Bivariate model for a meta analysis of diagnostic test accuracy

Modeling
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This might be an easier model to think about if you use a different representation of the data. Specifically, for a given study you can re-code the 2x2 table of counts into two vectors, both containing 0/1, where one reflects the outcome of the diagnostic test (D) and the other reflects the truth/gold-standard (T) So, if you had a 2x2 of:

TP: 1
FP: 2
TN: 3
FN: 4

Then you’d have vectors:

D T
1 1
1 0
1 0
0 0
0 0
0 0
0 1
0 1
0 1
0 1
0 1

Then, for a given study, and using R’s formula syntax, you have a generalized linear model:

study_fit = glm(
    data = study_data
    , formula = D ~ 1+T
    , family = binomial
)

Where, if you make T a factor and use sum contrasts, the intercept parameter will reflect bias of the Diagnostic test while the effect of T will reflect the sensitivity (in the signal detection theory sense; I hate how medical stats adopted the same term for a different quantity in the same realm) of the Diagnostic test.

From there, the formulation of a meta-analysis can be achieved by treating the different studies as “random effects”, so if you had the above data for all studies combined together (again, making T a factor with sum contrasts) with a third vector identifying the study, then using lme4/brms formula notation you’d do:

meta_fit ~ glmer( 
    data = study_data
    , formula = D ~ 1+T + ( 1+T | Study)
    , family = binomial
)

(Using glmer there only bc I’m not confident in my brms)

In which case you’ve implemented a model where there’s an across-study mean bias, an across-study mean sensitivity, then each study gets its own regularized bias and sensitivity as correlated variates.

Finally, if you want to characterize the posterior’s implications for the TP/FP/TN/FN cell probabilities, these are deterministic transforms of the bias & sensitivity.

Note that if you have lots of data and encounter slow sampling, there are tricks to speed up binomial outcomes by having the likelihood evaluated as counts, but I didn’t want to confuse things by starting with that.

Note that the above signal detection theoretic model implies a shared variance magnitude between the latent distributions; if that’s not appropriate then check out this blog series on more nuanced approaches.