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Checking individuals and sampling populations
with imperfect tests
Giulio D'Agostini and Alfredo Esposito
arXiv:2009.04843 [q-bio.PE]
For more info and R/JAGS scripts
see here
Abstract:
In the last months, due to the emergency of
Covid-19, questions related to the fact
of belonging or not to a particular class
of individuals (`infected or not infected'),
after being tagged as `positive' or `negative'
by a test, have never been so popular.
Similarly, there has been strong interest in
estimating the proportion of a population expected to hold
a given characteristics (`having or having had the virus').
Taking the cue from the many related discussions
on the media, in addition to those to which we took part, we
analyze these questions from a probabilistic perspective
(`Bayesian'), considering several effects that
play a role in evaluating the probabilities of interest.
The resulting paper, written with didactic intent,
is rather general and not strictly related to
pandemics: the basic ideas of Bayesian inference are
introduced and the uncertainties on the performances
of the tests are treated using the metrological concepts
of `systematics', and are propagated into
the quantities of interest following the rules of probability theory;
the separation of `statistical'
and `systematic' contributions to the uncertainty
on the inferred proportion of infectees allows to optimize
the sample size;
the role of `priors', often overlooked, is stressed,
however recommending the use of `flat priors', since
the resulting posterior distribution can be `reshaped'
by an `informative prior' in a later step;
details on the calculations are given, also deriving useful
approximated formulae, the tough work being however done
with the help of direct Monte Carlo simulations and
Markov Chain Monte Carlo,
implemented in R and JAGS (relevant code provided in appendix).
and 
to
: Bayes' rule applied to `numbers'
Inf
Pos
and
and 
with our `standard parameters'
and 
and 
in the case of
`anomalous' number of positives
![[*]](crossref.png)
) and
(
reported in
Fig. 
E
,
with detailed contributions to them, using the approximated
Eqs. (
and 
using a Beta prior (see Sec. 