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21
Appendix
Underre porti ng
The underr eportin g in da ta col lection is a fairly common prob lem in social science s, publ ic heal th, c riminolo gy, and
microe conomi cs. It occur s whe n the coun ting of some event of inte rest is for some reas on i ncompl ete or t here are
errors in recording the out comes . Exam ples are unemplo yment data, infec tious or chronic disea se data (e.g. HIV o r
diabetes), cr imes with an as pect of shame (e. g. sexualit y and domestic vi olence), err or counts in a production
process es or soft war e en ginee ring , and traffic accidents with minor damage [1]. An e stima ted preval ence of event s
based on the inc omplete counts i s likely t o be smaller than the true proportion of events in t he population. Several
inference te chniques based o n binomial, beta - binomial, and regression models have been proposed for estimati ng
the actual co unt values [2]. How ever, in all those tec hniques the reportin g probability (unde rreporting rate) is
assumed to be a constant parameter over ti me that is estimated based on the sample counts.
A very simi lar probl em ex ists in preli minar y or pilo t cl ini cal i nvest igat ions , ep idemio logic al s urvey s, a nd lo ngit ude
studies where the objective is to estimate any possible clinical effect of a treatment or prevalence of a particular
disease in a population of pat ients, but the prevalence of events can only be esti mated by selecti ng a sample of
patients from t he population [3].
In all these situations, the preva lence of the events are estimated ba sed on a random sample o f events from the
population, unde r the assumption that the sample set con tains the same characteristics and distribu tions of the actual
population, inc luding those of the underreport ed and missing c ases.
Furthermore, it i s often r equired t o perform a sample - size calculation based on con fidence intervals in order to
provide a preci se estimate wit h a large margi n of certai nty and to make s ure that t he estimated pr oportion is close to
the actual pro portion with a high probability [3]. Co nfidence intervals fo r the proportions es timated based on
samples from large populations and finite populations can be calculated by using the normal approximation to the
binomial distr ibution as f ollows:
For large populatio ns:
𝑝 ± 𝑧
! ! ! / !
𝑝 ( 1 − 𝑝 )
𝑁
For finit e populat ions:
𝑝 ± 𝑧
! ! ! / !
𝑝 ( 1 − 𝑝 )
𝑁
.
𝑁
!"#$% !"#$% ! !
𝑁
!"#$%&'(")
where N is the siz e of sample, 𝑝 =
!
!
is the estim ate of the proportion of events of interests in the sample and
𝑁
!"#$%&'(")
is the size of p opulation in case of finite populations [3 ].
In this study, w e estimated the prevalence of adverse events by m aking sure that we have a significantly large
enough number of samples to provide confident estimates. Our estimations are obtained under the assumption that
the charac teristics and distribu tions of the obs erv ed events are not significantly different from those in the actual
population and would not significantly change after including the underreported cases. We are curre nt ly
investigating the extension of the proposed inferen ce tech niques in [1][2] to estimat e the actual number o f adverse
events with considering a variable reporting probability over time.
!
[1] Neubauer, G. an d Fri edl, H. , “ Model l ing sample sizes o f frequencies,” Proceedings of the 2 1
st
International
Works hop on Stat ist ica l Mo dell ing , 3 - 7 July 2006, Galway, Ireland.
[2] Neubauer, G., Dj uras G., Frie dl H. , “ Models for under repor ting: A Be rnoul li sampli ng ap proac h for rep orted
counts,” Austri an Journal of S tatist ics , Vol. 40 (2011 ), No. 1 & 2, 85 – 92