Estimating Parameters

For a given data set, and a particular model, the problem of estimating the best fit dose-response parameters is one that can be approached by using maximum

TABLE 4.1. Empirical Dose-Response Functions


Dose-Response Relationship (P\)


Log-probit Weibull

y/2h exp dx

likelihood methods (Haas, Rose et al., 1999). This is a standard problem in risk assessment, which has been widely faced in chemical risk assessment (Crump, 1981) as well as microbial risk assessment. The estimation may be made using various computer programs, as well as in a spreadsheet environment (Haas, 1994).

Problem of Low-Dose Extrapolation

Different dose-response models may fit a single data set. For most data sets, particularly when human subjects are used, relatively few subjects per dose are tested, and the average doses used are fairly high (typically to produce an expected proportion of responses in excess of 10%). Under these conditions several different dose-response models may provide acceptable fits and may appear quite similar within the range of observation; however, when these models are used to extrapolate to lower doses they may provide dramatically different estimates of risk.

As an example of this, data for the infectivity of multiple nontyphoid strains of Salmonella fit to the beta-Poisson and the three empirical dose-response models in Table 4.1 are shown in Figure 4.2. The original data may be found in the report by Haas, Rose et al., (1999). The adequacy of the fit of the four models is about the same (the beta-Poisson model provided the best fit and is the only mechanistically consistent model tested). There is a large scatter to the experimental data (due to small numbers of subjects at most doses); however, the fit of the data to all of the models is fairly similar within the dose range tested (top panel of Fig. 4.2). However, when the best-fit parameters for the models are used to compute the dose-response relationship at low average dose, there is a dramatic spread between the models. As shown in the lower panel of Figure 4.2, at a mean dose of 10 2 organisms, there is a five order of magnitude range to the extrapolated risk between models. In this particular case, the beta-Poisson model estimates the lowest risk (at low dose), whereas the highest risk is estimated by the Weibull model—however, this relative ordering of models will be different for different data sets.

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