Lecture notes on epidemiology

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1 Bostatstcs 2002, ÅS Lecture otes o epdemology The ams of epdemology: Epdemology s the study of the dstrbuto ad determats of health related states or evets specfed populatos ad the applcato of ths study to cotrol health problems (Mould) Typcally oe tres to fd assocatos betwee rsk factors (exposures) ad dseases by studyg the occurrece of dseases populatos The fal am s to coclude casual relatos Examples of epdemologcal fdgs (from Rothma): Smokg ad lug cacer Smokg ad other health related evets Passve smokg Low-level osg radato ad leukaema Sacchar ad bladder cacer Swe flu ad Gulla-Barré sydrome The effect of dethylstlbestrol o offsprg Tampos ad toxc-shock sydrome Coffee drkg ad pacreatc cacer Epdemologcal studes are ot geerally ameable to beg vestgated by radomsed trals ad observatoal studes are therefore the more practcal to study factors ad exposure whch ca ot be cotrolled by the vestgators (Mould) 1

2 Bostatstcs 2002, ÅS Hll s crtera for dstgushg betwee causal ad o-causal assocatos: 1 Stregth (the assocato should be suffcetly strog) 2 Cosstecy (t should be see dfferet populatos uder dfferet crcumstaces) 3 Specfcty (a cause should be specfc to ts effect, eg smokg leads to a specfc form of lug cacers) 4 Temporalty (cause precedes effect) 5 Bologcal gradet (greater dose should gve larger effect) 6 Plausblty (see amal models) 7 Coherece (ot coflct to kow bology) 8 Expermetal evdece (f possble a radomsed expermet should be carred through) 9 Aalogy (the cause make sese) Type of studes: Studes may be classfed as prospectve or retrospectve I a prospectve study measures of exposures ad covarates are made before lless occur I a retrospectve study these measuremets are made after the case have already occurred Cross-sectoal study A study that cludes all persos, the populato, at the tme of ascertamet, or a represetatve sample of such persos Dsease ad exposure are observed smultaeously 2

3 Bostatstcs 2002, ÅS Cohort study Two groups (or more) groups of people that are free from the dsease ad that dffer accordg to the extet of ther exposure to a potetal cause of the dseases are studed durg a tme terval Case-Cotrol study A case cotrol study cludes people wth a dsease ad a sutable cotrol group of people uaffected by the dsease The occurrece of the possble cause (exposure) s compared betwee cases ad cotrols A relevat stuato s to thk of a case-cotrol study as a cohort study where exposure s vestgated for all cases ad for a sample of the o-cases Measures of dsease frequecy: Prevalece s the umber of cases of a dsease a defed populato The prevalece ca be measured by a prevalece rate, e, the proporto of the populato that has the dsease at a specfc tme Ths ca be measured by a crosssectoal study Icdece s the umber of ew cases of dsease a populato The cdece ca oly be measured by cosderg a populato durg a tme terval It ca be measured by the cdece rate The cdece rate s defed as (The umber of ew cases the populato durg the tme cosdered)/(the total umber of years that people are uder rsk do develop the dsease) Cumulatve cdece s the umber of people a cohort that gets the dsease durg the tme of study The relato betwee prevalece ad cdece s somewhat complcated It depeds o the durato of the dsease state A approxmate relato s P = I D Where D s the mea durato whch dseased people carry the dsease 3

4 Bostatstcs 2002, ÅS If D s creased (eg the mortalty s lowered by ew medces or f the dsease s dagosed at a earler state) the prevalece rate creases eve f the cdece rate s the same Epdemologcal are most ofte cocered wth the study of cdeces The ma am s to compare cdeces for exposed ad o-exposed people Measures of effect ad assocato: Icdece rates exposed ad o-exposed people ca be compared several way Obvous alteratves are dffereces ad ratos Let I 1 be the cdece rate for exposed people ad I 0 be the cdece rate of o-exposed people The the two alteratves are I 1 - I 0 ad I 1 /I 0 If p 1 s the probablty that a exposed perso gets ll durg a tme terval ad p 0 s the correspodg probablty that a uexposed perso gets ll the we ca measure the dfferece ether as a rsk dfferece p 1 - p 0, Or a rsk rato p 1 /p 0 Stll aother alteratve s the odds rato {p 1 /(1-p 1 )} /{p 0 /(1-p 0 )} These measures gves dfferet results all cases other the p 1 = p 0 I that case the dfferece s 0 ad the rsk rato ad the odds rato are both equal to 1 It should be observed that these comparsos say lttle of the absolute values of the rsk Other measures (whch relates to a certa populato) s attrbutable rsk or etologcal fracto 4

