Estimation Techniques

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Estimation Techniques for Dose-response
Functions Presented by Bahman Shafii,
Ph.D. Statistical Programs College of
Agricultural and Life Sciences University of Idaho
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Acknowledgments

• Research partially funded by USDA-ARS Hatch
  Project
• IDA01412, Idaho Agricultural Experiment Station.
• Collaborators
• William J. Price Ph. D., Statistical Programs,
• University of Idaho.
• Steven Seefeldt, Ph. D., USDA -ARS,
• University of Alaska Fairbanks.

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Introduction

• Dose-response models are common in
  agricultural research.
• They can encompass many types of problems
• Modeling environmental effects due to exposure
  to chemical or temperature regimes.
• Estimation of time dependent responses such as
  germination, emergence, or hatching.
• (e.g. Shafii and Price 2001 Shafii, et
  al. 2009)
• Bioassay assessments via calibration curves and
  quantal estimation. (e.g. Shafii and Price
  2006)

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• 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.

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

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

Response
Dose
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Bioassay and Calibration

• Given a dose-response curve and an observed
• response
• What dose generated the response?
• What is the probability of a dose given an
• observed response and the calibration curve?
• This problem fits naturally into a Bayesian
  framework.

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• Typical dose-response estimation assumes that
  the
• functional form or tolerance distribution,
• is known, e.g. a sigmoidal shape.
• In some cases, however, it may be advantageous
  to
• relax this assumption and restrict estimation
• to a family of dose-response forms.
• The dose-response population consists of a
• mixture of subpopulations which can not be
• sampled separately.
• The dose-response series exhibits a more
  complex
• behavior than a simple sigmoidal shape,
• e.g. hormesis.

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• Objectives
• Outline estimation methods for dose-
• response models.

• Traditional approaches.
• Probit - Least Squares.

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

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• Objectives
• Demonstrate solutions for calibration of an
  unknown dose with a binary response
• assuming

• A known dose-response form.
• Standard MLE estimation.
• Standard Parametric Bayesian estimation.

• A family of dose-response forms.
• Nonparametric Bayesian estimation.

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Estimation Methods

• Traditional Approach
• Probit Analysis - Least Squares

• A linearized least squares estimation (Bliss,
  1934 Fisher, 1935
• Finney, 1971)
• Probiti F -1(pij) b0 b1dosei eij
  
  (1)

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• ? 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.

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• Modern Approaches
• 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 b1dosei )
  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.

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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.

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• Modern Approaches (cont)
• Nonlinear Regression - Iterative Least 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.

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• 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 PROC NLIN.
• differential sums of squares, or
• specialized SAS codes PROC IML.

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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(?).

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• Generalized Nonlinear Model - Maximum
  Likelihood
• Maximize L(?) ? P ƒ(? 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.

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• Generalized Nonlinear 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 qk are the parameters of the kth
  treatment.
• Comparison of parameter values is then possible
  through
• single and multiple degree of freedom contrasts.

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• Generalized Nonlinear Model
• Limitations
• percentile solution may still be based on
  inversion or
• Fiellers theorem.
• inferences based on normal theory
  approximations.
• standard errors and confidence intervals
  asymptotic.

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Modern Approaches (cont)

• Bayesian Estimation - Iterative Numerical
  Techniques

• Considers the probability of the parameters, q,
• given the data yij.
• Using Bayes theorem, estimate
• p(qyij) p(yijq)p(q)
  
  (6)
• ?p(yijq)p(q)dq

where p(qyij) is the posterior distribution of q
given the data yij, p(yijq) is the likelihood
defined above, and p(q) is a prior probability
distribution for the parameters q.
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• 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 function 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 PROC MCMC.

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• Bayesian Estimation - Iterative Numerical
  Techniques

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

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Calibration Methods

• Tolerance Distribution Logistic
• The response yij/Ni at dose i 1 to k, and
  replication
• j 1 to r , is binomial with the proportion of
  success
• given by
• yij/Ni M/(1 exp(-b (dosei - g)))
  (7)
• where b is a rate related parameter and g is the
  
• dosei for which the proportion of success,
• yij/Ni , is M/2. M is the theoretical maximum
• proportion attainable.

