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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 Sc

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for Dose-response Functions

Bahman Shafii

Statistical Programs

College of Agricultural and Life Sciences

University of Idaho, Moscow, Idaho

- 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

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

Response

Dose

Dose

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

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

- Outline estimation methods for dose-
- response models.

- Traditional approaches.
- Probit - Least Squares.

- Modern approaches.
- Probit - Maximum Likelihood
- Generalized non-linear models.
- Bayesian solutions.

- 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

^

- Traditional Approach
- 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.

- 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.

- 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.

- 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.

- 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.

- 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.

- 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

- 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.

- 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.

- 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!

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