5 Bostatstcs 2002, ÅS Statstcal aalyss: Oe strata: I the smplest stuato the outcome of a cohort study ca be summarzed a 2x2- table: exposed ll Not ll A terestg hypothess s that the rsks of gettg ll are the same for exposed ad o-exposed people Ths correspods to a χ 2 -test of homogeety (or Fsher s exact test) If ths hypothess s rejected t s of terest to estmate the effect of the exposure Oe effect measure s the odds rato The odds-rato s estmated by the cross-product rato: Most commo s however to estmate the logarthm of the odds rato The MLestmate s l(11 ) - l(12) - l( 21) + l( 22) If the observatos the cells are large t s possble to prove that ths estmate s asymptotcally ormal dstrbuted wth the true value of the logarthm of the odds rato as mea ad a asymptotc varace that ca be estmated by 1/ /12 1/ 21 1/ 22 Ths ca be used to calculate a approxmate cofdece terval for the logarthm of the odds rato by the followg formula l( + 11) - l(12) - l( 21) + l( 22) z a 1/11 + 1/12 + 1/ 21 1/ 22 From ths a approxmate cofdece rego for the log odds rato ca be derved as 5

6 Bostatstcs 2002, ÅS ( z 1/ + 1/ + 1/ + ) exp 1/ a The reaso for calculate estmates ad cofdece tervals for log odds stead of the odds drectly s that the ormal approxmato ofte works better for the log of the cross-product rato tha for the cross-product rato tself I a case-cotrol study the observatos ca also be summarzed a 2x2 table, but wth a dfferet terpretato: exposed cases cotrols The same aalyss ca be appled Observe that a a case-cotrol study you ca estmate the odds-rato but ot the rsk rato However f the rsk s small these measure are almost detcal Several strata: Sometmes the aalyss has to be stratfed to dfferet groups of people I that case we have to summarze the observatos to several 2x2-tables (oe for each strata) The reaso for ths s that there may be some other factor that relates to the exposure ad to the dsease that we wat to cotrol for Makg a separate aalyss for strata whch such factors are costat does ths Typcal stratfcato varables are sex, age, smokg, drkg ad other lfe style factors Such factor may ether fluece the effect as beg ether effect modfers or cofouders Cofoudg s a systematc bas the estmate of the effect that follows from that the cases ad cotrols dffers wth regard to a rsk factor ot cosdered the study The aalyss have to cosder ths possblty as s see by the followg example Stratum 1 Exposed Cases Cotrols

7 Bostatstcs 2002, ÅS Ths gves the estmated odds rato 40 Stratum 2 Exposed Cases Cotrols Ths gves the estmated odds rato 43 Summg these two tables gves Strata Exposed Cases Cotrols Whch gves the estmated odds rato 33 That t ot possble to do a far result by just summarsg (collapsg) over strata s referred to as o-collapsblty It s easy to costruct other examples that gve eve more devatg estmates If the effect s dfferet dfferet strata ths effect modfcato has to be descrbed ad aalysed Formulatg a model for how the effect s modfed by varous covarates may do ths If the effect s the same all strata there s eed for a statstcal method to do testg ad estmatg smultaeously for several 2x2-tables: The ull hypothess of o effect ay of the strata ca be tested by a so-called Matel-Haeszel test 7

