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1998 Progress Report: Statistical Issues Related to the Implementation of Benchmark Dose Method

EPA Grant Number: R825385
Title: Statistical Issues Related to the Implementation of Benchmark Dose Method
Investigators: Patil, G. P. , Banga, S. , Sciullo, C. , Stiteler, W. M.
Current Investigators: Patil, G. P. , Banga, S. , Stiteler, W. M. , Taillie, C.
Institution: Pennsylvania State University - Main Campus
EPA Project Officer: Saint, Chris
Project Period: January 1, 1997 through December 31, 2002
Project Period Covered by this Report: January 1, 1997 through December 31, 1998
Project Amount: $299,823
RFA: Environmental Statistics (1996)
Research Category: Environmental Statistics

Description:

Objective:

Develop likelihood-based procedures for calculating confidence limits on risk function and effective dose (benchmark dose, BMD) for continuous responses with emphasis on skew (nonnormal) distributed responses. Assess the sensitivity to model mis-specification. Examine the statistical validity of BMD-determination by inversion of an upper confidence curve on the risk function.

Progress Summary:

A benchmark dose (BMD) for continuous responses may be defined as a lower confidence limit on the effective dose corresponding to a specified risk level r. However, calculating such a confidence limit is not straightforward. By contrast, it is technically easier to obtain confidence limits on the risk function R(d). One approach that has been suggested for BMD-determination is to first obtain a pointwise upper confidence curve U(d) on the risk function and then to invert this relationship by solving the equation U(d)=r. The solution d is purported to be the desired BMD, i.e., a lower confidence limit on the effective dose corresponding to the risk level r.

Project research to date has focused on the following issues:

  1. Development of a general Lagrange contour method for obtaining (asymptotic) likelihood-based confidence limits on any real-valued function of the parameters in a multi-parameter statistical model. The risk function and the effective dose are both examples of such real-valued functions.

  2. Proof that the contour method has exactly two solutions (corresponding to upper and lower confidence limits) for sufficiently large sample sizes. For finite sample sizes, more than two solutions are, of course, possible depending upon the model. For the normal homoscedastic model, there are exactly two solutions for all sample sizes.

  3. Development of starting values for iterative solution of the equations. This is accomplished by replacing the equations by the lowest order terms in their asymptotic expansions and solving the simplified approximating equations.

  4. Implementation of the contour method and assessment of its behavior for small sample sizes. Simulation is used to obtain achieved coverage levels which are compared with nominal levels. Comparison is also made with levels achieved by the MLE confidence limits (point estimate plus or minus a multiple of the standard error as determined from Fisher's information matrix). We also compute the coverage probabilities achieved by using the starting values described in item (3) above. To date, the simulations have been carried out for upper confidence limits on the risk function in the case of two models: (i) normal hornoscedastic model with the mean as a quadratic function of the dose, and (ii) gamma distributed responses with constant (but unknown) index parameter and the log of the mean as a quadratic function of the dose. Results indicate that asymptotic confidence limits computed from the contour method yield coverage probabilities that match the nominal levels to within a percentage point or two, and that convergence to nominal levels is quite rapid with increasing sample sizes. On the other hand, coverage achieved by MLE confidence limits can be off by 5 or 10 percentage points and converges only slowly with increasing sample sizes. Using the starting values as final values achieves a coverage that is much better than MLE, but inferior to the final iterative solution for small sample sizes. Thus, full iterative solution will usually be worth the computational effort unless the sample sizes are large.

  5. Investigate the statistical validity of the inversion method for determining BMD levels. Regardless of the distributional model, we have shown that the inversion method is conservative in the sense that the achieved coverage is at least as large as the nominal coverage. Depending upon the model, the overcoverage can be zero (i.e., the inversion method can be correct). A necessary and sufficient condition for overcoverage to be zero is that the upper confidence curve on the risk function be monotone increasing (at least beyond some initial dip) with probability one. A detailed investigation has been made for the normal hornoscedastic model whose mean is a polynomial function of the dose. Here an exact method, based on the noncentral t distribution, is available for obtaining upper confidence limits on the risk function. We have used the Abramowitz-Stegun approximation to the noncentral t distribution to carry out the inversion analytically and to study its properties. For t-based intervals with the normal homoscedastic model, the conclusions are: (i) the inversion overcoverage vanishes identically when the mean is a straight line function of the dose; but (ii) the inversion overcoverage is strictly positive when the mean is a quadratic function of the dose (Kodell-West model). Further, in case (ii), the achieved coverage can range anywhere from the nominal level all the way up to 100 percent (exclusively), depending upon the parameters of the model.

    An analytic expression, involving the bivariate noncentral t probability integral, has been obtained for the overcoverage probability in case (ii). A Mathernatica program has been written for efficient evaluation of this integral.

  6. Investigate the accuracy of the Abramowitz-Stegun approximation to the noncentral t-distribution (used in item 5, above). Results show that the approximation is highly accurate (effectively exact for most practical purposes) for degrees of freedom as small as 10. By comparison, the more familiar approximation based on a normal distribution with the same mean and variance is poor unless degrees of are very large.

Future Activities:

Work in progress includes the following:

  1. Assessment of the contour method for additional models. The lognormal and reciprocal gamma are envisioned.

  2. Assessment of the contour method when the direction of adversity is to the right. In our assessments to date, the direction of adversity has been to the left. For symmetric models like the normal homoscedastic, the formalism is the same regardless of the direction. But this is not the case with skew models like the gamma. A key question is whether our starting values continue to work well in the right hand tail of the distribution because good starting values are critical for convergence.

  3. Extension of the contour method to direct determination of (asymptotic) BMD levels rather than inverting an upper confidence curve on the risk function.

  4. Assessing the effect of model mis-specification. A principal goal of our research has been to extend the BMD methods to non-normal models. But this raises two related questions: (i) Suppose the data are actually non-normal but you analyze the data as if it followed a normal model. What are the consequences for your BMD determinations? In other words, how important is it to have non-normal methodology available? (ii) Given that a non-normal model is appropriate, how accurately must that model be specified? For example, what are the consequences if you analyze the data using a lognormal model when the data actually follow a gamma model?

  5. General proof (currently a conjecture) that the inversion method is asymptotically correct, i.e., that the overcoverage goes to zero as the sample size becomes large. We can prove this for the normal homoscedastic model with confidence limits derived from the noncentral t distribution. For this model, we are also studying the rate at which the overcoverage converges to zero (it appears to be exponential) and are developing large deviation approximations for the overcoverage.

Journal Articles:

No journal articles submitted with this report: View all 6 publications for this project

Supplemental Keywords:

NOAEL, LOAEL, likelihoood ratio, deviance, dose response model. ,

Progress and Final Reports:
Original Abstract
Final Report

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The perspectives, information and conclusions conveyed in research project abstracts, progress reports, final reports, journal abstracts and journal publications convey the viewpoints of the principal investigator and may not represent the views and policies of ORD and EPA. Conclusions drawn by the principal investigators have not been reviewed by the Agency.

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