Two dimensions of measurement error: Classical and Berkson error in residential radon exposure assessment

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1 Journal of Exposure Analyss and Envronmental Epdemology (2004)14, r 2004 Nature ublshng Group All rghts reserved /04/$ Two dmensons of measurement error: Classcal and Berkson error n resdental radon exposure assessment I.M. HEID, a H. KU CHENHOFF, b J. MILES, c L. KREIENBROCK d AND H.E. WICHMANN a a GSF-Natonal Research Center for Envronment and Health, Instute of Epdemology, Neuherberg, Germany b Department of Statstcs, Ludwg-Maxmlans-Unverstta t Mu nchen, Munch, Germany c Natonal Radaton rotecton Board, Oxford, UK d Insttute for Bometry, Epdemology and Informaton rocessng, Hanover School of Veternary Medcne, Hanover, Germany Measurement error n exposure assessment s unavodable. Statstcal methods to correct for such errors rely upon a vald error model, partcularly regardng the classfcaton of classcal and Berkson error, the structure and the sze of the error. We provde a detaled lst of sources of error n resdental radon exposure assessment, stressng the mportance of (a) the dfferentaton between classcal and Berkson error and (b) the clear defntons of predctors and operatonally defned predctors usng the example of two German case control studes on lung cancer and resdental radon exposure. We gve ntutve measures of error sze and present evdence on both the error sze and the multplcatve structure of the error from three data sets wth repeated measurements of radon concentraton. We conclude that modern exposure assessment should not only am to be as accurate and precse as possble, but should also provde a model of the remanng measurement errors wth clear dfferentaton of classcal and Berkson components. Journal of Exposure Analyss and Envronmental Epdemology (2004) 14, do: /sj.jea Keywords: measurement error, Berkson error, error models, error sources, radon, case control studes. Introducton Radon, a ubqutous radoactve gas, s the second leadng cause of lung cancer after smokng n the general populaton (NAS, 1994). Epdemologcal studes on lung cancer and resdental radon exposure have been conducted n many countres worldwde to obtan relatve rsk (RR) estmates and to descrbe the exposure dsease relatonshp. Despte the sze and the qualty of these studes, the results range from no ncreased rsk to a sgnfcantly ncreased relatve lung cancer rsk of about1.10 per 100 Bq/m 2 radon gas concentraton (ershagen etal., 1994). For example n Western and Eastern parts of Germany, two case control studes were conducted durng the 1990s ncludng over 2500 patents dagnosed wth lung cancer from hosptals (cases) and an adequate group of about 4000 dsease-free partcpants recruted va populaton regstry (populaton controls). The partcpants were ntervewed wth regard to ther long-tme resdental, smokng, and occupatonal hstory. The radon concentratons n the bedroom and the lvng room of ther homes were measured by alpha track 1. Address all Correspondence to: I.M. Hed, GSF-Natonal Research Center of Envronment and Health, Insttute of Epdemology, Ingolstädter Landstr. 1, Neuherberg, Germany. E-mal: hed@gsf.de Receved 27 February 2003; accepted 28 October 2003 detectors over 1 year. Based on these measurements and ntervew nformaton on the tme of the rooms occupancy and on each home s resdency, the resdental radon exposure was assessed retrospectvely. The reported RR estmates per 100 Bq/m 3 and 95% confdence ntervals were 0.98 (0.82, 1.17) and 1.04 (0.96, 1.12) for the Westand the Eaststudy, respectvely, based on measurements n the homes nhabted atndex date and 0.97 (0.82, 1.14) and 1.10 (0.98, 1.24) for the West and the East study, respectvely, based on all measurements n homes nhabted up to 15 years pror to ntervew (Wchmann et al., 1998, 1999; Kreenbrock et al., 2001; Kreuzer etal., 2003). Measurementerrors n exposure assessmentare unavodable, wth resdental radon exposure beng no excepton (Ba verstam and Swedjemark, 1991; Lubn et al., 1995), and nduce bas on RR estmates. Methods to correct for such errors are avalable, but requre a model for the error n the assessed exposure, and qute dfferent results emerge dependng on the error type (classcal or Berkson,.e. error ndependentfrom true exposure or ndependentfrom observed exposure), on the structure (addtve or multplcatve), or the sze (Carroll et al., 1995). Usng the example of resdental radon exposure, we dentfy the error components and classfy them wth regard to classcal- or Berkson-type error. Further, we show that the applcablty of an error component depends on the practcal varable chosen to represent the predctor of the dsease n the

