Local&Bayesianoptimaldesigns in binary bioassay

Transcription

1 Local&Bayesianoptimaldesigns in binary bioassay D.M.Smith Office of Biostatistics & Bioinformatics Medicial College of Georgia. New Directions in Experimental Design (DAE 2003 Chicago) 1

2 Review of binary bioassay. Local and Bayesian optimal designs A range of (GLM) link functions & prior distributions. Usually no closed form solutions for Bayesian optimal designs, so many of the investigations are numerical. 2

3 Single substance binary bioassays. Probability of response π has the form π(x; θ) =F (α + βx) =F (z), z = α + βx θ T =(α, β) vector of unknown parameters. F () is a cumulative distribution function An alternative parametrization is z = β(x µ), θ T =(µ, β), where µ = α β. GLM terminology, z is the linear predictor and the inverse function F 1 () is the link function. Assumed that x = log(dose) and β 0. 3

4 Figure 1: pdf & cdf normal tolerance distribution mean µ = α β,sdσ = 1 β 4

5 Link function Logit Probit Φ normal c.d.f. Double exponential s = sign(z) Double reciprocal s = sign(z) F (z) 1/(1 + exp( z)) Φ(z) (1 + s) s 2 2 exp( z ) (1 + s) s ( ) z Complementary log log 1 exp[ exp(z) ] Skewed logit m = 1/3, 2/3, 3/2, 3 1/(1 + exp( z)) m 5

6 Bioassays comparing two substances. Parameter of interest is the potency of one substance compared to the other. Analysed using the parallel line model π(x; θ) =F (z) = F (α + β(x ρm)) or F (z) = F (β(x µ ρm)) θ T =(α, β, ρ) or θ T =(µ, β, ρ) m is a factor taking the value 0 or 1 representing the two substances. ρ is the log(potency) and is the horizontal distance between the two parallel lines. 6

7 Figure 2: comparative binary bioassay 7

8 Prior distributions Fixed values, locally optimal designs. Independent uniform distributions or, more generally, independent beta distributions. Multivariate normal distributions. Others e.g. bivariate normal for α(µ) & β, ρ an independent beta distribution for comparative bioassay of two substances. 8

9 Criterion functions All are functions of the expected information matrix. Let ξ = x 1, x 2,, x k w 1, w 2,, w k m 1, m 2,, m k denote a design with distinct support points x 1,,x k, weights w i at x i,where0 w i 1 and w i =1,andm i = 0 or 1 depending on substance. For θ T = (α, β, ρ), the elements of the information matrix I(ξ; θ) are i αα = w i fi 2 π i (1 π i ) i ββ = w i (x i ρm i ) 2 fi 2 π i (1 π i ) i βρ = β m i w i (x i ρm i )fi 2 π i (1 π i ) i αβ = w i (x i ρm i )fi 2 π i (1 π i ) i αρ = β m i w i fi 2 π i (1 π i ) i ρρ = β 2 m i w i fi 2 π i (1 π i ) where f i = F (α + β(x i ρm i )) and the summations are for i = 1,...,k. 9

10 Various criterion functions, φ(ξ) considered. Criterion φ(ξ) log (deti(ξ; θ)) I(ξ; θ) 1 ρρ log (deti(ξ; θ)) + log(i(ξ; θ) αα ) Associated directional derivative functions h(x) (General Equivalence Theorem). Directional derivative h(x) tr(i(x; θ) I(ξ; θ) 1 ) 3 (I(ξ; θ) 1 I(x; θ) I(ξ; θ) 1 ) ρρ + φ(ξ) tr(i(x; θ) I(ξ; θ) 1 ) I(x; θ) αα I(ξ; θ) 1 αα 2 Used to check optimality. If design optimal h(x) has a maximum value of zero, attained at, and only at, the support points of the design. φ(ξ) andh(x) for locally optimal designs. For Bayesian designs replaced by their expectation over the prior distribution of the parameters. Designs conveniently expressed as response probability π, or the linear predictor z. 10

11 Closed form solutions are not generally available for Bayesian optimal designs. Solutions have been found for simple cases. Prior distributions for the parameters leads to (Bayesian) designs where the range of doses around the support points showing efficiencies close to maximal widens. Also, the number of support points increases as the spread of the prior distributions increases. 11

12 Example Ashton (1972): Chrysanthemum aphid data Rotenone Deguelin dose x r n π dose x r n π Fitting parallel line model by non linear function minimisation using a logit link function gives ˆα = 4.629, ˆβ = and ˆρ = where Rotenone is taken as the standard (i.e. α = α R ). The variance/covariance matrix is ˆα ˆβ ˆρ ˆα ˆβ ˆρ

