Design of HIV Dynamic Experiments: A Case Study

Transcription

1 Design of HIV Dynamic Experiments: A Case Study Cong Han Department of Biostatistics University of Washington Kathryn Chaloner Department of Biostatistics University of Iowa

2 Nonlinear Mixed-Effects Models Participants (people, animals, etc.) are similar, but different. Individual parameters θ i are realizations from a population. y i θ i F, θ i G Usually, y ij θ i ind N ( f(θ i,x j ),σ 2), θ i iid N(µ, Σ). 1

3 Nonlinear Mixed-Effects Models Example: Perelson et al., 1996, Science 271: Ritonavir administered to five patients; plasma HIV RNA measurements within one week. 2

4 V 0 = baseline viral load; δ = rate of loss of infected cells; c = rate of virion clearance. Mathematical modelling V (t) =V 0 [ c 2 (c δ) 2e δt c2 (c δ) 2 ] (c δ) 2 e ct cδ c δ te ct 3

5 Design issues for nonlinear mixed-effects models Choose a sampling schedule to achieve high efficiency in parameter estimation. Classical optimal design criteria are based on Fisher information, which is impractical to compute for nonlinear mixed-effects models. 4

6 A pragmatic approach Prior distribution π (for data analysis): σ 2 G(α, β) µ N(η, Λ) Σ 1 W (Ω,ν) Interested in estimating µ. 5

7 Quadratic loss function in estimating µ: 1. L(µ, a) =(µ a) (µ a), posterior risk= k Varπ (µ k y) 2. L(µ, a) =(µ a) D(µ a), D =diag(d kk ) p.s.d. with tr(d) =1, posterior risk= k d kkvar π (µ k y) Designer s loss = posterior risk. 6

8 A possibly different prior distribution, ω, for design σ 2 G(γ,δ) µ N(ζ, K) Σ 1 W (Ψ,χ) Under ω, adesignξ induces a distribution G ξ on y. Find a design to minimize preposterior risk Var π (µ k y)dg ξ (y) or d kk Var π (µ k y)dg ξ (y) k k 7

9 Some references on using different prior distributions: Etzioni and Kadane, JASA 88, Lindley and Singpurwalla, JASA 86, Tsai and Chaloner, Technical Report, School of Statistics, University of Minnesota. 8

10 Theorem 1. For all y in the sample space, E π (µ 2 k y) <. This theorem establishes existence of posterior variances. Need conditions for the integrability of posterior variances. Recall partial ordering of symmetric matrices: for A and B symmetric, A > B A B p.d. A B A B p.s.d. 9

11 Theorem 2. The following conditions on π and ω imply k Var π (µ k y)dg ξ (y) <. a. K 1 Λ 1 and ζ = η; ork 1 > Λ 1. b. γ = α and β δ; orγ>αand β>δ. c. χ = ν and Ψ 1 Ω 1 ;orχ>νand Ψ 1 > Ω 1. Recall π : σ 2 G(α, β), µ N(η, Λ), Σ 1 W (Ω,ν) ω : σ 2 G(γ,δ), µ N(ζ, K), Σ 1 W (Ψ,χ) 10

12 Note the conditions allow π and ω to be identical. Loosely speaking, the prior for design, ω, needs to be more informative than, or as informative as, the prior for analysis, π. Based on a case study of Bayesian inference for a population HIV dynamic model (Han, Chaloner, and Perelson, 2002, Case Studies in Bayesian Statistics 6, Springer-Verlag, ), examine a finite set of alternative designs. Could the design have been better with the same resources? 11

13 HIV dynamic model: y ij V 0i,c i,δ i,σ 2 ind N log V 0i log c i V 0,c,δ,Σ iid N log δ i ( x ij,σ 2), log V 0 log c, Σ log δ where x ij = [ c 2 log V 0i +log i it j (c i δ i ) 2e δ c2 i (c i δ i ) 2 ] (c i δ i ) 2 e c it j c iδ i t c i δ j e c it j i. 12

14 The 8 sampling schedules for an HIV dynamic experiments Schedule Day Hour 13

15 Assume 240 measurements are allowed: 15 subjects if schedules 1-7 are used, 30 if schedule 8 is used. Ignore the overhead cost associated with recruiting subjects. 14

16 π: σ 2 G(4.5, 9.0) µ N ( ) ( ), diag ( ) Σ 1 W (diag ( ), 3.0) ω: σ 2 G(81, 0.5) µ N ( ) ( ), diag ( ) Σ 1 W (diag ( ), 100) 15

17 Note that ω and π satisfy the conditions in Theorem 2. Given data, can estimate the posterior variances of log c and log δ (Theorem 1). Can integrate the posterior variances wrt the marginal distribution of data (Theorem 2). 16

18 Estimates of E[Var(log c y)] and E[Var(log δ y)] Schedule 1 Schedule 2 Schedule 6 Schedule V(log c y) V(log delta y) 17

19 Preposterior risk Schedule Risk D=diag(0 1 0) Schedule Risk D=diag( ) Schedule Risk D=diag( ) Risk Risk Risk k d kk Var π (µ k y)dg ξ (y). D=diag( ) Schedule Schedule Risk D=diag( ) D=diag( ) Schedule Schedule Risk D=diag( ) D=diag( ) Schedule Schedule Risk D=diag(0 0 1) 18

20 Generalization Application to PK studies. 2. Multivariate responses: viral load and CD Mean function not available in closed form. 4. Model discrimination. 19