Hierarchical Bayesian Model for

Certification of a Country as “Free” From an Animal Disease

Eric A. Suess, Dept. of Statistics, Calif. State Univ., Hayward

Ian Gardner, Dept. of Med. and Epi., School of Vet. Med., Univ. of Calif., Davis

Wes Johnson, Div. of Statistics, Univ. of Calif., Davis

Background

• Risk analysis for trade in animal products.

• Countries are interested in new trading opportunities and maintaining current trade.

• Is the country disease free?

• Needed for risk analysis to make policy decisions.

Disease Freedom

• “Disease freedom” – requires a perfectly sensitive test – All animals – All negative tests

• Level of Disease Freedom may not mean total freedom in all situations.– prevalence < threshold

Quantitative Approach

• Model: two-stage cluster sample

AnimalHerdCountry

Country D+ ?

Herd D+ ? All Herd D-

Animal D+ ? All D- All D-

Latent Data – Unknown Truth (conditional)

Yes

Yes No

No

No

T+ T- T+ T+T- T-

Observed

Quantitative Approach

We want to incorporate…

• Prior history of the Country

• Imperfect Test Sensitivity and Specificity

Inferences:

• Country

• Proportion of diseased herds

• Within-herd prevalence

Bayesian Statistics

• Bayes’ Rule

)(

)()|()|(

TP

DPDTPTDP countrycountry

country

Bayesian Approach

• Prior, Model, Posterior Analysis

• Latent Data (“true” status not observed)

• Markov Chain Monte Carlo – Gibbs sampler (a simulation technique)– Adaptive Rejection Sampling

Beta Priors

Expert opinion

mode and percentiles

beta(a,b)

x

y

0.85 0.90 0.95 1.00

05

10

15

mode = 0.985

lower 95% = 0.90

Example: Newcastle Disease

Data:• 260 flocks, 30 animals per flock

– 0 194 flocks

– 1 50

– 2 9

– 3 3

– 10 2

– 14 1

– 22 1

Example: Newcastle Disease

Priors on model probabilities:

• Country diseased = 0.50 uniform(0,1)

• Within herd-level prevalence = 0.30 beta(4.4,9)

• Herd-level prevalence = 0.05 beta(17,70)– Allowed to vary within flocks

• Sensitivity = 0.995 beta(103,1.5)

• Specificity = 0.995 beta(190,1.9)

Gibbs Sampler: assume country diseased

Reps = 200,000 Burn In =150,000

Results: Country D

Probability the country is diseased:

0.71 BCI: (0.16, 0.99)

Proportion of diseased flocks:

0.02 BCI: (0.01, 0.04)

Average within flock prevalence:

0.32 BCI: (0.21, 0.46)

Sensitivity:

0.987 BCI: (0.956, 0.999)

Specificity:

0.995 BCI: (0.988, 0.998)

Conclusions

• Bayesian approach.– Uses Prior Knowledge– Latent data– Imperfect test

• Variable within herd-level prevalence.

• Country/Herd/Animal Inference.

• Priors for others risk analysis.

Further Work

• Variable sample size within herds.

• Application to continuous surveillance.

References

• Audigé and Beckett (1999). A quantitative assessment of the validity of animal-health surveys using stochastic modeling. Prev. Vet. Med. 38, 259-276.

• Gohm, Thür, Audigé, and Hoffmann (1999). A survey of Newcastle disease in Swiss lay-hen flocks using serological testing and simulation modeling. Prev. Vet. Med. 38, 277-288.