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Annex D. Correlation Between TD50and MTD

In this annex, we derive an analytical expression for the correlation between the TD50 and the MTD. To this end, suppose that the probability P(d) of a tumor occurring in an animal exposed to dose D = MTD satisfies the Weibull model

in (2.5), where the background parameter α > 0 and the shape parameter k > 0 are known. This is a generalization of the one-stage model used by Bernstein et al. (1985) in which k = 1.

Suppose that x of the n animals exposed to dose D develop tumors. Since and k are assumed known, β may be estimated by

where p0 = P(0) describes the spontaneous response rate. This leads to an estimate

of the TD50.

The estimate of β is appropriate for r ≤ x ≤ n-1. The lower limit of x = r is the minimum value of x that would lead to a statistically significant result at a nominal significance level of 0 < γ < 1; the value of r is determined from the fact that in the absence of a treatment effect at dose D, x follows the binomial distribution Bin (50, P(D)). The upper limit of x = n-1 is included since β, and hence TD50, is undefined for x = n.

The constraint x ≤ n-1 implies that



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OCR for page 166
APPENDIX F 166 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. ANNEX D. CORRELATION BETWEEN TD50 AND MTD In this annex, we derive an analytical expression for the correlation between the TD50 and the MTD. To this end, suppose that the probability P(d) of a tumor occurring in an animal exposed to dose D = MTD satisfies the Weibull model in (2.5), where the background parameter α > 0 and the shape parameter k > 0 are known. This is a generalization of the one-stage model used by Bernstein et al. (1985) in which k = 1. Suppose that x of the n animals exposed to dose D develop tumors. Since and k are assumed known, β may be estimated by where p0 = P(0) describes the spontaneous response rate. This leads to an estimate of the TD50. The estimate of β is appropriate for r ≤ x ≤ n-1. The lower limit of x = r is the minimum value of x that would lead to a statistically significant result at a nominal significance level of 0 < γ < 1; the value of r is determined from the fact that in the absence of a treatment effect at dose D, x follows the binomial distribution Bin (50, P(D)). The upper limit of x = n-1 is included since β, and hence TD50, is undefined for x = n. The constraint x ≤ n-1 implies that

OCR for page 166
APPENDIX F 167 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. whereas x ≤ r implies We wish to find the correlation between Y = logeTD50 and X = log eD. (Although the correlation will be identical using logarithms to the base 10, the derivation of the correlation given here is simpler using natural logarithms.) Suppose now that W = TD50 follows a uniform distribution on the interval [a,b], reflecting the fact that given the value D of the MTD, the estimated value of the TD50 is unrelated to the MTD. Suppose further that X follows some distribution with mean µ and variance σ2. Although Bernstein et al. (1985) observed that the empirical distribution of X is approximately normal, the correlation between Y and X does not depend on the distribution of X other than through its variance σ2. To calculate corr (Y, X), note that and where

OCR for page 166
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 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. and APPENDIX F Thus we have where µ = E(X), we have with V(X) = σ2. Noting that This leads to the desired result: 1/2 ≤ ρ ≤ 1. In the limiting case as n → ∞, (D. 13) reduces to The values of the correlation coefficient ρ in (D.13) as a function of 168 It can be shown that h2 - h12 + 1 ≥ 0, so that 0 < ρ ≤ 1. It can also be shown that ρ ↓ [σ2/(σ2 + 1)]1/2 as k ↓ 0, and that r 1 as k → ∞. Thus [σ2/(σ2 + 1)]