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
back-transformed estimates. The discrepancies in total weights predicted by
The CDA method is fully multivariate and is AR are the most serious issue in its use. The
appropriate for the multivariate nature of composi- alternative provided by CDA is a more natural
tional data; hypotheses can be tested not only framework for analysing compositional data, but
between treatments for a particular response variable there are some difficulties in calculating the S.E.D.
y but also across response variables and, at least between predictions on the proportional or original
approximately, for comparisons of treatments for weight scale when implementing this approach.
linear combinations of the back-transformed pro- There are similar difficulties with the AR method.
portions. For example, one could check whether the The CDA is a more complete framework for infer-
difference between muscle and fat proportions was ence as it is based on a multivariate model rather
constant across treatments. This is not possible for than a series of univariate analyses. It is a more
the usual implementation of AR which is a series of natural framework than AR for modelling part–part
univariate analyses, although it is possible to extend relationships as it preserves the symmetry among
this to a multivariate framework. components. Although links exist between CDA and
Several multivariate generalisations of AR have AR the CDA equations are not uniquely determined
been proposed. Joliceur 1963 proposed using the by them.
first principal component of the covariance matrix of CDA overcomes difficulties associated with AR as
the logarithm of the component variables to define a a tool in the analysis of carcass dissection data. It
series of pairwise relationships among the D com- does not suffer from the discrepancies between
ponents of the composition. The characterisation predicted totals and the total of the body being
depended on D parameters u , i 5 1, . . . ,D on which
i
predicted that arise in AR i.e. predicted component was based a D 3 D array with typical member
a 5
ij
proportions not summing to 1. Also, predicted cos
u cosu which determined the pairwise rela-
i j
components are constrained to lie between 0 and 1. tionships among the components with interpretation
CDA provides a simple framework for handling similar to that for the difference between CDA slope
developmental questions and the value of the regres- coefficients Eq. 4. However, there was nothing in
sion coefficients are simply interpretable in terms of the method that would force the predicted com-
relative changes in proportions of a developing body. ponents to sum to unity, the starting problem for the
Interpretations and inferences that are best made on current work. Another development was the intro-
the transformed scale y, such as the interpretation duction of ingenious multiphasic models e.g. Koops
of the regression coefficients, lead to relatively and Grossmann, 1991 to model growth as the sum
simple presentation of results. Questions on the of a number of components or phases, each with a
effects of treatments on the components or pro- specific growth form logistic, Gompertz etc. but
portions in a composition are best addressed by with different parameters reflecting different rates of
transformation of predicted values to the scale of the development. These phases are not explicitly ob-
data or the proportional scale. served but are jointly estimated in a multiparameter
Computing standard errors of difference for back- model, with several parameters estimated for each
D . Muldowney et al. Livestock Production Science 67 2001 241 –251
249
phase. It is not obvious from the mathematical models and CDA as detailed in Appendix A. How-
development that, were the separate phases to be ever, this link does not simplify inferences when
estimated for a particular body measure, the com- analysis is performed solely on the AR scale.
ponents would sum to the body measure, as would The AR coefficients derived in the analysis for the
be desirable for reasons advanced above. current paper 0.743, 0.619, and 2.094 for muscle,
CDA has the added advantage that it is par- bone and fat, respectively differ slightly from those
ticularly suitable for the study of part–part relation- of Keane et al. 1990 for the same data set 0.726,
ships between carcass components. Traditional AR 0.625, and 2.057 for muscle, bone and fat, respec-
deals with this symmetric idea asymmetrically, with tively. In the earlier analysis fat included cod fat
one component being nominated the ‘x’ indepen- scrotal fat and the components were regressed on
dent variable and the other the ‘y’ dependent actual side weight. In contrast, in the present study
variable, without any compelling rationale for which carcass side weight was taken as the sum of the
component is so denominated. Were the roles re- dissected muscle, bone including other tissue and
versed a different regression equation would emerge, fat subcutaneous and intermuscular components
which is unsatisfactory. CDA, through the logratio from the joints, which excludes cod fat. These small
transformation, models the difference between the differences between the earlier and present analysis
logarithms of the components as a function of other do not affect the interpretation. In both analyses the
variables which may include total body size, time estimated regression slope coefficients for the muscle
and or other variables and deals with the relationship and bone components are both less than 1 indicating
in a single equation. that these components decreased as proportions of
An original argument by Haldane against the side as the carcass side weight increased. The
allometric relationship cited in Huxley, 1932 was coefficient for the fat component 2.094 is greater
that if each of the parts of the organism e.g. each than 1 indicating that fat increased as a proportion of
muscle bears an allometric relation to the whole side, as side weight increased.
organism, then the aggregate of a number of parts say all the component muscles aggregated to give
total muscle cannot have an allometric relation with
5. Conclusion
