Center for Statistical Ecology and Environmental Statistics 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 Department of Statistics The Pennsylvania State University University Park PA 16802 G P Patil Distinguished Professor and Director Tel 814 865 9442 Fax 814 865 1278 Email gpp stat psu edu http www stat psu edu gpp An Equal Opportunity University Continuous Dose Response Modeling and Risk Analysis with the Gamma and Reciprocal Gamma Distributions Senin Banga Ganapati P Patil and Charles Taillie Center for Statistical Ecology and Environmental Statistics Department of Statistics The Pennsylvania State University University Park PA 16802 Abstract 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 e ect But Banga Patil and Taillie 2000b 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 o by as much as 15 percentage 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 Prepared with partial support from the O ce 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 o cial endorsement should be inferred 1 Keywords Benchmark dose Confidence limits Deviance Likelihood contour method Likelihood ratio Maximum likelihood estimation 1 Introduction A model based approach to the development of risk assessment methodology is an appealing alternative to the NOAEL LOAEL approach Chen and Gaylor 1992 Crump 1984 Stiteler and Durkin 1990 For continuous responses however it is usually not apparent how a given response value should be dichotomized into adverse or not adverse One solution Chen and Gaylor 1992 Crump 1995 Gaylor and Slikker 1990 Glowa 1991 Kodell and West 1993 West and Kodell 1993 involves a so called abnormal point a response value that lies in the direction of adversity but is su ciently far from the control mean that its occurrence in unexposed subjects would be considered unusual Gaylor and Slikker consider the case where the abnormal point is directly specified and is a known parameter of the problem Kodell and West take the abnormal point to be a specified number of standard deviations from the unknown control mean in this approach the abnormal point is an unknown parameter In the present paper the abnormal point is defined to be a specified percentile of the control distribution For example when the direction of adversity is toward smaller responses the abnormal point might be taken as the 5th percentile of the control distribution For normal distributions the percentile definition of abnormal point is equivalent to that of Kodell and West for more general distributions the percentile definition has the advantage that it transforms in the same way as the response variable For a specified exposure level the risk is the probability of an adverse response Accordingly the risk is a tail area of the response distribution for the given dose level see Figure 1 When parametric models are specified for the response distributions then a parametric expression can be derived for the risk as a function of the dose which determines the ordinate of the dose response curve Likelihood methods can then be employed to estimate the risk function and to obtain upper confidence limits on the risk Inversion of this upper confidence curve is often used to obtain the benchmark dose level but see Sciullo Patil and Taillie 2000 Banga Patil and Taillie 2000c This program has been carried out by Kodell and 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 other distribution then the inferences are generally not consistent and coverage probabilities can depart markedly from nominal levels even with large sample sizes Banga Patil and Taillie 2000b have documented this e ect when data arising from a gamma reciprocal gamma or