1. Bayesian Analysis of Dose-Response Calibration Curves Bahman Shafii William J. Price Statistical Programs College of Agricultural and Life Sciences University of Idaho, Moscow, Idaho, USA

2. Dose-response curves are used to model: Time effects Germination, emergence, hatching Environmental effects Temperature, chemical, distance exposures Bioassay Calibration curves Estimation of quantities Introduction

3. The Response Data: Continuous Normal, Log Normal, Gamma, etc. Discrete Binomial, Multinomial, Poisson Curve Estimation Linear or non-linear techniques.

4. Given: Dose-response Curve Observed Response What dose generated the response? The question is naturally expressed in terms of Bayes Theorem: What is the probability of a dose given an observed response and the calibration curve?

5. Objectives Present potential Bayesian solutions for estimating an unknown dose with a binomial response under the following assumptive conditions: i) the dose-response curve is known. ii)the dose-response curve is estimated (known with error).

6. Methods Logistic Dose Response Model Commonly used ; (Berkson, 1944) The response, y ij, is binomial with the proportion of success given by:  i = 1/(1 + exp(-  (dose i -  ))) (1) where  is a rate related parameter and  is the dose i for which the proportion of success,  i, is 0.5.

7. A Bayesian estimate is : p(  |y ij ) = p(y ij |  ) · p(  ) (2)  p(y ij |  ) · p(  )d  where p(y ij |  ) is a likelihood for the data set y ij evaluated over the parameters  = [ ,  ], p(  ) is a prior distribution for the parameters in , and p(  |y ij ) is the posterior distribution of  given the data y ij.

8. The likelihood, p(y ij |  ), is given by: L(  i )   ij (  i ) y ij (1 -  i ) (N - y ij ) (3) The prior probability, p(  ), is user specified. Priors for  and , however, can be difficult to specify. The upper bound for  is open ended. The range for  may also be open ended.

9. The logistic model given in (1), however, can be reparameterized such that the required prior distributions are easier to define. Specifically, it is noted that at dose = 0 and dose = D Max, the logistic model reduces to:   = 1/(1 + exp(  )) and   = 1/(1 + exp(-  D Max -  )) (4) yielding:

10.  * = D Max * ln((1-   )/   )/(ln((1-   )/   ) - ln((1-   )/   ))   = ln((1-   )/   )/    (Price, et al., 2003)  (5) Under maximum entropy, prior distributions for   and   are assumed uniform. (Jaynes, 2003)

11. i. Dose-response curve known Given: Observe M successes in N trials Logistic dose-response,  i, given in (1) Parameters    and   known without error The probability that dose equals x given M, N, and  is: p(x|M,N,  ,   )  p(M|x,N,  ,   ) · p(x)   i M (1 -  i ) N-M p(x) (7) where p(x) is a prior probability for x.

12. i. Dose-response curve known (cont.) Assuming a uniform prior on x, say within the range of calibration doses, a closed form solution for the unknown dose is: x = (ln(N-M)/M)/  ) + , (8) and a (1-  ) credible interval can be derived from the posterior distribution in (7) as: p( L  x  U) = 1- . (9) ^ ^

13. ii. Dose-response curve estimated Given: Observe M successes in N trials Logistic dose- response,  i, given in (1). Dose-response “calibration” data : y ij, dose i ; Parameters    and   known with error. If M is independent of y ij and x independent of  the probability that dose equals x given M, N, and y ij is derived from the joint distribution of: p( x | M) and p(     |y ij )

14. ii. Dose-response curve estimated p(x|M,N,y ij )   p(M|x)·p(x)·p(     |y ij ) d  (10) where p( M | x) is given by  i M (1 -  i ) N-M, p(x) is the prior distribution for x, and p(     |y ij ) is the posterior distribution given in (6). This essentially filters the posterior distribution for dose in (7) through p(     |y ij ). Given prior distributions for x, estimation can be carried out using either numerical or simulation techniques such as MCMC.