2 Hed et al. Two dmensons of measurementerror specfc study, the operatonally defned predctor (Carroll et al., 1995). We provde ntutve measures of the sze of the error and analyse three data sets wth repeated radon concentraton measurements to provde further nformaton on error structure and sze. Methods redctor and Operatonally Defned redctor The error n the exposure assessment s the dfference between the observed exposure from the true exposure. The components of error that are applcable depend upon the predctor and the operatonally defned predctor. Generally, the predctor of nterest s gven by the epdemologcal objectve of the study. However, each nvestgator has to defne a practcal varable, whch s measurable and a vald surrogate for the predctor: the operatonally defned predctor. In the German radon studes, the objectve was to quantfy the RR of lung cancer due to resdental radon exposure. From a methodologcal pont of vew, any varable can be plugged as exposure nto the exposure dsease-model. However, from the bologcal pont of vew, several stages of the dsease-causng process are dstngushed and the term exposure s one of the three terms employed (Armstrong, 1990): (1) the concentraton, c(t), a measure of the agent s densty at tme pont t; (2) the exposure, a measure for the agent s mass accumulatng durng tme perod T n the envronmentof an ndvdual, R T cðtþdt, or, f the exposure s to have the same unt as the concentraton and s thus gven per tme unt, R T cðtþdt=t; and (3) the effectve organ dose from the exposure experenced durng tme perod T. Usng the example of resdental radon studes, we elucdate the complexty of ths ssue. The true radon gas concentraton n an envronment s the concentraton of 222 Rn at a certan tme pont. The unt s Bequerel per cubc metre (Bq/m 3 ). However, what s actually measured s the average concentraton durng the exposure of the detector, RN(detector). In the German radon studes, the detectors were exposed for 1 year and the radon gas concentraton n the th home of a study partcpant s assessed as the average between bedroom and lvng room concentratons, weghted by the relatve occupancy tme, RN (detector) ¼ 0.5(w RN B (detector) þ (1 w )RN L (detector), where w denotes the percentage of tme spent n the bedroom. The lung cancer predctor true resdental radon exposure s mostly defned as the envronmental (external) exposure of an ndvdual per year to radon gas n the resdences nhabted durng a t me perod T, whch s relevantfor the cancerogeness at the ndex date, RN(T). A measurable proxy for ths, that s, an operatonally defned predctor, s derved by usng RN (detector) as vald proxy for the average radon concentraton durng the resdency of the t hhome, RN (resdency), and by computng the tme-weghted mean (TWM) concentraton, that s, the mean across all homes nhabted durng the relevant tme perod, T, weghted by the resdency tme n the th home, T, TWMðTÞ ¼ RN ðresdencyþt T : The unts Bequerel-years per cubc metre and year, whch equals Bequerel per cubc metre [Bqa/(m 3 a) ¼ Bq/m 3 ]. Another proxy s the cumulatve radon exposure per year accountng for absolute home occupancy, that s, the percentage of tme spent n the home, O, RNCUMðTÞ ¼ O RN ðresdencyþt T : The unts, agan, Bq/m 3 ; however, due to the fact that O s on average 50%, RNCUM(T) s abouthalf of TWM(T). A thrd proxy for resdental radon exposure s the average radon concentraton n the current home (.e. the home nhabted at ndex date) durng the tme of resdency, RN 1 (resdency), abbrevated by RN 1, f the resdency tme n the current home covers a good proporton of T. Naturally, the quanttes RN (resdency), T, and O can only be observed wth a certan error. Wth the observed quanttes denoted by RN (resdency) *, T *, and O *, t he observed proxes for resdental radon exposure are TWMðTÞ ¼ RN ðresdencyþ T T O RNCUMðTÞ RN ðresdencyþ T ¼ ; or RN : An alternatve lung cancer predctor to resdental radon exposure s the alpha dose, that s, the energy mparted to the lung tssue by alpha partcles from the radoactve decay of radon and radon progeny n the resdences durng the tme perod T, D(T), whch dffers from RN(T) by certan factors (Jacob, 1989, 1964, ICR, 1994): (a) the equvalence factor descrbng the equlbrum between radon and radon progeny gven the envronmental condtons (temperature, compresson), (b) a factor descrbng the amount of radon progeny actvty deposted n the lungs (dependng on partcles n the ar and the ndvdual s nhalaton depth and frequency), and (c) a factor descrbng the dose delvered to the senstve cells n the lungs (dependng on where the progeny are deposted and the depth of the senstve cells). Mostradon studes use TWM(T) as operatonally defned predctor for resdental radon exposure, where T covers the 5 35 years pror to ndex date. Approprate weghtng of the radon gas concentratons dependng on tme snce exposure s then necessary, snce exposures before 15 years are sad to lose potental to nduce lung cancer (Lubn et al., 1994). The German studes were analysed based on two operatonally T 366 Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5)

