D . Muldowney et al. Livestock Production Science 67 2001 241 –251
243
1995, the composition of sand Johnsson, 1990, of weight and, for b ,0, it decreases relative to the Dth
i
fossils Reyment and Kennedy, 1991 as well as proportion for increased carcass weight.
several examples cited in Aitchison 1986. The relationship between two components, neither
In subsequent sections of this paper the CDA of which is the reference component, can also be
theory is developed for application to the study of evaluated.
How the
ith component
proportion carcass composition, it is then applied to a set of data
changes by comparison with the jth rather than the and the results compared with those from AR.
Dth component proportion as total carcass weight Various aspects of the use of CDA are discussed.
changes is assessed by examining the sign of the estimate of b 2b ; positive, negative and zero
i j
values being interpreted as above. This follows readily from the CDA equations for the two com-
ponent proportions:
2. Material and methods
E y 5 a 1 b logw
i i
i
CDA is first described in general, then for a specific case where interest is focused on a single
E y 5 a 1 b logw
j j
j
component of composition. And so
2.1. Description of CDA x
i
] E y 2 y 5 log
i j
S D
x Let x 5 w w for i 5 1, . . . ,D D 5number of
j i
i
components represent the component proportions of 5
a 2 a 1 b 2 b logw 4
i j
i j
D
some whole, where o
x 5 1. The component
i 51 i
proportions can be transformed to produce D 21 new The intercept for the ratio of the ith to the jth
variables y , . . . , y using the logratio transforma-
ˆ ˆ
proportion is estimated by a 2 a and the coefficient
1 D 21
i j
ˆ ˆ
tion: y 5 log x x for i 51, . . . ,D 21. This trans-
s d
of logw is estimated by b 2 b .
i i
D i
j
formation uses the Dth or final component propor- The CDA equations can be used to predict com-
tion as the denominator for the transformation, but position at a given carcass weight. For a given total
the method will yield the same results irrespective of ˆ
weight, predictions y are made from the CDA
i
which proportion is taken as the denominator Ait- models for i 5 1, . . . ,D 2 1 and these predicted
chison, 1986. Once the data have been transformed values are back-transformed to the proportional x
to this y scale, provided that the relevant assumptions scale using the back-transformation:
hold usually that the y values follow a multivariate
ˆy
i
e normal distribution, any statistical modelling can be
ˆ ]]]]]]]
x 5
i ˆ
ˆ y
y
1 D 21
done on the y scale. The simplest CDA equation that e 1 ? ? ? 1 e
1 1
may be fitted relating the ith logratio y to the log
i
ˆ ˆ
ˆ for i 5 1, . . . ,D 2 1 and x 5 1 2 x 2 ? ? ? 2 x
.
D 1
D 21
of carcass weight is: In these back-transformation equations the numerator
E y 5 a 1 b logw 3
i i
i
is always less than the denominator for each pre- dicted proportion, thus constraining each predicted
where E y is the expected value of y . In this
i i
proportion to lie between 0 and 1. The predicted model, a is the intercept and b measures how the
i i
component proportions also sum to 1 as is shown ith proportion changes relative to the Dth proportion
below. The sum of the predicted proportions other as logw changes. If b equals 0 the ratio of the ith
i
than the Dth is: to the Dth component proportion remains constant as
ˆ ˆ
y y
logw increases, and if b equals 0 for all i, body
1 D 21
i
e 1 ? ? ? 1 e ˆ
ˆ ]]]]]]]
x 1 ? ? ? 1 x 5
5 change is isometric b equal to 0 has the same
1 D 21
ˆ ˆ
y y
i
1 D 21
e 1 ? ? ? 1 e 1
1 interpretation as the coefficient of logw being equal
to 1 in AR. For b .0 the ith proportion increases which is less than 1. The final predicted proportion,
i
ˆ relative to the Dth proportion for increased carcass
x , is obtained by subtraction as:
D
244 D
. Muldowney et al. Livestock Production Science 67 2001 241 –251
ˆ ˆ
ˆ x 5 1 2 x 1 ? ? ? 1 x
2.3. Application to experimental data
D 1
D 21 ˆ
ˆ y
y
1 D 21
e 1 ? ? ? 1 e 2.3.1. Experimental design and data
]]]]]]] 5
1 2
ˆ ˆ
y y
1 D 21
e 1 ? ? ? 1 e 1
1 The data used to evaluate CDA were from the
dissection of 162 beef carcasses described by Keane 1
]]]]]]] 5
6
ˆ ˆ
y y
et al. 1990, but excluding the data from the 27
1 D 21
e 1 ? ? ? 1 e 1
1 animals in the pre-finishing slaughter group. The
experiment was replicated with animals born in three and so summing all D predicted proportions Eqs. 5
consecutive years using 18 Hereford3Friesian HE, and 6 gives 1.
