Relating models to data: A review P.D. O’Neill University of Nottingham

Relating models to data: Relating models to data: A reviewA review

P.D. O’NeillP.D. O’Neill

University of NottinghamUniversity of Nottingham

CaveatsCaveats

Scope is strictly limited Review with a view to future challenges

OutlineOutline1.1. Why relate models to data?Why relate models to data?

2.2. How to relate models to dataHow to relate models to data

3.3. Present and future challengesPresent and future challenges




1. Why relate models to data?1. Why relate models to data?

1. Scientific hypothesis testing1. Scientific hypothesis testing

e.g. Can within-host heterogeneity of susceptibility to HIV explain decreasing prevalence?

e.g. Did control measures alone control SARS in Hong Kong?


2. Estimation2. Estimation

e.g. What is R0?

e.g. What is the efficacy of a vaccine?


3. What-if scenarios3. What-if scenarios e.g. What would have happened if

transport restrictions were in place sooner in the UK foot and mouth outbreak?

e.g. How much would school closure prevent spread of influenza?


4. Real-time analyses4. Real-time analyses

e.g. Has the epidemic finished yet?

e.g. Are control measures effective?


5. Calibration/parameterisation5. Calibration/parameterisation

e.g. What range of parameter values are sensible for simulation studies?




2. How to relate models to data2. How to relate models to data

2.1 Fitting deterministic models Options include

(i) “Estimation from the literature”(ii) Least-squares / minimise metric(iii) Can be Bayesian (Elderd, Dukic and

Dwyer 2006)


2.2 Fitting stochastic models

Available methods depend heavily on the model and the data.



(i) Explicit likelihood

e.g. Longini-Koopman model for household data (Longini and Koopman, 1982)


P (Avoid infection from outside) = q

P (Avoid infection from housemate) = p

Given data on final outcome in (independent) households, can formulate likelihood L (p,q)

SEIR model within household



(i) Explicit likelihood (continued)

Related household models examples:

Bayesian analysis (O’Neill at al., 2000)

Multi-type models (van Boven et al., 2007)


2.2 Fitting stochastic models(i) Explicit likelihood (continued) Methods include Max likelihood (e.g. Longini and Koopman, 1982)EM algorithm (e.g. Becker, 1997)MCMC (e.g. O’Neill et al., 2000)Rejection sampling (e.g. Clancy and O’Neill, 2007)



(ii) No explicit likelihood

Can arise due to model complexity and/or insufficient data

2. How to relate models to data2. How to relate models to dataEver-infectedNever-infected

Sample Unseen

Two-level mixing model

2. How to relate models to data2. How to relate models to dataIndividual-based

transmission models involve

unseen infection times

2. How to relate models to data2. How to relate models to dataEven detailed data from

studies generally only give

bounds on unseen infection

times – e.g. infection occurs

between last –ve test and first

+ve test


2.2 Fitting stochastic models(ii) No explicit likelihood Solutions include: Use a simpler approximating model e.g. use pseudolikelihood, e.g. Ball, Mollison and

Scalia-Tomba, 1997



Explicit interactions between households



-> independent households model

In a large population, households are approximately independent

2. How to relate models to data2. How to relate models to data2.2 Fitting stochastic models(ii) No explicit likelihood Solutions include: Use a simpler approximating model e.g. discrete-time model instead of a continuous time model

(e.g. Lekone and Finkenstädt, 2006)


2.2 Fitting stochastic models(ii) No explicit likelihood Solutions include: Direct approach – e.g. Martingale methods

(Becker, 1989)


2.2 Fitting stochastic models(ii) No explicit likelihood Solutions include: Data augmentation: add in “missing data” or extra

model parameters to formulate a likelihood

2. How to relate models to data2. How to relate models to data2.2 Fitting stochastic models(ii) No explicit likelihood: Data augmentation (continued) Common example - model describes individual-to-individual transmission- observe times of case ascertainment, test results, etc, but

not times of infection/exposure- augment data with missing infection/exposure times


Exposure time

Infectivity starts

Not observed Observed data

Infectivity ends

= -ve test

TI

TE

Höhle et al. (2005)

= +ve test


2.2 Fitting stochastic models(ii) No explicit likelihood: Data augmentation

(continued)

Data-augmentation methods include MCMC (e.g. Gibson and Renshaw, 1998; O’Neill

and Roberts, 1999; Auranen et al., 2000) EM algorithm (e.g. Becker, 1997)


2.2 Fitting stochastic models(ii) No explicit likelihood: Data augmentation

(continued)

Data-augmentation methods can also be used in less “obvious” settings

e.g. final size data for complex models



Augment parameter space using links to describe potential infections

Data

Demiris and O’Neill, 2005




3. Present & future challenges3. Present & future challenges

3.1 Large populations/complex models

Current methods often struggle with large-scale problems.

e.g: Large population, Many missing data, Many hard-to-estimate parameters/covariates

3. Present & future challenges3. Present & future challenges3.1 Large populations/complex models

e.g. UK foot & Mouth outbreak 2001

Keeling et al. (2001) stochastic discrete-time model, parameterised via likelihood estimation and tuning/ simulation.

Attempting to fit this kind of model using “standard” Bayesian/MCMC methods does not work well.