15. All programs and graphics carried out using SAS. Sample programs and data are available at: http://www.uidaho.edu/ag/statprog

16. Data Effects of organic pesticide on egg hatch of black vine weevil (BVW). 20 BVW eggs placed in a petri dish with the pesticide. 9 doses (concentrations) of pesticide used. 0 to.03 g. Each dose replicated 10 times. The number of eggs failing to hatch recorded (success). Three experiments conducted, each varying in dose range. Demonstration

17. Bayesian Logistic Model Estimation # Unhatched Eggs Dose (g) 0 10 20 0.000.010.020.03 Credible Regions Parameter Estimate Lower Upper    0.01750 0.01280 0.02320    0.99995 0.99990 0.99998  * 0.00864 0.00832 0.00891  * 466.800 432.547 502.796

18. 1) Observe M successes in N trials in a new experiment. 2) Logistic model assumed and parameters assumed known. What was the dose associated with this new observation? i. Dose-response curve known

19. P(x|M) M : 5 N : 20 0.00420.00690.0085 P(x|M) 0.00720.00880.0103 P(x|M) Dose 0.0000.0040.0080.0120.0160.0200.0240.028 0.01140.01370.0198 Unknown Dose L 95 xU 95 ^ Dose-response Curve Known M : 10 N : 20 M : 19 N : 20 Unknown Dose L 95 xU 95 ^ Unknown Dose L 95 xU 95 ^

20. 1) Observe M successes in N trials in a new experiment. 2) Logistic model assumed and estimated ( parameters known with error). What was the dose associated with this new observation? ii. Dose-response curve estimated

21. P(x|M) 0.00400.00680.0086 P(x|M) Dose 0.0000.0040.0080.0120.0160.0200.0240.028 M : 19 N : 20 0.01140.01390.0199 Unknown Dose L 95 U 95 x ^ Dose-response Curve Estimated P(x|M) 0.00710.00880.0104 M : 10 N : 20 Unknown Dose L 95 U 95 x ^ M : 5 N : 20 Unknown Dose L 95 U 95 x ^

22. Dose-response Curve Known & Estimated (M = 10, N=20) 0.0060.0070.0080.0090.0100.011 0.012 P(x|M) Dose Known Estimated Obs. = 1580 Unknown Dose L 95 x U 95 0.00720.00880.0103 0.00710.00880.0104 ^ Known Estimated

23. Dose-response Curve Known & Estimated (M = 10, N=20) 0.0020.0040.0060.0080.0100.0120.0140.016 P(x|M) Dose Known Estimated Obs. = 310 Unknown Dose L 95 x U 95 0.00650.0090.0114 0.00580.0090.0122 ^ Known Estimated

24. Entropy (Shannon, 1948) uniquely quantifies the level of information in a distribution. H = -  p(x)·ln(p(x)) The ratio of entropy values from two distributions, say H 1 and H 2, can give a relative measure of their respective information. E R = H 1 /H 2 If H 2 represents a dose distribution from the known case, i.e. perfect information, and H 1 represents the corresponding estimated case, then E R will give some measure of the distance between the two distributions as well as the efficiency of the estimated case.

25. Dose-response Curve Known & Estimated 0.0020.0040.0060.0080.0100.0120.0140.016 P(x|M) Dose E R = 0.939 P(x|M) Dose M : 50 N : 100 Known Estimated 0.0020.0040.0060.0080.0100.0120.0140.016 E R = 0.876 P(x|M) Dose 0.0020.0040.0060.0080.0100.0120.0140.016 E R = 0.670 M : 10 N : 20 Known Estimated M : 400 N : 800 Known Estimated

26. Concluding Remarks Determining an unknown dose from calibration information can be naturally posed as a Bayesian problem. Dose estimation can be carried out both with and without calibration error. Calibration error will subsequently increase estimated interval limits. Increases in sampling intensity for the unknown dose cannot overcome calibration error. It is important to concentrate sampling effort on the definition, estimation, and development of the calibration model.

27. References Berkson, J. 1944. Application of the Logistic function to bio-assay. J. Amer. Stat. Assoc. 39, pp 357-65. Jaynes, E. T. 2003. Probability Theory. Cambridge University Press, Cambridge, UK. pp. 727. Price, W. J., B. Shafii, K. B. Newman, S. Early, J. P. McCaffrey, M. J. Morra. 2003. Comparing Estimation Procedures for Dose-response Functions. In Proceedings of the Fifteenth Annual Kansas State University Conference on Applied Statistics in Agriculture, CDROM SAS Inst. Inc. 2004. SAS OnlineDoc® 9.1.3. Cary, NC: SAS Institute Inc. Shannon, C., 1948. The Mathematical Theory of Communication. Bell System Technical Journal, 27: 379, 623.

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