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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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slide1

Estimation Methods

for Dose-response Functions

Bahman Shafii

Statistical Programs

College of Agricultural and Life Sciences

University of Idaho, Moscow, Idaho

slide2

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
slide3

The response distribution:

    • Continuous
      • Normal
      • Log Normal
      • Gamma, etc.
    • Discrete - quantal responses
      • Binomial, Multinomial (yes/no)
      • Poisson (count)
slide4

Response

Response

Dose

Dose

  • The response form:
    • Typically expressed as a nonlinear curve
      • increasing or decreasing sigmoidal form
      • increasing or decreasing asymptotic form
slide5

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.
slide6

Objectives

    • Outline estimation methods for dose-
      • response models.
  • Traditional approaches.
    • Probit - Least Squares.
  • Modern approaches.
    • Probit - Maximum Likelihood
    • Generalized non-linear models.
    • Bayesian solutions.
slide7

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

^

  • 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)
slide8

  •  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.
slide9

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.
slide10

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.
slide11

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.
slide12

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.
slide13

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().
slide14

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.
slide15

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.
slide16

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.
slide17

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.

slide18

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.
slide19

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
slide21

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.
slide22

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.
slide23

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.
slide24

“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?

slide25

“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!