Estimation Methods for Dose-response Functions Bahman Shafii Statistical Programs College of Agricultural and Life Sc

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Estimation Methods for Dose-response Functions Bahman Shafii Statistical Programs College of Agricultural and Life Sciences University of Idaho, Moscow, Idaho. Introduction. Dose-response models are common in agricultural research. They can encompass many types of problems:.

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## Estimation Methods for Dose-response Functions Bahman Shafii Statistical Programs College of Agricultural and Life Sc

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Presentation Transcript

Estimation Methods

for Dose-response Functions

Bahman Shafii

Statistical Programs

College of Agricultural and Life Sciences

University of Idaho, Moscow, Idaho

Introduction

• Dose-response models are common in agricultural research.
• They can encompass many types of problems:
• Time effects
• germination, emergence, hatching
• exposure times
• Environmental effects
• temperature exposure
• chemical exposure
• depth or distance from exposure
• Related Problems - Bioassay
• standard curves and determination of unknown quantities

The response distribution:

• Continuous
• Normal
• Log Normal
• Gamma, etc.
• Discrete - quantal responses
• Binomial, Multinomial (yes/no)
• Poisson (count)

Response

Response

Dose

Dose

• The response form:
• Typically expressed as a nonlinear curve
• increasing or decreasing sigmoidal form
• increasing or decreasing asymptotic form

Estimation

• Curve estimation.
• Linear or non-linear techniques.
• Estimate other quantities:
• percentiles.
• typically: LD50, LC50, EC50, etc.
• percentileestimation problematic.
• inverted solutions.
• unknown distributions.
• approximate variances.

Objectives

• Outline estimation methods for dose-
• response models.
• Probit - Least Squares.
• Modern approaches.
• Probit - Maximum Likelihood
• Generalized non-linear models.
• Bayesian solutions.

Methods

• where
• pij = yij / N and yij is the number of successes out of N
• trials in the jth replication of the ith dose.
• b0 and b1 are regression parameters and ei is a random
• error; eij ~ N(0,s2).
• Minimize: SSerror =  (pij - probit)2

^

• Probit Analysis - Least Squares
• A linearized least squares estimation (Bliss, 1934 ; Fisher, 1935;
• Finney, 1971):
• Probiti = F -1(pij) = b0 + b1*dosei + eij (1)

•  is a convenient CDF form or “tolerance
• distribution“, e.g.
• Normal:pij = (1/2) exp((x-)2/2
• Logistic:pij = 1 / (1 + exp( -b1( dosei - b0 ))
• Modified Logistic:pij = C + (C-M) / (1 + exp( -b1(dosei -b0))
• (e.g. Seefeldt et al. 1995)
• Gompertz: pij = b0 (1 - exp(exp(-b1(dose))))
• Exponential:pij = b0 exp(-b1(dose))
• SAS: PROC REG.

Modern Approach

• Probit Analysis - Maximum Likelihood
• The responses, yij, are assumed binomial at each dose i
• with parameter pi. Using the joint likelihood, L(pi) :
• Maximize: L(pi) P (pi)yij (1 - pi)(N - yij) (2)
• for data set yij where pi = F (b0 + b1*dosei ) and b0, b1,
• and dosei are those given previously.
• The CDF, F, is typically defined as a Normal, Logistic, or
• Gompertz distribution as given above.
• SAS: PROC PROBIT.

Probit Analysis

• Limitations:
• Least squares limited.
• Linearized solution to a non-linear problem.
• Even under ML, solution for percentiles approximated.
• inversion.
• use of the ratio b0/b1 (Fieller, 1944).
• Appropriate only for proportional data.
• Assumes the response F-1(pij) ~ N(m, s2).
• Interval estimation and comparison of percentile
• values approximated.

Modern Approaches (cont)

• Nonlinear Regression - IterativeLeast Squares
• Directly models the response as:
• yij = f(dosei) + eij (3)
• where yij is an observed continuous response, f(dosei)
• may be generalized to any continuous function of dose
• and eij ~ N(0, s2).
• Minimize: SSerror =  [ yij - f(dosei) ]2.
• SAS: PROC NLIN.

Nonlinear Regression - Iterative Least Squares

• Limitations:
• assumes the data, yij , is continuous; could be discrete.
• the response distribution may not be Normal,
• i.e.eij ~ N(0, s2).
• standard errors and inference are asymptotic.
• treatment comparisons difficult in SAS.
• differential sums of squares.
• specialized SAS codes ; PROC IML.

Modern Approaches (cont)

• Generalized Nonlinear Model - Maximum Likelihood
• Directly models the response as:
• yij = f(dosei) + eij
• where yij and f(dosei) are as defined above.
• Estimation through maximum likelihood where the
• response distribution may take on many forms:
• Normal: yij ~ N(i, ) ,
• Binomial: yij ~ bin(N, pi) ,
• Poisson: yij ~ poisson(i) , or
• in general: yij ~ ƒ().

