A Growth-Curve Model Approach to Multivariate Quantal Bioassay

The typical quantal bioassay is used to estimate that concentration which will affect a given response in lOOp% of the subjects of interest. When p = .50 we speak of EC50 for median effective concentration, or LC50 for a lethal effect. More explicitly, such a controlled laboratory experiment usually involves subjecting nj experimental units to a concentration cj and recording the number, Zj, of experimental units affected by this concentration level. For an increasing sequence of concentrations Cj, j = 1, 2, . . . , d, with different corresponding sets of nj subjects, a sequence, zj, of the number of subjects responding, is obtained. When a lethal effect occurs, the ratios zj/nj = pj, say, become mortality rates and the problem is to estimate the concentration, LC50, which corresponds to p = .50. Various univariate methods can be used to solve this problem, the most popular being probit analysis; see, for example, Finney (1971) or Hubert (1984). If the responses are recorded after a time of t hours, the parameter to be estimated is referred to as 'the t-hour LC50'. When the experiment is monitored over a sequence of time points, the corresponding response variables become functions of concentration and time. However, the application of univariate probit analysis at each time point neglects the dependency in time and also the possible interaction of time and concentration on the response. The analysis of indirect quantitative assays with correlated data has been described by Box and Hay (1953). Their analysis assumes a repeated-measures model; that is, the correlations between observations on the same experimental unit are assumed to be equal. Elashoff (1981) studied the effect of heterogeneity of variances and correlations on this analysis. V0lund (1980) treated the case in which these correlations are arbitrary, thereby extending the work of Rao (1954). Multivariate indirect quantal assays have been studied by Kolakowski and Bock (1981) and by Kooijman (1981). Kolakowski and Bock (1981) assumed a latent structural model with independence among the responses over time. Kooijrman (1981) assumed that the mortality counts follow a multinomial distribution with the probability of death being a function of time and concentration. Such a model does not differentiate between sampling units and experimental units. For example, if concentrations of copper are administered to tanks of 20 fish, the model requires the assumption that there is no error variation among tanks of fish. Some work on the inclusion of this extra variation in the model for experiments involving litters of subjects has been done by Segreti and Munson (1981).