Livestock Production Science 67 2001 241–251 www.elsevier.com locate livprodsci
Compositional data analysis in the study of carcass composition of beef cattle
a a ,
b
D. Muldowney , J. Connolly , M.G. Keane
a
Department of Statistics , University College Dublin, Belfield, Dublin 4, Ireland
b
Teagasc , Grange Research Centre, Dunsany, Co. Meath, Ireland
Received 10 June 1999; received in revised form 12 October 1999; accepted 7 May 2000
Abstract
Allometric regression AR has been widely used to model changes in the body composition of animals. However, predicted body component proportions based on AR equations do not necessarily sum to 1 and this discrepancy is
confounded with treatment effects on components of the composition. Predicted component proportions are not bounded to lie between 0 and 1. An alternative method, compositional data analysis CDA, which avoids these difficulties is proposed
for beef carcass dissection data. For a composition consisting of D components e.g. muscle, fat and bone a new set of D 2 1 variables is created based on the logarithm of the ratios of components to one of the components e.g. logmuscle
bone and logfat bone. Any statistical analysis can be applied on this scale, subject to the assumptions for that method of analysis being true. Regression models with simple interpretations in terms of animal development can be fitted to these
logratio variables. Some inferences and interpretations are best made on the scale of component proportions. Predictions made from the models on the logratio scale may be back-transformed to give compositions on the proportional scale which
obey the constraints that the component proportions sum to 1 and individually cannot exceed 1. The method generalises readily to multiple regression models involving factors and variables. CDA provides a fully multivariate framework for
dealing with carcass dissection data within which questions on the effects of treatments and covariates on component composition and the differences between components can be addressed. It is a more natural vehicle than AR for analysing
part–part relationships as it respects the symmetry between the components being compared. A simple relationship between CDA and AR models is developed.
2001 Elsevier Science B.V. All rights reserved.
Keywords : Beef cattle; Compositional data analysis; Allometric regression; Linear regression; Carcass composition
1. Introduction way, have for long been sought. Huxley 1924,
1932 related the growth of a part of the body w ,
i
Patterns of tissue growth in animals have been where i is the ith body part to growth of the whole
widely studied and methods of describing them w by the allometric equation:
mathematically, but also in a biologically relevant Elogw 5 c 1 d logw
1
i i
i
Corresponding author. Tel.: 1353-1-706-7103; fax: 1353-1-
where Elogw is the expected value of logw for
706-1186.
i i
E-mail address : john.connollyucd.ie J. Connolly.
a given value of logw, c is the intercept and d is
i i
0301-6226 01 – see front matter
2001 Elsevier Science B.V. All rights reserved. P I I : S 0 3 0 1 - 6 2 2 6 0 0 0 0 2 0 0 - 1
242 D
. Muldowney et al. Livestock Production Science 67 2001 241 –251
the slope of the relation. It was claimed that this ponent proportions sum to different totals for two
equation is frequently suitable for the description of treatments e.g. 1.02 and 0.99 then, in the com-
organ or part to whole body relationships based on parison of treatments for a particular component e.g.
the assumption that the changes in the size of the muscle with mean estimated proportions 0.62 and
organs or parts during growth is more dependent on 0.65 for the two treatments, how much of the
the absolute size of the whole than on the time taken difference 0.65–0.62 between treatment means is
to reach that size Berg and Butterfield, 1976. due to the treatment effect and how much is due to
Examination of the effects of rate of growth separ- the bias arising from the non-summation of the
ately from that of size has been described by predicted component proportions to unity?
Seebeck 1983. Attempts to provide a rationale for A second difficulty with AR is that individual
observed allometric rules of scaling within organisms predicted proportions of composition are not con-
continue Banaver et al., 1999; West et al. 1999. strained to lie between 0 and 1. This can be seen
Many studies of the development of beef animals more clearly by rewriting the AR equation in terms
have used allometric regression AR to examine of proportions, by subtracting logw from each side,
how changes in body composition through time are as:
affected by levels of various factors, for example Elogw w 5 logx 5 c 1 d 2 1log w
2
i i
i i
breed, nutrition or weight Berg et al., 1978a,b; Keane et al., 1990; Keane, 1994. Linear regression
In this equation, x is the weight of the ith com-
i
has also been used e.g. Nour et al., 1981. Allomet- ponent as a proportion of the total weight, and so a
ric regression, in this context, generally describes the predicted proportion should lie between 0 and 1.
relationship between the logarithm of the weight of Predicted proportions are obtained using Eq. 2 by
components of a body logw and the logarithm of taking the exponential of values predicted from the
i
the weight of the whole body logw. It is not equation. Eq. 2 is linear in logw, and unless
necessary for w to be the weight of the whole animal d 5 1 predictions from it are unbounded. For d . 1,
i i
or carcass and AR has been used to describe both logx increases unboundedly as w increases. The
i
part to whole and part to part relationships Huxley, exponential of these values, the predicted proportions
1932. For example, the growth of individual carcass x , are also unbounded and not constrained to lie
i
muscles can be described e.g. psoas, semiten- between 0 and 1 as is required for a proportion.
dinosus can be described relative to the growth of Thus, depending on the estimated parameter values
ˆ total carcass muscle w or to each other.
ˆc and d , and the value of w, a predicted component
i i
Two logical difficulties arise with the AR ap- proportion greater than 1 could occur. Note that
proach. Firstly, for a carcass of weight w the weights although the first difficulty does not arise when linear
of the component parts sum to w exactly. It is rather than allometric regression is used the second
desirable that predicted weights of carcass com- one does.
ponents from models of carcass composition should The objective of this study is to explore the
also share this property to avoid confounding effects importance of difficulties in the AR approach and to
of treatments with artifacts due to model specifica- evaluate an alternative method of analysis for carcass
tion. From the AR equations the ith component can dissection data, based on compositional data analysis
ˆ be predicted for a given weight w as w 5
CDA Aitchison, 1986. This serves the same
i ˆ
ˆc 1d logw
i i
ˆ ˆ
e , where c and d are the estimates of the
purposes as AR when used on carcass dissection
i i
parameters of the AR equation. However, there is data, but constrains predicted component proportions
nothing in the method that will constrain the pre- to sum to 1, and to lie between 0 and 1. Although
dicted components to sum to the total w. This means CDA does not appear to have been applied to carcass
ˆ ˆ
that the predicted component fractions, x 5 w w, dissection data, its usefulness in analysis of com-
i i
are not constrained to sum to 1. Apart from this positional data, which should obey the sum to unity
constituting a lack of elegance in the analysis it constraint on its components, has been recognised in
introduces inferential difficulties in the comparison other areas. These include habitat and dietary selec-
of treatments in experiments. If the estimated com- tion by animals Aebischer et al., 1993; Carroll et al.,
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