18 Friesian FR and 18 Charolais3Friesian CH The full range of multiple regression models,
steers per year. The animals were all reared together including factors and variates and their interactions
from shortly after birth to about 18 months of age can be used in CDA models on the y scale. In
and 400 kg live weight. They were then allocated to particular, interactions of treatment factors with
either a high High or a medium Medium energy logw implies different rates of development differ-
level diet, offered ad libitum, until slaughter at one ent b values of a component for different treatments.
of three target carcass weights: light 260 kg for the Treatments may also affect the size of intercepts.
three breeds, normal 300 kg for HE and FR, and 320 kg for CH and heavy 340 kg for HE and FR,
2.2. Development of a single component and 380 kg for CH. This gave a 3 breed types32
dietary energy
concentrations33 slaughter
So far the exposition has focused on part–part weights factorial arrangement of treatments with
relationships among the components of the com- nine animals per treatment group.
position. If a particular component the ith is of Carcasses were dressed, and the left side of each
primary importance and the composition of the was jointed and dissected into the muscle, bone, fat
remainder is of secondary or no importance then it is and ‘other’ tissue connective tissue and ligamentum
simpler to consider just the relationships for that nuchae components, as described by Williams and
component relative to the other components com- ¨
Bergstrom 1977. The ‘other’ tissue component was bined. The logratio transformation can be taken with
added to the bone component in the analysis. respect to the sum of the other components to give a
single response variable on the logratio scale as: 2.3.2. Analysis of experimental data
In the experimental data, there were D 53 pro- x
i
]] y 5 log
7
S D
i
portions, the muscle, bone and fat components as 1 2 x
i
fractions of the side of the carcass. For easier reference these will be represented by x , x and x ,
and the coefficient b of the CDA regression:
m b
f i
respectively. Two new variables were formed: y 5
1
logx x and y 5 logx x and these were taken E y 5 a 1 b logw
m b
2 f
b i
i i
as the two response variables in the analysis of the can be interpreted as measuring the relative change
data. A model consisting of breed type, levels of in the component within the whole. Thus, if b 5 0,
dietary energy concentration and their interaction,
i
the component maintains the same relative size in the and the logarithm of side weight logside as a
body of which it is a component as logw changes, covariate was fitted to y and y . The model may be
1 2
while b . 0 b , 0 means that it is becoming an written as:
i i
increasing decreasing fraction of the body as logw y
5 m 1 a 1 d 1 ad
1 b logw 1
e 8
ijkl i
ij ik
ijk i
ijkl
increases. If it is also of interest to study develop- ment relationships among a subset of components,
where, for the ith logratio y , m is an overall average
i i
this can be carried out independently using the subset and a
for j 51, 2 and 3, reflect the effects of the
ij
to define a subcomposition and applying CDA within three breeds, HE, FR and CH, d
for k 51 and 2
ik
the subcomposition Aitchison, 1986. reflect the effects of the two levels of dietary energy
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
