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To explore the extent to which this result is influenced by sample size, similar calculations were performed for a series of experiments with sample sizes ranging from n=50 to n=1000 animals per group (Table 1). These results indicate that the correlation between the log10(TD 50) and the log10(MTD) remains high, even at the larger sample sizes for which the allowable range of potency values becomes much wider. Even in the limiting case of n = ∞, we have ρ = 0.944 (see annex D).

3.4 Model Dependency

Bernstein et al. (1985), Crouch et al. (1987), and Reith and Starr (1989ab) all used a one-stage model to characterize the carcinogenic potency. The one-stage model does not accommodate the majority of dose-response curves which exhibit curvilinearity. Kodell et al. (1990) argue that this limits the range of estimates of potency, and so artificially increases the correlation between the estimates of potency and the

TABLE 1 Correlation Between Carcinogenic Potency and the Maximum Tolerated Dose as a Function of Sample Sizea

Sample Size n

Range of Experimental Outcomes, x/n

Range of Potency Estimates (upper limit ÷ lower limit)

Correlationc

Minimum

Maximum

50

0.200

0.98

32

0.965

100

0.150

0.99

79

0.957

500

0.122

0.998

247

0.950

1000

0.116

0.999

349

0.949

b

0.1

1.0

0.944

aBased on a one-stage model and the assumptions in annex D.

bLimiting case as n → ∞.

cρ=Corr(logTD50,log10MTD)



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APPENDIX F 128 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. To explore the extent to which this result is influenced by sample size, similar calculations were performed for a series of experiments with sample sizes ranging from n=50 to n=1000 animals per group (Table 1). These results indicate that the correlation between the log10(TD50) and the log10(MTD) remains high, even at the larger sample sizes for which the allowable range of potency values becomes much wider. Even in the limiting case of n = ∞, we have ρ = 0.944 (see annex D). 3.4 Model Dependency Bernstein et al. (1985), Crouch et al. (1987), and Reith and Starr (1989ab) all used a one-stage model to characterize the carcinogenic potency. The one- stage model does not accommodate the majority of dose-response curves which exhibit curvilinearity. Kodell et al. (1990) argue that this limits the range of estimates of potency, and so artificially increases the correlation between the estimates of potency and the TABLE 1 Correlation Between Carcinogenic Potency and the Maximum Tolerated Dose as a Function of Sample Sizea Correlationc Sample Size n Range of Experimental Range of Outcomes, x/n Potency Estimates (upper limit ÷ lower limit) Minimum Maximum 50 0.200 0.98 32 0.965 100 0.150 0.99 79 0.957 500 0.122 0.998 247 0.950 1000 0.116 0.999 349 0.949 ∞b 0.1 1.0 ∞ 0.944 aBasedon a one-stage model and the assumptions in annex D. bLimiting case as n → ∞. cρ=Corr(logTD ,log MTD) 50 10

OCR for page 128
APPENDIX F 129 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. MTD. Under the one-stage model, the potency β is related to the dose D = MTD and the added risk R(D) by For a population of chemicals, this relationship provides a linear regression of logeß versus loge(1/D) with a slope of unity. The error term, loge[-loge(1 - R (D))], expresses the variation in R(D). Since the extra risk at the MTD is likely to fall in the range of 0.10 to 0.98, the variation about 1/MTD is limited to a range of approximately loge[-loge(1 - 0.10)] = -2.25 to loge[-loge(1 - 0.98)] = 1.36. This corresponds to the approximate 30-fold range noted by Bernstein et al. (1985). Using (2.3), the relationship in (3.3) may be re-expressed as so that the TD50 is restricted to this same range. Kodell et al. (1990) suggest relaxing the linear restraints of the one-stage model and using the Weibull model in (2.5) to accommodate curvature. The Weibull model includes the one-stage model as a special case when k = 1. The TD50 and MTD are related by where k varies from chemical to chemical to accommodate either convexity (upward curvature, k > 1) or concavity (downward curvature, k < 1) in dose-response. This permits additional variation and reduces the correlation between loge(TD50) and loge (MTD) obtained with k fixed at unity. Bailar et al. (1988) demonstrate that an appreciable portion of the National Cancer Institute/ National Toxicology Program bioassays exhibit downward curvature (k < 1) (cf. Williams & Portier, 1992). Clearly, the one-stage model limits the values of potency estimates to a rather narrow range determined by the MTD, which contributes to the observed correlation between loge(TD50) and loge (MTD). We expect that the MTD will still be highly correlated with the TD50 derived from a Weibull model, although the degree of correlation will be somewhat less than is observed with the one-stage model. This expectation is confirmed by the correlation coefficients between