Large data set and many missing data can cause problems for standard (and also non-standard) MCMC



e.g. Measles data

Cauchemez and Ferguson (2008) discuss the problems that arise when fitting a standard SIR model to large-scale temporal aggregated data in a large population using standard methods.



Problems of this kind are usually tackled via approximations (e.g. of the model itself).

Challenge: Can generic non-approximate methods be found?


3.2 Data augmentation

Comment: this technique is surprisingly powerful and is (probably) under-developed.



e.g. Cauchemez and Ferguson (2008) use a novel MCMC data-augmentation scheme using a diffusion model to approximate an SIR epidemic model.



e.g. For final size data, instead of imputing a graph describing infection pathways, could instead impute generations of infection (joint work with Simon White).

This can lead to much faster MCMC algorithms.

3. Present & future challenges3. Present & future challengesEver-infectedNever-infected


Imputing edges in graph

3. Present & future challenges3. Present & future challengesEver-infectedNever-infected


1

2

2

2

3

4

4

5

Infection chain = {1, 3, 1, 2, 1}



e.g. Augmented data can also (sometimes) be used to bound quantities of interest.

Clancy and O’Neill (2008) show how to obtain stochastic bounds on R0 and other quantities by considering “minimal” and “maximal” configurations of unobserved infection times in an SIR model.



Observed removal times

Imputed infection times

x x x x x

x





xxxxx

x

Soon as possible





x x x x x

x

Late as possible

Can show that “Soon as possible” maximises R0 but that minimal value is not necessarily given by “Late as possible” – use Linear Programming to find actual solution.

General idea also applicable to final outcome data


3.3 Model fit and model choice

Various methods are used in the literature to assess model fit, e.g.

Simulation-based methods; use of Bayesian predictive distribution; standard methods where applicable; Bayesian p-values


3.3 Model fit and model choice

Likewise for model choice methods include AIC, RJMCMC

Challenge Better understanding of pros and cons of such methods

ReferencesReferencesB. D. Elderd, V. M. Dukic, and G. Dwyer (2006) Uncertainty in predictions of disease spread and public health responses to

bioterrorism and emerging diseases. PNAS 103, 15693-15697

I.M. Longini, Jr and J.S. Koopman (1982) Household and community transmission parameters from final distributions of infections in households. Biometrics 38, 115-126.

P.D. O'Neill, D. J. Balding, N. G. Becker, M. Eerola and D. Mollison (2000) Analyses of infectious disease data from household outbreaks by Markov Chain Monte Carlo methods. Applied Statistics 49, 517-542.

M. Van Boven, M. Koopmans, M. D. R. van Beest Holle, A. Meijer, D. Klinkenberg, C. A. Donnelly and H.A.P. Heesterbeek (2007) Detecting emerging transmissibility of Avian Influenza virus in human households. PLoS Computational Biology 3, 1394-1402.

D. Clancy and P.D. O'Neill (2007) Exact Bayesian inference and model selection for stochastic models of epidemics among a community of households. Scandinavian Journal of Statistics 34, 259-274.

N.G. Becker (1997) Uses of the EM algorithm in the analysis of data on HIV/AIDS and other infectious diseases. Statistical Methods in Medical Research 6, 24-37.

F.G. Ball, D. Mollison and G-P. Scalia-Tomba (1997) Epidemic models with two levels of mixing. Annals of Applied Probability 7, 46-89.

M. Höhle, E. Jørgensen. and P.D. O'Neill (2005) Inference in disease transmission experiments by using stochastic epidemic models. Applied Statistics 54, 349-366.

References…References…N. G. Becker (1989) Analysis of Infectious Disease Data. Chapman and Hall, London.

G. Gibson and E. Renshaw (1998). Estimating parameters in stochastic compartmental models using Markov chain methods. IMA Journal of Mathematics Applied in Medicine and Biology 15, 19-40.

P.D. O’Neill and G.O. Roberts (1999) Bayesian inference for partially observed stochastic epidemics. Journal of the Royal Statistical Society Series A 162, 121-129.

K. Auranen, E. Arjas, T. Leino and A. K. Takala (2000) Transmission of pneumococcal carriage in families: a latent Markov process

model for binary longitudinal data. Journal of the American Statistical Association 95, 1044-1053.

P.E. Lekone and B.F. Finkenstädt (2006) Statistical Inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study. Biometrics 62, 1170-1177.

M.J. Keeling, M.E.J. Woolhouse, D.J. Shaw, L. Matthews, M. Chase-Topping, D.T. Haydon, S.J. Cornell, J. Kappey, J. Wilesmith, B.T. Grenfell (2001). Dynamics of the 2001 UK Foot and Mouth Epidemic: Stochastic Dispersal in a Heterogeneous Landscape. Science 294, 813-817.

S. Cauchemez and N.M. Ferguson (2008). Likelihood-based estimation of continuous-time epidemic models from time-series data: application to measles transmission in London. Journal of the Royal Society Interface 5, 885-897.

D. Clancy and P.D. O'Neill (2008) Bayesian estimation of the basic reproduction number in stochastic epidemic models. Bayesian Analysis, in press.

Documents

Relating models to data: A review P.D. O’Neill University of Nottingham