Download Hierarchical Bayesian Model for Certification of a Country as “Free

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
Country
Herd
Animal
Latent Data – Unknown Truth (conditional)
Country D+ ?
Yes
No
Herd D+ ?
All Herd D-
No
Yes
Animal D+ ?
No
All D-
All DObserved
T+
T-
T+
T-
T+
T-
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
P( Dcountry | T ) 
P(T | Dcountry) P( Dcountry)
P(T )
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
15
Expert opinion
10
mode and percentiles
y
mode = 0.985
lower 95% = 0.90
0
5
beta(a,b)
0.85
0.90
0.95
x
1.00
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:
Specificity:
0.987 BCI: (0.956, 0.999)
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.