y are predicted means for a treatment
and ,0.001, respectively.
1 2
from the models for the first and second logratio, Predicted intercepts for breed and diet and the
ˆ respectively. The bone proportion is taken as the
estimated regression coefficients b values are
i
common divisor in the creation of y and y but any given in Table 1 Although there is interaction in the
1 2
of the proportions can be taken as the common model we present the main effects here for simplicity
divisor depending on what particular ratios are of of exposition. The regression coefficients for the
interest Aitchison, 1986. muscle bone and fat bone ratios are 0.124 and
Computation of standard errors of difference for 1.475, respectively. These coefficients are both posi-
these back-transformed proportions is approximate tive indicating that the muscle bone ratio and the
and their computation is complex, relying on the fat bone ratio increased as the side weight increased.
Table 1 The intercepts for breed and diet and the regression coefficients for the covariate, logside Eq. 7 from the compositional analysis of y and
1
y
2
ˆ Variable
Breed Intercept
Diet Intercept
b S.E.
i
y 5 logx x
1 m
b a
a
HE 0.690
Medium 0.656
0.124 0.038
b b
FR 0.645
High 0.699
a
CH 0.699
S.E.D. 0.0132
0.0105 y 5 logx x
2 f
b a
a
HE 2
6.939 Medium
2 7.216
1.475 0.108
b b
FR 2
7.164 High
2 7.121
c
CH 2
7.417 S.E.D.
0.0365 0.02838
a,b,c
Intercepts with different superscripts are significantly different P ,0.05.
246 D
. Muldowney et al. Livestock Production Science 67 2001 241 –251 Table 3
A coefficient for logmuscle fat is estimated by
Predicted composition g kg of muscle, bone and fat for 111 and
ˆ ˆ
b 2 b 50.124–1.4755 21.351 see Eq. 4. This
1 2
202 kg side weights for each of the three breeds using com-
negative coefficient indicates that the muscle fat
positional data analysis CDA
ratio decreased as the side weight increased. The
Side weight kg Component
FR HE
CH S.E.D.
intercepts for breed or diet for logmuscle fat can
111 Muscle
654 640
689 41
be estimated similarly to the estimation of the
Bone 192
179 192
43
coefficient for logmuscle fat. For example, the
Fat 154
180 119
18
intercept for logmuscle fat for HE is estimated as
202 Muscle
555 528
607 33
0.690226.93957.629.
Bone 151
137 157
39
The intercepts and regression coefficients in Table
Fat 294
334 236
10
1 can be used to predict the muscle, bone and fat proportions for a breed, averaged over diet or for
diet, averaged over breed, for various side weights. Table 1 for 111 and 202 kg carcass side weights
For example, to predict component proportions for a for each of the three breeds averaged over dietary
150-kg side weight logarithm55.011 Hereford3 energy level Table 3. These were the minimum and
Friesian animal, averaged over diet, first predict maximum side weights in the data set. The heavier
ˆ ˆ
ˆ y and
y as
y logmuscle bone50.6910.124 animals had lower predicted proportions of muscle
1 2
1
ˆ 5.01151.3114 and y logfat bone5 26.9391
and bone and higher proportion of fat for all breeds.
2
1.4755.01150.4522. These predicted values can be Muscle proportion was higher in CH than the other
back-transformed Eq. 9 to give predicted muscle, two breeds and the advantage over HE was accen-
bone and fat proportions g kg of 591, 159 and 250, tuated for the heavier weight. Bone fraction was
respectively, which sum to 1000. similar for all breeds at a given weight and fat was
Comparable parameter estimates for the data lower for CH, the differential over HE being greater
analysed using AR are given in Table 2. The muscle, at the high side weight.
bone and fat components were analysed using a model with the same explanatory factors and vari-
3.2. Comparison of compositional data analysis ables as above.
with allometric regression In further predictions of composition, the propor-
tional contributions of muscle, bone and fat com- To compare the two approaches AR and CDA,
ponents were predicted using the CDA equations the muscle, bone and fat component weights were
Table 2 The intercepts for breed and diet and the regression coefficients for the covariate, logside when muscle, bone and fat components were
analysed using allometric regression ˆ
Variable Breed
Intercept Diet
Intercept b S.E.
i a
Muscle HE
0.757 Medium
0.803 0.743 0.0267
b
FR 0.793
High 0.799
c
CH 0.858
S.E.D. 0.0089
0.0071
a
Bone HE
0.067 Medium
0.147 0.619 0.034
b
FR 0.147
High 0.100
b
CH 0.159
S.E.D. 0.0113
0.010
a
Fat HE
2 6.872
Medium 2
7.069 2.094 0.0877
b
FR 2
7.017 High
2 7.021
c
CH 2
7.258 S.E.D.
0.0306 0.0224
a,b,c
Intercepts with different superscripts are significantly different P ,0.05.