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• A convenient generalization of (1) will allow g
  to
• represent any dose at which yij/Ni Q

yij/Ni MC / (C exp(-b (dosei - g)))
(8)
Where the constant C Q/(M Q). Note that, if
Q M/2, then C 1 and equation (8)
reverts to the standard form given in
(7). Equation (8), therefore, permits an unknown
dose at a given response, Q, to be
estimated through parameter g.
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• Maximum Likelihood

• Given the binomial responses, yij/Ni, a joint
• likelihood may be defined as
• L(pi yij/Ni) ? Pij (pi)yij (1 -
  pi)(Ni - yij) (9)

• Where the binomial parameter ,pi , is defined by
  (8)
• and the associated parameters, q M, b, g,
  are
• estimated through maximization of (9). Ni
  and yij
• are the total number of trials and number of
  
• successes, respectively.
• Inferences on g are carried out assuming g
  N(mg, sg).
• SAS PROC NLMIXED

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• Bayesian Parametric

• A Bayesian posterior distribution for q is
  given by
• pr(q yij/Ni) ? pr(yij/Ni q) pr(q)
  (10)

• where pr(yij/Ni jq) is the likelihood shown in
  (9) and pr(q)
• is a prior distribution for the parameters q
  M, b, g. Estimation of q is carried out
  through numerically intensive techniques such as
  MCMC. (e.g. Price and Shafii 2005)
• Inference on g is obtained through integration
  of (10) over the parameter space of M and b.

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• Bayesian Nonparametric

• This methodology was first proposed by
  Mukhopadhyay (2000) and
• followed by Kottas et al. (2002).
• The technique considers the dose-response
  series as a
• multinomial process with parameters P p1, p2,
  p3, pk.

• Assuming the responses, yij/Ni, are binomial, a
  likelihood can
• then be defined as
• L(P yij/Ni) ? Pij (pi)yij (1 - pi)(Ni - yij)
  (11)

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• If the random segments between true response
  rates, pi ,
• are distributed as a Dirichlet Process (DP), a
  joint prior
• distribution on the pi may then be defined by
• pr(P) ? Pi (pi pi -
  1)(li - 1) (12)
• where li a F0(dose i) F0(dose i 1 ) , a
  is a precision
• parameter , and F0 is a base tolerance
  distribution.
• The precision parameter, a, reflects how
  closely the final estimation follows the base
  distribution. Low values indicate less
  correspondence , while larger values indicate a
  tighter association.
• The base distribution, F0(.), defines a family
  of tolerance distributions.

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• A posterior distribution for P can then be
  defined by
• combining (11) and (12) as
• pr(P yij/Ni) ? Pij (pi)yij (1 -
  pi)(Ni - yij) Pi (pi pi - 1)(li - 1)
• (13)
• Estimation of this posterior is again carried
  out numerically using techniques such as MCMC.
• Inference on an unknown dose, g , at a known
  response p0 y0/N0, is obtained through
  sampling of the posterior given in (13) .

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Concluding Remarks

• Dose-response models have wide application in
  agriculture.

• They are useful for quantifying the relative
  efficacy of treatments.

• Probit models of estimation are limited in
  scope.

• Generalized nonlinear and Bayesian models
  provide the most
• flexible framework for dose-response estimation.
• 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.

• Generalized nonlinear models sufficient in most
  
• situations.

• Bayesian estimation is preferred when
  estimating
• extreme percentiles.

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Concluding Remarks (cont)

• Bioassay is an import part of dose-response
  analysis.

• Determining an unknown dose can be problematic
  for
• some parametric functional forms.

• Dose estimation fits naturally in a Bayesian
  framework.

• Methodology proposed here uses a base tolerance
  
• distribution.
• Should be used and interpreted with caution.
• Standard model assessment techniques still
  apply.
• Introduces more uncertainty into the estimation
  situation.

• Some dose-response data may not follow typical
• sigmoidal patterns.

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References
Bliss, C. I. 1934. The method of probits.
Science, 792037, 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.
9218-227. Kottas, A., M. D. Branco, and A.
E. Gelfand. 2002. A Nonparametric Bayesian
Modeling Approach for Cytogenetic Dosimetry.
Biometrics 58, 593-600.
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References
Mukhopadhyay, S. 2000. Bayesian Nonparametric
Inference on the Dose Level with Specified
Response Rate. Biometrics 56, 220-226. Price,
W. J. and B. Shafii. 2005. Bayesian Analysis of
Dose-response Calibration Curves. Proceedings
of the Seventeenth Annual Kansas State
University Conference on Applied Statistics in
Agriculture CDROM, April 25-27, 2005.
Manhattan Kansas. Shafii, B. and W. J. Price.
2001. Estimation of cardinal temperatures in
germination data analysis. Journal of
Agricultural, Biological and Environmental
Statistics. 6(3)356-366. Shafii, B. and W. J.
Price. 2006. Bayesian approaches to dose-response
calibration models. Abstract Proceedings of
the XXIII International Biometrics Conference
CDROM, July 16 - 21, 2006. Montreal, Quebec
Canada. Shafii, B., Price, W.J., Barney, D.L.
and Lopez, O.A. 2009. Effects of stratification
and cold storage on the seed germination
characteristics of cascade huckleberry and
oval-leaved bilberry. Acta Hort. 810599-608.
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Questions / Comments