Bayesian Analysis of

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Bayesian Analysis of Dose-Response Calibration
Curves Bahman Shafii William J.
Price Statistical Programs College of
Agricultural and Life Sciences University of
Idaho, Moscow, Idaho, USA
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Introduction

• Dose-response curves are used to model
• Time effects
• Germination, emergence, hatching
• Environmental effects
• Temperature, chemical, distance exposures
• Bioassay
• Calibration curves
• Estimation of quantities

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• The Response Data
• Continuous
• Normal, Log Normal, Gamma, etc.
• Discrete
• Binomial, Multinomial, Poisson

• Curve Estimation
• Linear or non-linear techniques.

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• Given
• Dose-response Curve
• Observed Response
• What dose generated the response?
• The question is naturally expressed in terms of
  
• Bayes Theorem
• What is the probability of a dose given an
• observed response and the calibration
• curve?

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• Objectives
• Present potential Bayesian solutions for
• estimating an unknown dose with a binomial
• response under the following assumptive
• conditions
• i) the dose-response curve is known.
• the dose-response curve is estimated
• (known with error).

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Methods

• Logistic Dose Response Model
• Commonly used (Berkson, 1944)
• The response, yij, is binomial with the
  proportion
• of success given by
• pi 1/(1 exp(-b (dosei - g))) (1)
• where b is a rate related parameter and g is the
  
• dosei for which the proportion of success,
• pi , is 0.5.

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• A Bayesian estimate is
• p(qyij) p(yijq) p(q)
  (2)
• ?p(yijq) p(q)dq
• where p(yijq) is a likelihood for the data set
  yij evaluated over the parameters q b , g,
  p(q) is a prior distribution for the parameters
  in q, and p(qyij) is the posterior distribution
  of q given the data yij.

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• The likelihood, p(yijq), is given by
• L(pi) ? Pij (pi)yij (1 - pi)(N
  - yij) (3)
• The prior probability, p(q), is user
  specified.
• Priors for b and g, however, can be difficult
  to specify.
• The upper bound for b is open ended.
• The range for g may also be open ended.

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• The logistic model given in (1), however, can
  be reparameterized such that the required prior
• distributions are easier to define.
  Specifically, it is noted that at dose 0 and
  dose DMax , the
• logistic model reduces to
• q1 1/(1 exp(bg))
  and
• q2 1/(1 exp(-b (DMax -
  g)) (4)
• yielding

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• g DMax ln((1- q1)/ q1)/(ln((1- q1)/ q1) -
  ln((1- q2)/ q2))
• b ln((1- q1)/ q1)/g
• 
  (Price, et al., 2003) (5)

• Under maximum entropy, prior distributions for
  q1 and q2 are assumed uniform. (Jaynes, 2003)

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• i. Dose-response curve known
• Given
• Observe M successes in N trials
• Logistic dose-response, pi , given in (1)
• Parameters q1 and q2 known without error
• The probability that dose equals x given M, N,
  and q is
• p(xM,N,q1,q2 ) ? p(Mx,N, q1, q2 ) p(x)
• ? piM (1 -
  pi)N-M p(x) (7)
• where p(x) is a prior probability for x.

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• i. Dose-response curve known (cont.)
• Assuming a uniform prior on x, say within the
  range of calibration doses, a closed form
  solution for the unknown dose is
• x (ln(N-M)/M)/ b) g ,
  (8)
• and a (1- a) credible interval can be
  derived from the posterior distribution in (7)
  as
• p( L ? x ? U) 1- a .
  (9)

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• ii. Dose-response curve estimated
• Given
• Observe M successes in N trials Logistic
  dose- response, pi , given in (1).
• Dose-response calibration data yij, dosei
  Parameters q1 and q2 known with error.
• If M is independent of yij and x independent of
  q, the probability that dose equals x given M,
  N,
• and yij is derived from the joint
  distribution of
• p( x M) and p(q1,
  q2yij)

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• ii. Dose-response curve estimated
• p(xM,N,yij) ? ?p(Mx)p(x)p(q1,
  q2yij) dq (10)
• where p( M x) is given by piM (1 - pi)N-M ,
  p(x) is the prior distribution for x, and p(q1,
  q2yij)
• is the posterior distribution given in (6).
  