D . Muldowney et al. Livestock Production Science 67 2001 241 –251
247 Table 4
Predicted components weights kg of muscle, bone and fat for 111 and 202 kg side weights for each of the three breeds using allometric regression AR and compositional data analysis CDA
Side weight Component
Friesian Hereford3Friesian
Charolais3Friesian S.E.D. CDA
kg AR
CDA AR
CDA AR
CDA 111
Muscle 73.03
72.62 70.45
71.04 77.92
76.48 4.56
Bone 21.41
21.29 19.75
19.92 21.66
21.26 4.79
Fat 17.19
17.09 19.87
20.03 13.50
13.25 2.02
Total 111.63
111 110.07
110.99 113.08
110.99 202
Muscle 113.97
112.18 109.94
106.68 121.6
122.64 6.73
Bone 31.00
30.51 28.61
27.76 31.36
31.63 7.87
Fat 60.25
59.31 69.63
67.56 47.33
47.73 2.01
Total 205.22
202 208.18
202 200.29
202
predicted for each of the three breeds using the two accounted discrepancies when estimating differences
methods for a 111- and 202-kg carcass side weight between treatments.
Table 4. For CDA, the predicted weights of each It may be objected that these comparisons were
component summed to the correct total apart from a carried out at the extremes of weight but these were
minimal effect of rounding for all breeds and both chosen to examine the limits of the consequences of
side weights, as expected from the theory. For AR, bias which would be expected to occur at one or
the total of predicted weights for the lighter carcass other of the extremes. However, were the predictions
differed from 111 by 0.63, 20.93 and 2.08 kg for made out at the mean logside weight of 5.004,
FR, HE and CH, respectively, and for the heavier corresponding to a side weight of 149 kg, the
carcass the differences were 3.22, 6.18, and 21.71 discrepancies in the AR prediction of total side
kg for the three breeds, respectively. These dis- weight would be 20.57, 20.80, and 21.01 kg for
crepancies are not the same for each breed and so Friesian, Hereford and Charolais, respectively, not
they will be confounded with breed differences in inconsiderable relative to the size of breed differ-
components. They are not negligible in size when ences in composition. Predictions at the three
compared with estimated treatment differences or the weights shows that there is no consistent pattern in
S.E.D. between treatments. For example, the differ- the direction or magnitude of the error in the sum of
ence in muscle weight between FR and HE predicted the predicted components from AR, but these dis-
by AR for the heavy carcass is 4.03 kg 113.97– crepancies can be sizeable when compared with
109.94 whereas the difference between predicted treatment effects and with the S.E.D. between treat-
total side weights is 22.96 kg 205.22–208.18. ments.
Since muscle contributes about 60 of the weight of these animals most of the discrepancy is likely to
3.3. Development of a single component lodge in the estimates of muscle and so the dis-
crepancy is likely to be about half of the size of the The data were re-analysed using CDA but with
estimated treatment difference. Furthermore, the only two components, muscle and non-muscle. The
discrepancy is appreciable compared with the size of bone and fat components were combined to form
the S.E.D. between treatments 6.73 for CDA but non-muscle, which as a proportion of side weight is
similar measures of variability would apply for AR.. 12x . One y variable was created:
m
For fat weight, the allocation of the discrepancy x
m
]] would tend to reduce the size of the fat difference
y 5 log
S D
1 2 x
m
estimated by AR by about 0.9 kg, not negligible compared with the size of the S.E.D. for fat weight
and the same model as above, with separate terms 2.01. It is not desirable to have such large un-
for breed and diet and their interaction, and logside
248 D
. Muldowney et al. Livestock Production Science 67 2001 241 –251
logarithm of carcass side weight as a covariate was transformed proportions or components of total side
fitted. The resulting regression coefficient 20.676 weights as in Table 4 poses a problem when using
60.069 shows that the muscle non-muscle ratio either AR or CDA. In both cases, standard errors and
declines as side weight changes. The estimated standard errors of difference are available for the
model allows prediction of the muscle proportion for predicted values on the transformed scale, but it is
different side weights for each breed3diet combina- debatable how best to estimate those associated with
tion as above. the back-transformed values as in Table 4. The
method proposed here for computing approximate S.E.D. values for the back-transformed predicted
4. Discussion component proportion can also be used for AR
