EXAMPLE OF FITTED MULTISTAGE MODEL

An example of the maximum likelihood estimate (MLE) of the multistage model to a hypothetical data set is shown in Figure 1.  The hypothetical dose-response data for Figure 1 are 0 individuals responding out of 50 individuals at risk (i.e., 0/50) at dose 0 (the controls), 0/50 at one-fourth of the maximum tolerated dose (MTD), 15/50 at one-half the MTD, and 45/50 at the MTD.  As suggested in Figure 1, the best fit of the multistage model usually passes close to the data points (that is, the estimated probabilities of the specified response occurring at the dose levels are usually close to the observed proportion of responses at these dose levels).  Although the fitted multistage model usually passes close to the observed data, its flexibility to pass exactly through each data point is limited by the structure (functional form) of the multistage model family and the usual regulatory restriction that the parameters in the model be zero or greater.  As indicated in the example in Figure 1 and emphasized in Figure 2, the observed proportion of animals with the specified response at lower doses (e.g., the dose MTD/4 in Figures 1 and 2) may be zero but the limited flexibility in the multistage dose-response model may prohibit the estimated model from estimating the response probability to be zero (or to be the same as the response probability at dose zero) until the dose is equal to zero.  That is, the form of the multistage model commonly used by federal and state regulatory agencies is forced to estimate that there is some increased risk at every dose greater than zero regardless of the observed data.

>> Figure 1

>> Figure 2