2.Probabilities & Bayesrule

Considerascenariowhereyouareinchargeofanalysingthedataandmodellingapandemic. Weconsideragivendisease(let'scallit'VIRUS'),whichhasanunknownprevalencein thepopulation(wewillassumethat[0,1]istheproportionofthepopulationthathas thedisease).Wewillwritetheprobabilitythatapersonisdiseasedasp(D)=.

(a)Yourlabhasdevelopedafasttestingproceduretodetectthisdisease.Inordertoevaluate theaccuracyandreliabilityofthistest,youhaveconductedtrialson132subjects,and comparedtheresultsofyourtestwithperfectlyaccurate(supposedlymoreexpensive) diagnostic.Theresultsofthosetrialsarecollatedinthefollowingtable:

positive negative

diseased 28 3

healthy 12 89

(i)UsingBayesformula,andthetrialdatainthetable,provideanestimateoftheprobabilities:

p(D|T),thatasubjectwhotestedpositiveistrulydiseased;and

p(D|T),thatasubjectwhotestednegativeisactuallydiseased. [4]

(ii)Takingintoconsiderationthetestaccuracyandreliabilityasevidencedinthetrials, wouldthistestbeappropriateforthefollowingsituations:

1.regulartestingofpeopleworkingwithvulnerablepopulations;

2.decidingonwhethertoadministeratreatmentwithseveresideeffects;or

3.applyingtothewholepopulationtofindalldiseasedindividuals (justify youranswers).

[3]

(iii)

Youadministeratestwithprobabilitiesp(D|T)=0.7andp(D|T)=0.01toasample of1000subjectsdrawnrandomlyfromthepopulation.Thetestreturns980negatives and 20positives.

Fromthisdata,calculateanestimateoftheprevalencep(D)explainingyourreasoning.

[4]

(b)Letusconsiderthatyouareexperimentingwithavaccineagainstthedisease.Youhave 1000subjectsingroupAwhotakethevaccineand1000ingroupBwhotakeaplacebo.Let usassumethatyoutestthesubjectsinbothgroupsdaily,andafteronemonthyouobtainthe followingresults:2subjectsfromgroupAtestedpositiveatsomepointduringthemonth, and40subjectsfromgroupB.Inthispartwewillassumethatweareusingatestwiththe followingstatistics:

theprobabilityofhavingthediseaseiftestedpositiveisp(D|T)=0.7

theprobabilityofhavingthediseaseiftestednegativeisp(D|T)=0.01.

(i)Accountingforthelimitationsofthetest,howmanysubjectsingroupAandBdidpossiblycatchthediseaseduringthismonth? [5]

(ii)Theefficacyofavaccineistypicallycalculatedas

p(D|V ) p(D|V )

Ev =

p(D|V)

Useyourresultsfromabovetocalculatetheefficacyofthevaccine.Discusswhat wouldhappenifyourtestwerelessaccurate:Whatwouldhappenifp(D|T)wouldbe lower?Ifp(D|T)wouldbehigher? [4]