Generalized Nonlinear Model - Maximum Likelihood

• Maximize: L() Pƒ( | yij) (4)
• Nonlinear estimation.
• Response distribution not restricted to Normal.
• May also incorporate random components into the model.
• Treatment comparisons easier in SAS.
• Contrast and estimate statements.
• SAS: PROC NLMIXED.

Generalized Non-linear Model - Inference

• Formulate a full dummy variable model encompassing k
• treatments.
• The joint likelihood over the k treatments becomes:
• L(k) Pijkƒ(k | yijk) (5)
• where yijk is the jth replication of the ith dose in the kth
• treatment and qk are the parameters of the kth treatment.
• Comparison of parameter values is then possible through
• single and multiple degree of freedom contrasts.

Generalized Nonlinear Model

• Limitations
• percentile solution may still be based on inversion or
• Fieller’s theorem.
• inferences based on normal theory approximations.
• standard errors and confidence intervals asymptotic.

Modern Approaches (cont)

• Bayesian Estimation - Iterative Numerical Techniques
• Considers the probability of the parameters, q,
• given the data yij.
• Using Bayes theorem, estimate:
• p(q|yij) = p(yij|q)*p(q) (6)
• p(yij|q)*p(q)dq

where p(q|yij) is the posterior distribution of q

given the data yij, p(yij|q) is the likelihood defined

above, and p(q) is a prior probability distribution

for the parameters q.

Bayesian Estimation - Iterative Numerical Techniques

• Nonlinear estimation.
• Percentiles can be found from the distribution of q.
• The likelihood is same as Generalized Nonlinear Model.
• flexibility in the response distribution.
• f(dosei) any continuous funtion of dose.
• Inherently allows updating of the estimation.
• Correct interval estimation (credible intervals).
• agrees well with GNLM at midrange percentiles.
• can perform better at extreme percentiles.
• SAS: No procedure available.

Bayesian Estimation - Iterative Numerical Techniques

• Limitations
• User must specify a prior probability p(q).
• Estimation requires custom programming.
• SAS: Datastep, PROC IML
• Custom C program codes
• Specialized software: WinBUGS
• Computationally intensive solutions.
• Requires statistical expertise.
• Sample programs and data are available at:
• http://www.uidaho.edu/ag/statprog

Concluding Remarks

• Dose-response models have wide application in agriculture.
• They are useful for quantifying the relative efficacy of various
• treatments.
• Probit models are limited in scope.
• Generalized nonlinear and Bayesian models provide the most
• flexible framework for estimating dose-response.
• Can use various response distributions
• Can use various dose-response models.
• Can incorporate random model effects.
• Can be used to compare treatments.
• GNLM: full dummy variable modeling.
• Bayesian methods: probability statements.

Concluding Remarks

• Both GNLM and Bayesian methods give similar percentile
• estimates for midrange percentiles.
• Generalized nonlinear models sufficient in most situations.
• Software available.
• Bayesian estimation is preferred when estimating extreme
• percentiles.
• Custom programming required.

References

• Bliss, C. I. 1934. The method of probits. Science, 79:2037, 38-39
• Bliss, C. I. 1938. The determination of dosage-mortality curves from small
• numbers. Quart. J. Pharm., 11: 192-216.
• Berkson, J. 1944.Application of the Logistic function to bio-assay. J.
• Amer. Stat. Assoc. 39: 357-65.
• Feiller, E. C. 1944. A fundamental formula in the statistics of biological
• assay and some applications. Quart. J. Pharm. 17: 117-23.
• Finney, D. J. 1971. Probit Analysis. Cambridge University Press, London.
• Fisher, R. A. 1935. Appendix to Bliss, C. I.: The case of zero survivors.,
• Ann. Appl. Biol., 22: 164-5.
• SAS Inst. Inc. 2004. SAS OnlineDoc, Version 9, Cary, NC.
• Seefeldt, S.S., J. E. Jensen, and P. Fuerst. 1995. Log-logistic analysis of
• herbicide dose-response relationships. Weed Technol. 9:218-227.

“Top Ten Things A Statistician

Does Not Want to Hear”

10. I have never had a course in statistics, but how hard can it be?

9. I don’t have a design!

8. I should have talked to you before I ran the experiment, but.....

7. Why should I replicate? I might get a different answer!

6. I should have randomized what?

“Top Ten Things A Statistician

Does Not Want to Hear”

5. Could you have this by tomorrow?

4. Halfway through the experiment, we changed.....

3. Can you make it so that the p-value is less than.....?

2. I have 20,000 observations from this one cow!

1. Do you have a minute?

Thank you!