A Multivariate Generalization of Probit Analysis

Abstract

A generalized form of probit analysis is applied to the problem of the inference of the value of all unobservable continuous variable from quantal variables. The observational units are assumed to belong to known groups that are homogeneous with respect to the value of the continuous variable, and responses within and between units in the same group are assumed to be independent. The use of the model in bioassay and epidemiology is discussed. It is applied here to data on tooth eruption in order to estimate 'developmental age'. 1. lntroduction In probit analysis, a quantal variable is assumed to be controlled statistically by a continuous variable, the values of which are fixed by the investigator. In some applications, e.g. bioassay, the values of the continuous variable are fixed by experimental manipulation; in others, e.g. epidemiology, they are fixed by selection of subjects at specified levels of the continuous variable. The purpose of probit analysis is to estimate parameters for, and test the goodness of fit of, the integrated normal-error function relating the probability of the quantal response to the value of the continuous variable. In this paper, we apply a generalization of probit analysis, derived from models originally proposed by Lawley (1943), to multiple quantal variables under the control of a single continuous variable. Because of the presence of multiple quantal variables, the analysis can proceed on weaker assumptions and extend to a wider range of applications than does conventional probit analysis. The technique requires that the values of the continuous variable be fixed by the selection of subjects. In this case, we find that to estimate the parameters of the probit model we need assume only that the subjects are selected in relatively homogeneous groups. The value of the variable for each group is not assumed to be known in advance, as is the case in experimental applications, but is estimated along with the parameters of the error functions.