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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:
• 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)
• The response form:
• Typically expressed as a nonlinear curve
Response
• increasing or decreasing sigmoidal form
• increasing or decreasing asymptotic form
Dose
• Estimation
• Curve estimation.
• Linear or non-linear techniques.
• Estimate other quantities:
• percentiles.
• typically: LD50, LC50, EC50, etc.
• percentile estimation problematic.
• inverted solutions.
• unknown distributions.
• approximate variances.
• Objectives
• Outline estimation methods for dose-
response models.
• Traditional approaches.
• Probit - Least Squares.
• Modern approaches.
• Probit - Maximum Likelihood
• Generalized non-linear models.
• Bayesian solutions.
Methods
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
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).
^ 2
• Minimize: SSerror =  (pij - probit)
(1)
• F is a convenient CDF form or “tolerance
distribution“, e.g.
• Normal:
pij =(1/2s) exp((x-)2/s2

• 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 i. Using the joint likelihood, L(i) :
Maximize:
L(i) 
P ( )
i
yij
(1 - i)(N - yij)
(2)
for data set yij where i = 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(, s2).
• Interval estimation and comparison of percentile
values approximated.
Modern Approaches (cont)
• Nonlinear Regression - Iterative Least Squares
• Directly models the response as:
yij = f(dosei) + eij
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.
(3)
• 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:
Binomial:
Poisson:
in general:
yij ~ N(i, s) ,
yij ~ bin(N, i) ,
yij ~ poisson(i) , or
yij ~ ƒ().
• Generalized Nonlinear Model - Maximum Likelihood
• Maximize:
L() 
P ƒ( | y )
ij
(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) 
Pijk ƒ(
k
| yijk)
(5)
where yijk is the jth replication of the ith dose in the kth
treatment and k 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, ,
given the data yij.
• Using Bayes theorem, estimate:
p(|yij) =
p(yij|)*p()
(6)
p(y |)*p()d
ij
where p(|yij) is the posterior distribution of 
given the data yij, p(yij|) is the likelihood defined
above, and p() is a prior probability distribution
for the parameters .
• Bayesian Estimation - Iterative Numerical Techniques
• Nonlinear estimation.
• Percentiles can be found from the distribution of .
• 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().
• 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
Summary of Estimation Methods
Estimation
Method
SAS
Probit Least
PROC REG
Squares
Probit Maximum
PROC
PROBIT
Likelihood
Nonlinear
PROC NLIN
Regression
Generalized
Nonlinear
Regression
Bayesian
Estimation
Approximate Computational Treatment Flexible Flexible
Inferences
Intensity
Comparisons Models Distribution
Yes
Low
No
No
No
Yes
Low
Yes
No
No
Yes
Low
No
Yes
No
PROC
NLMIXED
Yes
Low
Yes
Yes
Yes
none
No
High
Yes
Yes
Yes
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!
Questions / Comments