Modelling and optimisation of group dose-response challenge - PowerPoint PPT Presentation

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Modelling and optimisation of group dose-response challenge experiments David Price Supervisors: Nigel Bean, Joshua Ross, Jonathan Tuke July 9, 2013 David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Experimental Design ◮ An experiment is a scientific procedure undertaken to make a discovery, test a hypothesis or demonstrate a known fact ◮ Procedure ◮ Subjects ◮ Time(s) to observe experiment David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Optimal Experimental Design Control Statistical Experiment Variables Criterion David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Statistical Criterion ◮ Given the log-likelihood ℓ = log( L ( θ )) The Fisher information is �� ∂ℓ � � ∂ℓ �� I i,j = E ∂θ i ∂θ j ◮ Related to the variance of parameter estimates David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Optimality Criterion Many different optimality criteria that look at minimising the variance of the parameter estimates. E-Optimality: Maximise the smallest eigenvalue of I Minimise variance of parameter estimate with largest variance A-Optimality: Maximise trace of I Minimise average variance of parameter estimates D-Optimality: Maximise determinant of I Minimise the generalised variance of the parameter estimates David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Frequentist Experimental Design ◮ Choose design that satisfies chosen criterion ◮ Locally optimal ◮ Require knowledge about the parameter in order to determine the optimal design David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Frequentist Example ◮ Exponential lifetimes; F ( t ) = 1 − exp( − tθ ) I ( θ, t ) = t 2 exp( − tθ ) 1 − exp( − tθ ) ∂ I ∂t = t exp( − tθ )(2 − 2 exp( − tθ ) − tθ ) (1 − exp( − tθ )) 2 ◮ Maximised when t ≈ 1 . 5936 /θ David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Bayesian Optimal Experimental Design ◮ Allows incorporation of prior knowledge into design ◮ Choose a utility function that we wish to maximise ◮ Expected Kullback-Leibler divergence � � � p ( θ | y , d ) � U ( d ) = log p ( y , θ | d ) d y dθ p ( θ ) ◮ Maximise our gain in information David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Optimal Experimental Design for Markov Chains ◮ Why Markov Chains? ◮ Open field of research ◮ Becker and Kersting [1983] ◮ Cook et al. [2008] ◮ Pagendam and Pollett [2010] ◮ Pagendam and Ross [2013] ◮ Pagendam and Pollett [2013] David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Dose-response challenge experiments Dose 1 Not Infected Chicken 1 Dose 2 Not Infected Chicken 2 Dose 3 Infected Chicken 3 David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Dose-response relationship 1 0.9 Proportion of Colonized Hosts 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 log 10 Dose David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Dose-response relationship 1 0.9 Proportion of Colonized Hosts 0.8 Slope 0.7 0.6 0.5 0.4 0.3 0.2 ID 50 0.1 0 0 2 4 6 8 10 log 10 Dose David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Group dose-response experiment with transmission Dose 1 Not Infected Chicken 1 Dose 2 Infected Chicken 2 Dose 3 Infected Chicken 3 David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Dose-response relationship with transmission 1 0.9 Proportion of Colonized Hosts 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 2 4 6 8 10 log 10 Dose David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Dose-response relationship with transmission 1 0.9 Proportion of Colonized Hosts 0.8 Slopes 0.7 0.6 0.5 0.4 0.3 0.2 ID 50 0.1 0 0 2 4 6 8 10 log 10 Dose David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Modelling dose-response challenge experiments ◮ Conlan et al. [2011] first to account for transmission ◮ Two-stage process; dose-response and transmission ◮ Need to take into account latency period of dose-response ◮ Create SEIR model (Susceptible, Exposed, Infected, Removed) David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Optimal Design for some epidemic models ◮ Pagendam [2010] and Pagendam and Pollett [2013] looked at optimal design for experimental epidemics ◮ Locally optimal design of the SIS epidemic ◮ Likelihood evaluation computationally expensive David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Recall: SIS epidemic ◮ State space is number of infected individuals ( 0 , . . . , N ) ◮ Transition rates are ◮ q i,i +1 = βi ( N − i ) N ◮ q i,i − 1 = µi ◮ Estimate parameters ( ρ, α ), where ρ = µ β and α = β − µ ◮ Nice physical interpretation David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References An SIS epidemic, α = 3 , ρ = 0 . 25 400 350 Infected Individuals 300 250 200 150 100 50 0 0 2 4 6 8 10 Time David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References An SIS epidemic, α = 3 , ρ = 0 . 25 400 350 Infected Individuals 300 ρ 250 200 α 150 100 50 0 0 2 4 6 8 10 Time David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References An SIS epidemic, α = 3 , ρ = 0 . 25 400 350 Infected Individuals 300 250 t 1 t 2 200 150 100 50 0 0 2 4 6 8 10 Time David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References An SIS epidemic, α = 3 , ρ = 0 . 25 400 350 Infected Individuals 300 250 t 1 t 2 200 150 100 50 0 0 2 4 6 8 10 Time David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Overestimating α 25 0.4 18 16 0.35 20 14 0.3 12 0.25 15 10 0.2 α 8 10 0.15 6 0.1 4 5 0.05 2 0 0 0 0 0.2 0.4 0 20 40 0 0.2 0.4 ρ α ρ Figure: Densities of MLE’s for ( ρ, α ) . True values are (0 . 25 , 3) . David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Does Bayes have a problem? ◮ Kullback-Leibler divergence looks to maximise the difference between the prior and posterior ◮ What if our posterior distribution for the ‘bad’ design is “further away” than the posterior for the ‘good’ design? David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Example 0.4 ‘Good’ likelihood 0.35 0.3 0.25 Density 0.2 0.15 0.1 0.05 0 0 5 10 15 α David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Example 0.4 ‘Bad’ likelihood 0.35 0.3 0.25 Density 0.2 0.15 0.1 0.05 0 0 5 10 15 α David Price ANZAPW 2013

Optimal Experimental Design Optimal Experimental Design for Markov Chains Optimal Design for some Epidemic Models References Example 0.4 Prior 0.35 0.3 0.25 Density 0.2 0.15 0.1 0.05 0 0 5 10 15 α David Price ANZAPW 2013