Due to individual variability, we do not expect a sharp threshold for an effect; that is, a single value with no mortality at lower dose, and 100% mortality at a dose above that level. If the range of concentrations is chosen properly, the usual situation will exhibit a sigmoidal response vs. either the dose or the logarithm of the dose (Figure 19.2a). That is, as dosage increases, the percent mortality will increase in a smooth, increasing curve. The logarithm of the dose is often used, especially when the mean of the frequency distribution is no more than two or three standard deviations above zero. This prevents the theoretical problem of the distribution predicting toxic effects at negative dosages.
Each point on Figure 19.2a is the percent response, P,-, at one dosage, d,-. In the Daphnia experiment example, this would be the result from one of the test tubes. If a very large number of test tubes were tested covering a much larger number of different dosages (d,-), a histogram could be prepared showing the frequency distribution for mortality, called a tolerance distribution, T(d). In typical toxicity tests there aren't enough dosage levels to do this. Instead, T(d) can be estimated by numerically differentiating the dose-response curve, P(d), using successive pairs of points (Pi; d,), (Pi+1, di+1), plotted at the midpoint of each pair:
A sigmoidal dose-response curve would result in a bell-shaped tolerance distribution as a function of dose. This leads naturally to an assumption of the normal distribution for the shape of T(d) (Figure 19.2b). If this is a valid assumption, the dose-response curve will be given by the integral of the normal distribution:
where x is either the dose, d, or the logarithm of the dose, as described above. The distribution has two parameters, a and p, related to the mean, dmean, and standard deviation,
s, of the tolerance distribution. Several methods are available for estimating them. They could be computed directly from the points of T(d) computed using the method of moments. The procedure is as follows. First, equation (19.1) is used to compute T(d). Then the following equations are used to compute the mean and standard deviation:
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