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
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
• The Response Data:
• Continuous
• Normal, Log Normal, Gamma, etc.
• Discrete
• Binomial, Multinomial, Poisson
• Curve Estimation
• Linear or non-linear techniques.
• 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?
•
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.
ii) the dose-response curve is estimated
(known with error).
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.
• A Bayesian estimate is:
p(q|yij) = p(yij|q) · p(q)
(2)
p(yij|q) · p(q)dq
where p(yij|q) 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(q|yij)
is the posterior distribution of q given the data yij.
• The likelihood, p(yij|q), is given by:
L(pi) 
P
y (1 - p )(N - y )
(p
)
ij
i
i
ij
ij
(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.
• 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))
yielding:
(4)
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)
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(x|M,N,q1,q2 )  p(M|x,N, q1, q2 ) · p(x)
 piM (1 - pi)N-M p(x)
where p(x) is a prior probability for x.
(7)
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)
ii. Dose-response curve estimated
• Given:
• Observe M successes in N trials Logistic doseresponse, 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, q2|yij)
ii. Dose-response curve estimated

p(x|M,N,yij)  p(M|x)·p(x)·p(q1, q2|yij) 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, q2|yij)
is the posterior distribution given in (6).
• This essentially filters the posterior distribution for
dose in (7) through p(q1, q2|yij).
• Given prior distributions for x, estimation can be
carried out using either numerical or simulation
techniques such as MCMC.
• All programs and graphics carried out using SAS.
• Sample programs and data are available at:
http://www.uidaho.edu/ag/statprog
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.
Bayesian Logistic Model Estimation
# Unhatched Eggs
20
10
0
0.00
0.01
Parameter Estimate
Dose (g)
0.02
Credible Regions
Lower
Upper
q1
q2
0.01750
0.99995
0.01280 0.02320
0.99990 0.99998
g*
b*
0.00864
466.800
0.00832 0.00891
432.547 502.796
0.03
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?
P(x|M)
P(x|M)
P(x|M)
Dose-response Curve Known
M:5
N : 20
Unknown Dose
L95
0.0042
U95
0.0085
Unknown Dose
M : 10
N : 20
M : 19
N : 20
0.000
^
x
0.0069
L95
^
x
U95
0.0072
0.0088
0.0103
Unknown Dose
0.004
0.008
0.012
0.016
Dose
L95
^
x
U95
0.0114
0.0137
0.0198
0.020
0.024
0.028
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?
Dose-response Curve Estimated
Unknown Dose
^
L95
x
U95
0.0040
0.0068
0.0086
M : 10
N : 20
Unknown Dose
^
L95
x
U95
0.0071
0.0088
0.0104
P(x|M)
P(x|M)
P(x|M)
M:5
N : 20
M : 19
N : 20
0.000
L95
0.0114
0.004
0.008
0.012
0.016
Dose
0.020
Unknown Dose
^
x
U95
0.0139 0.0199
0.024
0.028
Dose-response Curve Known
& Estimated (M = 10, N=20)
Obs. = 1580
P(x|M)
Known
Estimated
0.006
0.007
0.008
0.009
0.010
Dose
Unknown Dose
^
L
x
U
95
Known
Estimated
95
0.0072 0.0088 0.0103
0.0071 0.0088 0.0104
0.011
0.012
Dose-response Curve Known
& Estimated (M = 10, N=20)
Obs. = 310
P(x|M)
Known
Estimated
0.002
0.004
0.006
0.008
0.010
0.012
Dose
Unknown Dose
^
L
x
U
95
Known
Estimated
0.0065 0.009
0.0058 0.009
95
0.0114
0.0122
0.014
0.016
• 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.
Dose-response Curve Known
& Estimated
E R = 0.939
P(x|M)
Known
Estimated
M : 10
N : 20
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
Dose
P(x|M)
Known
Estimated
E R = 0.876
M : 50
N : 100
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
Dose
P(x|M)
Known
Estimated
E R = 0.670
M : 400
N : 800
0.002
0.004
0.006
0.008
0.010
Dose
0.012
0.014
0.016
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.
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.
Questions / Comments