Relating models to data: A review

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Relating models to data A review

• P.D. ONeill
• University of Nottingham

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Caveats

• Scope is strictly limited
• Review with a view to future challenges

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Outline

• Why relate models to data?
• How to relate models to data
• Present and future challenges

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Outline

• Why relate models to data?
• How to relate models to data
• Present and future challenges

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1. Why relate models to data?

• 1. 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?

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1. Why relate models to data?

• 2. Estimation
• e.g. What is R0?
• e.g. What is the efficacy of a vaccine?

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1. Why relate models to data?

• 3. 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?

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1. Why relate models to data?

• 4. Real-time analyses
• e.g. Has the epidemic finished yet?
• e.g. Are control measures effective?

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1. Why relate models to data?

• 5. Calibration/parameterisation
• e.g. What range of parameter values are
  sensible for simulation studies?

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Outline

• Why relate models to data?
• How to relate models to data
• Present and future challenges

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2. 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)

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2. How to relate models to data

• 2.2 Fitting stochastic models
• Available methods depend heavily on the model and
  the data.

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2. How to relate models to data

• 2.2 Fitting stochastic models
• (i) Explicit likelihood
• e.g. Longini-Koopman model for household data
  (Longini and Koopman, 1982)

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2. How to relate models to data
P (Avoid infection from housemate) p

SEIR model within household
P (Avoid infection from outside) q
Given data on final outcome in (independent)
households, can formulate likelihood L (p,q)
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2. How to relate models to data

• 2.2 Fitting stochastic models
• (i) Explicit likelihood (continued)
• Related household models examples
• Bayesian analysis (ONeill at al., 2000)
• Multi-type models (van Boven et al., 2007)

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2. How to relate models to data

• 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. ONeill et al., 2000)
• Rejection sampling (e.g. Clancy and ONeill,
  2007)

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2. How to relate models to data

• 2.2 Fitting stochastic models
• (ii) No explicit likelihood
• Can arise due to model complexity and/or
  insufficient data

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2. How to relate models to data
Ever-infected
Two-level mixing model
Never-infected
Sample
Unseen
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2. How to relate models to data
Individual-based transmission models involve
unseen infection times
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2. How to relate models to data
Even detailed data from studies generally only
give bounds on unseen infection times e.g.
infection occurs between last ve test and
first ve test
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2. How to relate models to data

• 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

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2. How to relate models to data
Ever-infected
Two-level mixing model
Never-infected
Explicit interactions between households
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2. How to relate models to data
Ever-infected
Two-level mixing model -gt independent households
model
Never-infected
In a large population, households are
approximately independent
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2. How to relate models to data

• 2.2 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)

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2. How to relate models to data

• 2.2 Fitting stochastic models
• (ii) No explicit likelihood
• Solutions include
• Direct approach e.g. Martingale methods
  (Becker, 1989)

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2. How to relate models to data

• 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

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2. How to relate models to data

• 2.2 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

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2. How to relate models to data
Infectivity starts
Infectivity ends

TI
TE
Exposure time
ve test
Not observed
Observed data
-ve test
Höhle et al. (2005)
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2. How to relate models to data

• 2.2 Fitting stochastic models
• (ii) No explicit likelihood Data augmentation
  (continued)
• Data-augmentation methods include
• MCMC (e.g. Gibson and Renshaw, 1998 ONeill and
  Roberts, 1999 Auranen et al., 2000)
• EM algorithm (e.g. Becker, 1997)

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2. How to relate models to data

• 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

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2. How to relate models to data
Ever-infected
Two-level mixing model
? Data
Never-infected
Augment parameter space using links to describe
potential infections
Demiris and ONeill, 2005
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Outline

• Why relate models to data?
• How to relate models to data
• Present and future challenges

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3. 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

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3. Present future challenges

• 3.1 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.

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3. Present future challenges
Large data set and many missing data can cause
problems for standard (and also non-standard) MCMC
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3. Present future challenges

• 3.1 Large populations/complex models
• 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.

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3. Present future challenges

• 3.1 Large populations/complex models
• Problems of this kind are usually tackled via
  approximations (e.g. of the model itself).
• Challenge Can generic non-approximate methods be
  found?

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3. Present future challenges

• 3.2 Data augmentation
• Comment this technique is surprisingly powerful
  and is (probably) under-developed.

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3. Present future challenges

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

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3. Present future challenges

• 3.2 Data augmentation
• 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.

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3. Present future challenges
Ever-infected
Two-level mixing model
Never-infected
Imputing edges in graph
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3. Present future challenges
Ever-infected
Two-level mixing model
Never-infected
2
Infection chain 1, 3, 1, 2, 1
1
2
3
4
2
5
4
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3. Present future challenges

• 3.2 Data augmentation
• e.g. Augmented data can also (sometimes) be used
  to bound quantities of interest.
• Clancy and ONeill (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.

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3. Present future challenges

• 3.2 Data augmentation

x
x
x
x
x
Observed removal times
x
Imputed infection times
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3. Present future challenges

• 3.2 Data augmentation

x
x
x
x
x
Observed removal times
Soon as possible
x
Imputed infection times
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3. Present future challenges

• 3.2 Data augmentation

x
x
x
x
x
Observed removal times
Late as possible
x
Imputed infection times
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
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3. Present future challenges

• 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

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3. Present future challenges

• 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

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References

• B. 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.

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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. ONeill 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.