Estimation of the multivariate logistic risk function: A comparison of the discriminant function and maximum likelihood approaches

Abstract We have considered the case in which the multivariate logistic model holds for risk of disease. That is, we assume the probability that a person with independent variable values X 1 ,… , x k develops the disease is given by P(X 1 ,…X k )=[1+e −a− ∑ i=1 k β i x i ] −1 for some constants α and β i . Truett, Cornfield and Kannel [1] have proposed the use of the linear discriminant function in estimating the coefficients α and β i . Their method is theoretically correct if the distribution of ( x 1 ,…, x k ) is multivariate normal both for the well and for the diseased populations, with equal variance-covariance matrices in the two populations, or if the linear compound α + Σβ i x i is univariate normal for the two populations, again with equal variance-covariance matrices, where the β i are now the population values of the linear discriminant coefficients. Walker and Duncan [3] suggest an iterative procedure which gives maximum likelihood estimates for the coefficients. On theoretical grounds the maximum likelihood method is preferable, since it does not assume any particular distribution for ( x 1 ,…, x k ) and it gives results which asymptotically converge to the proper values if the logistic model holds [5]. On the other hand, the discriminant function approach is computationally simpler and may be more convenient to apply for very large amounts or data or when computer facilities are limited. We have compared the two methods theoretically, for the case when all independent variables are of the attribute type, and empirically, using Framingham 12-yr incidence data for (1) coronary heart disease and (2) CHD death or myocardial infarction. Our results suggest the following if the logistic model holds but the normality assumptions on the independent variables are violated: 1. (a) β i which are zero will tend to be estimated as zero for large samples by the method of maximum likelihood, but not necessarily by the discrimination function method; 2. (b) if any β i are non-zero they will tend to be estimated as non-zero by either method, but the discriminant function approach will give asymptotically biased estimates for those β i and for α; 3. (c) empirically, the assessment of significance for a variable, as measured by the ratio of the estimated coefficient to its estimated standard error, is apt to be about the same whichever method is used; 4. (d) empirically, the maximum likelihood method usually gives slightly better fits to the model, as evaluated from observed and expected numbers of cases per decile of risk; 5. (e) there is a theoretical basis for the possibility that the discriminant function will give a very poor fit, even if the model holds. Of course we are not able to show that conclusion (c) will hold generally, even though it holds for our examples. Moreover, conclusion (a) warns that use of the discriminant function technique can give misleading results regarding significance. Hence, it would seem that use of the maximum likelihood method would be preferable, whenever practical, in situations where the normality assumptions are violated, especially when many of the independent variables are qualitative.