An Equal Opportunity University CONTINUOUS DOSE-RESPONSE MODELING AND RISK ANALYSIS WITH THE GAMMA AND RECIPROCAL GAMMA DISTRIBUTIONS by Senin Banga, G. P. Patil, and C. Taillie Center for Statistical Ecology and Environmental Statistics Department of Statistics The Pennsylvania State University University Park, PA 16802 EPA Project Officer: Chris Saint Prepared with partial support from the Office of Research and Development, United States Environmental Protection Agency, Washington, DC under a Cooperative Agreement Number R-825385. The contents have not been subjected to Agency review and therefore do not necessarily reflect the views of the Agency and no official endorsement should be inferred. Technical Report Number 2000-0401 TECHNICAL REPORTS AND REPRINTS SERIES April 2000 Center for Statistical Ecology and Environmental Statistics Department of StatisticsThe Pennsylvania State UniversityUniversity Park, PA 16802G. P. PatilDistinguished Professor and DirectorTel: (814)865-9442 Fax: (814)865-1278Email: [email protected]://www.stat.psu.edu/~gppContinuous Dose-Response Modeling and Risk Anal ysiswith the Gamma and Reciprocal Gamma DistributionsSenin Banga, Ganapati P. Patil, and Charles TaillieCenter for Statistical Ecology and Environmen tal StatisticsDepartment of StatisticsThe Pennsylvania State UniversityUniversity Park, PA 16802Abstract. Kodell and West (1993) describe two methods for calculating poin twise upperconfidence limits on the risk function with normally distributed responses and using a certaindefinition of adverse quantitative effect. But Banga, Patil, and Taillie (2000b) have shownthat these normal theory methods break down when applied to skew data. We accordinglydevelop a risk analysis model and associated likelihood-based methodology when the responsefollows either a gamma or reciprocal gamma distribution. The model supposes that theshape (index) parameter k of the response distribution is held fixed while the logarithm ofthe scale parameter is a linear model in terms of the dose level. Existence and uniquenessof the maximum likelihood estimates is established. Asymptotic likelihood-based upper andlower confidence limits on the risk are solutions of the Lagrange equations associated with aconstrained optimization problem. Starting values for an iterative solution a re obtained byreplacing the Lag range equations by the lowest order terms in their asymptotic expansions.Three methods are then compared for calculating confidence limits on the risk: (i) theaforementioned starting values (LRAL method), (ii) full iterative solution of the Lagrangeequations (LREL method), and (iii) bounds obtained using approximate normality of themaximum likelihood estimates with standard errors derived from the information matrix(MLE method). Simulation is used to assess coverage probabilities for the resulting upperconfidence limits when the log of the scale parameter is quadratic in the dose level. Resultsindicate that coverage for the MLE method can be off by as muc h as 15 percentage pointsand converges very slowly to nominal coverage levels as the sample size increases. Coveragefor the LRAL and LREL methods, on the other hand, is close to nominal levels unless (a)thesamplesizeissmall,sayN<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 thereciprocal gamma distribution.Prepared with partial support from the Office of Research and Development, Unite d States Environmen-tal Protection Agency, Washington, DC under a Cooperative Agreement Num ber R-825385. The contentshave not been subjected to Agency review and therefore do not necessarily reflect the views of the Agencyand no official endorsement should be inferred.1Keywords: Benc hmark dose; Confidence limits; Deviance; Lik elihood contour method;Likelihood ratio; Maximum likelihood estimation.1 IntroductionA model-based approach to the development of risk assessment methodology is an appealingalternative to the NOAEL/LOAEL approach (Chen and Gaylor, 1992; Crump, 1984; Stitelerand Durkin, 1990). For continuo us responses, however, it is usually not apparent how a givenresponse value should be di chotomized into “adverse” or “not adverse.” One solution (Chenand Gaylor, 1992; Crump, 1995; Gay lor and Slikker, 1990; Glowa, 1991; Kodell and West,1993; West and Kodell, 1993) involves a so-called abnormal point – a response value that liesin the direction o f adversity but is sufficiently far from the c ontrol mean that its occurrencein unexposed subjects would be considered unusual. Gaylor and Slikker consider the casewhere the a bnormal point is directly specifie d and i s a known parameter of the problem.Kodell and West take the abnormal point to be a specified num ber of standard deviationsfrom the (unknown) control mean; in this approach, the abnormal point is an unkno wnparameter. In the present paper, the abnormal point is defined to be a specified percentileof the control distribution. For example, when the direction of adversity is toward smallerresponses, the abnormal point might be taken as the 5thpercentile of the control distribution.For normal distributions, the percentile definition of abnormal point is equivalen t to that ofKodell and West; for more general distributions, the percentile definition has the advantagethat it transforms in the same way as the response variable.For a sp ecified exposure level, the risk is the probability of an ad verse response. Ac-cordingly, the risk is a tail area of the response distribution for the given dose level (seeFigure 1). When parametric models are specified for the response distributions, then a para-metric expression can be derived for the risk as a function of the dose, which determines theordinate of the dose response curve. Likelihood methods can then be employed to estimatethe risk function and to obtain upper confidence limits on the risk. Inversion of this upperconfidence curve is often used to obtain the benchmark dose level (but see Sciullo, Patil, andTaillie, 2000; Banga, Patil, and Taillie, 2000c). This program has been carried out by Kodelland West (1993) and West and Kodell (1993) when the responses are normally distributed.However, if normal theory methods are applied in a study where the data follow some otherdistribution, then the inferences are generally not consistent and