Continuous dose-response modeling and risk analysis with the gamma and reciprocal gamma distributions

Kodell and West (1993) describe two methods for calculating pointwise upper confidence limits on the risk function with normally distributed responses and using a certain definition of adverse quantitative effect. But Banga et al. (2000) have shown that these normal theory methods break down when applied to skew data. We accordingly develop a risk analysis model and associated likelihood-based methodology when the response follows either a gamma or reciprocal gamma distribution. The model supposes that the shape (index) parameter k of the response distribution is held fixed while the logarithm of the scale parameter is a linear model in terms of the dose level. Existence and uniqueness of the maximum likelihood estimates is established. Asymptotic likelihood-based upper and lower confidence limits on the risk are solutions of the Lagrange equations associated with a constrained optimization problem. Starting values for an iterative solution are obtained by replacing the Lagrange equations by the lowest order terms in their asymptotic expansions. Three methods are then compared for calculating confidence limits on the risk: (i) the aforementioned starting values (LRAL method), (ii) full iterative solution of the Lagrange equations (LREL method), and (iii) bounds obtained using approximate normality of the maximum likelihood estimates with standard errors derived from the information matrix (MLE method). Simulation is used to assess coverage probabilities for the resulting upper confidence limits when the log of the scale parameter is quadratic in the dose level. Results indicate that coverage for the MLE method can be off by as much as 15% points and converges very slowly to nominal coverage levels as the sample size increases. Coverage for the LRAL and LREL methods, on the other hand, is close to nominal levels unless (a) the sample size is small, say N < 25, (b) the index parameter is small, say k 1, and (c) the direction of adversity is to the left for the gamma distribution or to the right for the reciprocal gamma distribution.     

Direct calculation of likelihood-based benchmark dose levels for quantitative responses

A benchmark dose (BMD) for quantitative responses is a lower confidence limit (LCL) on the effective dose corresponding to a specified risk level r. A commonly adopted method for calculating the BMD is to obtain a pointwise upper confidence curve U(d) on the risk function and then invert this relationship by solving the equation U(d)=r. The solution d is taken to be the BMD. Sciullo et al. (2000) have shown that the coverage achieved by this inversion method is at least as great as the coverage achieved by U (·) but that there is otherwise no general relationship between the two coverage probabilities. The present paper develops a method for direct calculation of the BMD based on the asymptotic distribution of the likelihood ratio statistic. It is further shown that the direct method and the inversion method are equivalent when U (·) is also based on the likelihood ratio. Since the direct method is known to be asymptotically correct, it follows that the LR-based inversion method is also asymptotically correct. However, the direct method is computationally faster and easier to program. Finally, some simulation studies are conducted to assess the small sample coverage probabilities of the direct method when responses follow either a normal or a gamma distribution.     

Sensitivity of normal theory methods to model misspecification in the calculation of upper confidence limits on the risk function for continuous responses

Normal theory procedures for calculating upper confidence limits (UCL) on the risk function for continuous responses work well when the data come from a normal distribution. However, if the data come from an alternative distribution, the application of the normal theory procedures may lead serious over- or under-coverage depending upon the alternative distribution. In this paper we conduct simulation studies to investigate the sensitivity of three normal theory UCL procedures to departures from normality. Data from several gamma, reciprocal gamma, and lognormal distributions are considered. The normal theory procedures are applied to both the raw data and the log-transformed data.