Analysis of multiple response data by generalized linear models.

Abstract

Animal Production in Australia ANALYSIS OF MULTIPLE RESPONSE DATA BY GENERALIZED LINEAR MODELS P.J. NICHOLLS* In many fields of animal production research a response variable may be classified into one of several discrete categories, e.g. the number of progeny born per dam, or degree of severity of a disease. The dependence of the response on one or more independent variables may be expressed in terms of a linear model. However, as the response variable has a discrete non-normal distribution, the appropriate method of estimating model parameters is by maximum likelihood rather than least squares. A generalized linear models procedure has been developed for certain distributions, in which maximum likelihood estimates of parameters are obtained by an iterative weighted least squares method. The procedure involves specification of three components: the distribution of the observations, the linear model and the transformation linking means of the observations with predictors from the linear model. Terms in the model are fitted sequentially and an analysis of dewiance table can be constructed to provide x2 tests of parameters. For complex models and for markedly non-orthogonal data, several sequences of terms need to be fitted to derive appropriate x2 tests for all terms. Two methods of applying generalized linear models to multiple response data are i) a univariate approach with the response variable treated as an independent variable, where distributions of counts are assumed to be Poisson, with a logarithmic link transformation and ii) a multivariate approach due to G.H. Cooney in which the response variable retains its dependent status, having a multinomial distribution with a particular logit link transformation. While the two distributional assumptions are theoretically equivalent, the multinomial logit method fits a much simpler model, but does not allow the estimation of contrasts between response categories that is possible with the Poisson method. Both methods are available in the N.S.W. Department of Agriculture's REG, the program used for analyses in the following example. A selection experiment on Merino ewes at Trangie, N.S.W. compared the reproductive performance of two flocks, one selected for fertility (F) and the other randomly selected (R) (K.D. Atkins, pers. comm.). Ewes from three consecutive drops in each flock were classified into three response categories, those rearing nil, one or two lambs to weaning, for five lambings at two to six years of age. The frequencies of ewes in each category were recorded for each flock, drop and age combination. Using the multinomial logit method, with the effects and interaction of year of drop and age of ewe in the model, the flock effect was significant (x$=8.51, PcO.05) but the interactions of flock with drop and age were nonsignificant. The Poisson method was then applied with the two degrees of freedom for the response split into single-degree-of-freedom contrasts, a linear effect (none vs twin) and a quadratic effect((none + twin) vs single). The respective x: flock x response components were 8.43 (PcO.01) and 0.08 (P>O.50). The estimated proportions of ewes rearing nil, one and two lambs to weaning were 0.206# 0.516 and 0.278, respectively, for flock F and 0.259, 0.525 and 0.216 for flock R. The difference between the flocks for this response is therefore characterized by the difference in their relative proportions of dry and twinrearing ewes. * Department of Agriculture, Biometrical Branch, Sydney, N.S.W. 2000. 498