• This essentially filters the posterior
  distribution for dose in (7) through p(q1,
  q2yij).
• Given prior distributions for x, estimation can
  be carried out using either numerical or
  simulation
• techniques such as MCMC.

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• All programs and graphics carried out using
  SAS.
• Sample programs and data are available at
• http//www.uidaho.edu/ag/statprog

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Demonstration

• Data
• Effects of organic pesticide on egg hatch of
  black vine weevil (BVW).
• 20 BVW eggs placed in a petri dish with the
  pesticide.
• 9 doses (concentrations) of pesticide used.
• 0 to .03 g.
• Each dose replicated 10 times.
• The number of eggs failing to hatch recorded
  (success).
• Three experiments conducted, each varying in
  dose range.

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Bayesian Logistic Model Estimation
Unhatched Eggs
0.00
0.01
0.02
0.03
Dose (g)

Credible Regions Parameter Estimate
Lower Upper q1
0.01750 0.01280 0.02320 q2
0.99995 0.99990 0.99998 g
0.00864 0.00832 0.00891 b
466.800 432.547 502.796
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i. Dose-response curve known
1) Observe M successes in N trials in a new
experiment. 2) Logistic model assumed and
parameters assumed known.
What was the dose associated with this new
observation?
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Dose-response Curve Known
P(xM)
P(xM)
P(xM)
Dose
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ii. Dose-response curve estimated
1) Observe M successes in N trials in a new
experiment. 2) Logistic model assumed and
estimated ( parameters known with
error). What was the dose associated with
this new observation?
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Dose-response Curve Estimated
P(xM)
P(xM)
P(xM)
Dose
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Dose-response Curve Known Estimated (M 10,
N20)
Obs. 1580
Known
Estimated
P(xM)
0.006
0.007
0.008
0.009
0.010
0.011
0.012
Dose
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Dose-response Curve Known Estimated (M 10,
N20)
Obs. 310
Known
Estimated
P(xM)
Dose
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• Entropy (Shannon, 1948) uniquely quantifies the
  level of information in a distribution.
• H -? p(x)ln(p(x))
• The ratio of entropy values from two
  distributions, say H1 and H2, can give a
  relative measure of their respective
  information.
• ER H1/H2
• If H2 represents a dose distribution from the
  known case, i.e. perfect information, and H1
  represents the corresponding estimated case,
  then ER will give some measure of the distance
  between the two distributions as well as the
  efficiency of the estimated case.

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Dose-response Curve Known Estimated
E
0.939
R
P(xM)
Dose
E
0.876
R
P(xM)
Dose
E
0.670
R
P(xM)
Dose
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Concluding Remarks

• Determining an unknown dose from calibration
  information can be naturally posed as a Bayesian
  problem.
• Dose estimation can be carried out both with
  and without calibration error.
• Calibration error will subsequently increase
  estimated interval limits.
• Increases in sampling intensity for the unknown
  dose cannot overcome calibration error.
• It is important to concentrate sampling effort
  on the definition, estimation, and development
  of the calibration model.

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References

• Berkson, J. 1944. Application of the Logistic
  function to bio-assay. J. Amer. Stat. Assoc.
  39, pp 357-65.
• Jaynes, E. T. 2003. Probability Theory.
  Cambridge University Press, Cambridge, UK. pp.
  727.
• Price, W. J., B. Shafii, K. B. Newman, S.
  Early, J. P. McCaffrey, M. J. Morra. 2003.
  Comparing Estimation Procedures for Dose-response
  Functions. In Proceedings of the Fifteenth
  Annual Kansas State University Conference on
  Applied Statistics in Agriculture, CDROM
• SAS Inst. Inc. 2004. SAS OnlineDoc 9.1.3.
  Cary, NC SAS Institute Inc.
• Shannon, C., 1948. The Mathematical Theory of
  Communication. Bell System Technical Journal,
  27 379, 623.

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Questions / Comments