8 Bostatstcs 2002, ÅS Stratum 1: exposed cases a 1 b 1 cotrols c 1 d 1 Stratum exposed cases a b cotrols c d Stratum exposed cases a b cotrols c d The observed umber of exposed cases s O = a The expected umber of exposed cases s (accordg to hypergeometrc dstrbuto each strata) E = ( a + b )( a + c ) a + b + c + d The varace of O s (accordg to the hypergeometrc dstrbuto) V = ( a + b )( c + d )( a + c )( b + d ) 2 t ( t - 1) 8

9 Bostatstcs 2002, ÅS Where t = a + b + c + d The ull hypothess of o effect all strata s rejected f ( O - E) 2 V s large compared to a χ 2 (1)-percetle Matched case-cotrol studes: I order to mmse the possblty of troducg cofouders epdemologcal studes are sometmes matched Ths meas that cotrols are selected specfc to each case (or a group of cases) a way that makes them as close to the cases as possble as regards covarates that are ot uder study A typcal examples of ths s gve Rce secto 135 Ths descrbes a 1-1 matchg It s also possble to have more cotrols, so called 1-m matchg Data from matched studes should be aalysed by methods for smultaeous aalyss of several 2x2-tables Two remarks: I matched studes the precso gaed by creasg the umber of cotrols to each cases decreases rather fast There may be a dsadvatage of matched studes sce t prevets ay study of the effect of the varables that has bee used to defe the matchg Aalyss of matched case-cotrol studes The methods for estmatg matched case-cotrol studes are the same as for aalysg several 2x2-tables where same effect s assumed all strata I a matched study each case defe a stratum A estmate of the commo effect s obtaed through the Matel-Haeszel estmator: a d b c / t / t 9

10 Bostatstcs 2002, ÅS Logstc regresso I may stuato covarates ca be attach to each dvdual uder study Ths meas that we do ot oly observe the Beroull varable Y whch ca take the values 0 ad 1 accordg to f the dvdual s dseases or ot but we also have other measuremets (x 1,,x k ) whch may fluece the rsk of gettg ll I such cases there s eed for a model that descrbes the fluece of the covarates (depedet varables) o the dstrbuto of Y Such a model whch s smlar to ormal lear regresso assumes that exp Pr(Y = 1) = 1 + exp ( b0 + b1x1 + + bkxk ) ( b + b x + + b x ) k k Ths model s called a logstc regresso model Observe that the log odds s a lear combato of the covarates, e, Pr(Y = 1) l = b0 + b1x1 + K + b Ł Pr(Y = 0) ł k x k Oe of the covarates may be exposure, ether defed as a 0/1-varable (exposed or ot) or as a dose (that measure the amout of exposures I such case the correspodg regresso coeffcet measures the effect of the exposure I fact the estmate of the regresso coeffcet s the logarthm of the odds rato effect of exposure that dffers wth oe ut of the scale whch the exposure s measured Codtoal logstc regresso I matched case cotrol studes we have a case wth correspodg covarates ad oe or more cotrols wth possble dfferet covarates We ca the use a logstc regresso model to calculate a codtoal probablty that says how propable t s that amog a umber of people (the case ad the cotrols) wth a ceta set of covarate values t s that just the case have the dsease uder study Let cosder a 1-1 matchg stuato ad assume that the case has covarates x ad the cotrol covarate z (For smplcty we assume a oe-dmesoal covarate) The we ca calculate the codtoal probablty that the perso wth covarates x have the dsease gve that exactly oe of the two persos have the dsease Elemetary (but ot smple) calculatos gve the result 10

11 Bostatstcs 2002, ÅS exp exp( b1x) ( b x) + exp( b z) 1 1 The statstcal aalyss of the matched data ca be aalysed usg these probabltes to bult up a lkelhood ML estmates ca be foud ad lkelhood rato test derved Possble sources of Bas epdemologcal studes: Cofoudg Selecto bas (eg Healthy worker effect) Recall Bas (cases may report exposures more accurately tha cotrols) Publcato Bas (sgfcat results teds to be publshed easer tha studes wth sgfcace s ot foud) 11

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