13 Logit link function, values of previous slide as the parameters of a trivariate normal prior distribution give Bayesian optimal designs as follows. Local optimal designs included for purposes of comparison. Value of the associated criterion the expectation under the trivariate normal prior distribution. Local and Bayesian optimal designs log determinant Standard Test Criterion (Rotenone) (Deguelin) value local x i w i Bayesian x i w i variance of ˆρ Standard Test Criterion (Rotenone) (Deguelin) value local x i w i Bayesian x i β =6.776 w i Bayesian x i β not fixed w i log determinant sub matrix Standard Test Criterion (Rotenone) (Deguelin) value local x i w i Bayesian x i w i

14 Figure 3: h(x) function for logistic, variance of ˆρ, tri-variate normal prior 14

15 A design similar to that of original data (same support points but equal weights of 0.1) with the trivariate normal prior distribution gives criteria values of , and for the respective criteria. There is a clear benefit in using the optimal designs in preference to repeating this design. Use of the local optimal designs with the same trivariate normal prior distribution results in only very small increases in criterion values. The designs for the variance of ˆρ criterion illustrate that if β is assumed known, two-point Bayesian optimal designs can be found. For the log determinant and log determinant of a sub-matrix criteria, the Bayesian optimal designs with β assumed known are the same as the local optimal design for the variance of ˆρ. 15

16 Often, precise information is available for α and β but not for ρ. To illustrate how the designs change, Bayesian optimal designs were found for a logistic link; variance of ˆρ and log determinant of a sub matrix criteria; and uniform priors with ranges to α to β to ρ Bayesian optimal designs log determinant Standard (Rotenone) Test (Deguelin) x i w i variance of ˆρ Standard (Rotenone) Test (Deguelin) x i w i log determinant sub matrix Standard (Rotenone) Test (Deguelin) x i w i

17 Figure 4: logistic, variance of ˆρ, priors uniforms α, β, ρ 17

18 For a logistic link and a trivariate normal prior based on the aphid data, how two point designs (expressed as values of π) with equal weights compare with respect to criterion values. Table 19.2 of Finney (1971), criteria values Support points variance log determinant as π of ˆρ sub matrix Bayesian The Bayesian optimal design for the variance of ρ is π i 0.41, 0.66 standard w i 0.323, π i 0.37, 0.52 test w i 0.076,

19 Discussion Appropriate model, consider use of an optimal design. Precision could be improved. Drawback of optimal designs, often too few support points to assess the fit of the model. Practical usefulness of local optimal designs limited by their dependence on the unknown underlying parameter values. Benchmark for other designs. Often use will result in only a small increase in criterion value. Bayesian optimal designs. As spreads of the prior distributions increase, the number of support points increase. Numerical optimization of φ(ξ) can be difficult. Designs are asymptotic ones, presume a population of infinite size is available. In practice, finite size. The designs here approximate designs. Weights representing integer numbers exact designs. 19

20 Future work Transforma- Skewed logit power parameter. tions of response. Introduction of control mortality and/or immunity. Generalized Linear Models and related. Random effects and variance components. Cost:benefit comparisons exact and approximate designs. 20

21 References Ashton, W.D. (1972). The Logit Transformation. Charles Griffin & Company Ltd. Chaloner, K. (1993). A note on optimal Bayesian design for nonlinear problems. J.Statist.Plann.Inference, 37, Chaloner, K. & Larntz, K. (1989). Optimal Bayesian design applied to logistic regression experiments. J.Statist.Plann.Inference, 21, Fan, S. & Chaloner, K. (1999). Geometric Methods for Singular C Optimal Designs. Technical Report No. 629, University of Minnesota. Finney, D.J. (1971). Statistical method in biological assay. Charles Griffin & Company Ltd, second edition. Ford, I., Torsney, B. & Wu, C.F.J. (1992). The use of a canonical form in the construction of locally optimal designs for non linear problems. J.Roy.Statist.Soc. B, 54, Sitter, R.R. & Wu, C.F.J. (1993). Optimal Designs for Binary Response Experiments: Fieller, D and A Criteria. Scand.J.Statist., 20, Smith, D.M. & Ridout, M.S. (1998). Locally and Bayesian Optimal Designs for Binary Dose Response Models with Various Link Functions. In: COMPSTAT98. (ed. R.Payne and P.Green), Physica Verlag. Smith, D.M. & Ridout, M.S. (2003). Optimal designs for criteria involving log(potency) in comparative binary bioassays. J.Statist.Plann.Inference, 113, davismith@mail.mcg.edu 21