# Pooled Prevalence Calculator - Demonstration analyses

## Estimated true prevalence using two tests (unpooled) with a Gibbs sampler

This method uses a Bayesian approach and Gibbs sampling to estimate the true animal-level prevalence of infection based on testing of individual (not pooled) samples using two independent tests with imperfect sensitivity and/or specificity. The analysis requires prior estimates of true prevalence and test sensitivity and test specificity for both tests, as Beta probability distributions. It outputs posterior distributions for prevalence, sensitivity and specificity of both tests and several other parameters of interest. This method is similar to the one-test method, except that it allows incorporation of data from two tests used concurrently, and finds the best estimate that fits the combination of the prior information and the observed data. It also allows for uncertainty about the true values for sensitivity and specificity when calculating probability limits for the true prevalence estimate and the incorporation of prior information on the likely true prevalence based on pre-existing estimates or expert opinion. Because of the use of two tests, this method will often produce narrower probability limits about the prevalence estimate than the one-test method, particularly where there is considerable uncertainty about prior estimates.

For this analysis, the original values for stool sampling and serology for Strongyloides infection in Cambodian refugees from Joseph et al. (1996) were used, as listed in the table below, and 95% probability limits were calculated about the estimated prevalence.

Input Value
a (T1+/T2+) 38
b (T1+/T2-) 87
c (T1-/T2+) 2
d (T1-/T2-) 35
P alpha 1
P beta 1
Se 1 alpha 21.96
Se 1 beta 5.49
Sp 1 alpha 4.1
Sp 1 beta 1.76
Se 2 alpha 4.44
Se 2 beta 13.31
Sp 2 alpha 71.25
Sp 2 beta 3.75
Y1 start 35
Y2 start 30
Y3 start 2
Y4 start 10
Iterations 25000
Discard 5000

The prior Beta distributions defined above are equivalent to:

Distribution Alpha value Beta value 2.5% percentile Median 97.5% percentile Mean Mode Standard deviation
Prevalence 1 1 0.025 0.5 0.975 0.5 0.2887
Sensitivity (serology) 21.96 5.49 0.6346 0.8073 0.9242 0.8 0.8236 0.075
Specificity (serology) 4.1 1.76 0.3123 0.7235 0.9621 0.6997 0.8031 0.175
Sensitivity (stool) 4.44 13.31 0.0843 0.2406 0.469 0.2501 0.2184 0.1
Specificity (stool) 71.25 3.75 0.8909 0.954 0.9868 0.95 0.9623 0.025

The simulation was run for 25,000 iterations, with 5,000 iterations discarded to allow for convergence. Posterior probability distributions for prevalence, sensitivity, specificity and other parameters from the analysis are summarised below.

Prevalence Test 1 Se Test 1 Sp Test 1 PPV Test 1 NPV Test 2 Se Test 2 Sp Test 2 PPV Test 2 NPV Y1 Y2 Y3 Y4
Minimum 0.275 0.689 0.129 0.402 0.021 0.135 0.823 0.565 0.292 26 16 0 0
0.025 0.528 0.791 0.376 0.679 0.277 0.222 0.906 0.775 0.389 34 45 0 2
Median 0.772 0.889 0.695 0.918 0.639 0.305 0.96 0.907 0.519 38 77 2 10
0.975 0.921 0.954 0.955 0.992 0.858 0.425 0.989 0.973 0.705 38 87 2 25
Maximum 0.998 0.984 0.999 1 0.959 0.615 0.998 0.997 0.895 38 87 2 35
Mean 0.761 0.885 0.688 0.9 0.624 0.308 0.958 0.9 0.525 37 74 2 10
SD 0.099 0.042 0.16 0.084 0.15 0.051 0.021 0.051 0.079 1 11 1 6
Iterations 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000 20000