3 Two dmensons of measurementerror Hed et al. defned predctors: RN 1 and RNCUM(T), where T covers 5 15 years pror to ndex date, a homogeneous tme perod regardng the potental to cause cancer. For these three varables, TWM(T), RNCUM(T) andrn 1, as operatonally defned predctors for the true predctors resdental radon exposure or alpha dose, we establsh a lst of error components, formulate an error model, and provde, as far as possble, a sense of the plausble error sze. Fgure 1. Notaton of error model. Fgure 2. Error models: True predctor versus predctor measured wth (a) addtve or (b) multplcatve error (Fcttous data). Error Models In ths work, we deal wth random error (.e. zero expectaton), whch s nondfferental towards dsease status (.e. structure and sze ndependent of dsease status) and homoscedastc (.e. same structure and sze for all observatons). We elaborate partcularly on the dfferences between classcal-type error (.e. statstcally ndependent of the true varable) versus Berkson-type error (.e. statstcally ndependent from the observed varable) and between addtve versus multplcatve structure. Fgure 1 summarzes the notaton. Errors of the classcal type arse when a quantty s measured by some devce and repeated measurements vary around the true value. Error of the Berkson type s nvolved, when a group s average s assgned to each ndvdual sutng the group s characterstcs. The group s average s thus the measured value, that s, the value that enters the analyss, and the ndvdual latent value s the true value. Examples of Berkson error nclude the use of job-exposure-matrx entres nstead of ndvdual exposure measurements or the use of envronmental exposure measurements va fxed montors nstead of ndvdual dose measured va personal dosmetres (Tosteson et al., 1989). Thedfferencebetweenaddtveandmultplcatveclasscal error s elucdated n Fgure 2. For addtve error, the spread of true exposure gven measured exposure (vertcal spread of dots) s constant for the full range of the exposure: The graph shows a tube (Fgure 2a). For multplcatve error, the spread ncreases proportonally to measured exposure: The graph shows a trumpet (Fgure 2b). Snce the multplcatve error s addtve on the log-scale, all characterstcs of the addtve error are vald for the multplcatve error on the log-scale: The plot of the log of true exposure versus the log of exposure measured wth multplcatve error would provde the same pcture as Fgure 2a. Measures of Error Sze The error sze s usually gven as the standard devaton (SD) of the error on the orgnal scale for addtve error, s EA,oron the log-scale for multplcatve error, s log EM. For mult - plcatve error, alternatves are: (1) the geometrc standard devaton (GSD) of the error, exp(s log EM ), or (2) the coeffcent of varaton (CV) defned (a) as the error s SD on the orgnal pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff scale dvded by the mean on the orgnal scale, pffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff expðs logem Þ 1, or (b) as expðs 2 logem Þ expðs logem Þ 1, that s, the SD on the orgnal scale dvded by the geometrc mean (.e. the exponentated mean of the log of exposure) (GM). We compute a converson table for the dfferent measures. Further, we provde an ntutve way to grasp the sze of a classcal error by presentng the Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5) 367

4 Hed et al. Two dmensons of measurementerror range of measured predctor values that would most lkely be observed gven a true value, x. For multplcatve error, f error and true predctor are farly lognormally dstrbuted, 95% of the measured values on the orgnal scale le wthn [x(1/(gsd 2 )), x(gsd 2 )]. For addtve error, f error and true predctor are farly normally dstrbuted, 95% of the observed values le wthn [x 2s EA,x þ 2s EA ]. Fnally, we present the error varance as a proporton of the exposure varance observed n the German radon studes (on the orgnal scale for addtve error, on a log-scale for multplcatve error). Replcate Data Three data sets wth replcate radon concentraton measurements conducted by alpha track detectors n German dwellngs are avalable addressng dfferentssues. We explot ths nformaton to provde evdence about the error structure (addtve versus multplcatve) and error sze by plottng the data and applyng analyss-of-varance models (ANOVA) usng ROC MIXED by SAS s. Bedroom/lvng room measurements: For each study partcpant n the analysed sample of the German casecontrol studes, 1-year radon gas concentraton measurements n bedroom and lvng room of the current home are avalable. These nternal data allow the estmaton of the between-measurement-varablty, gven that the dfferences between rooms can be controlled for. Ths analyss s solely based on the controls (.e. about year measurements) to reflect the stuaton for the general publc. Year-by-year replcates: The German Federal Offce for Radaton rotecton has measured radon gas concentratons for several consecutve tme perods, each coverng about 1 2 months durng to montor changes n radon concentratons over tme. Two measurements were conducted under dentcal condtons for each tme perod n 11 arbtrarly selected houses ncludng basements of laboratores and houses wth very hgh radon levels n Schneeberg, an area of former uranum mnng. We computed the tmeweghted average radon concentraton of consecutve tme perods coverng 12 months (.e ¼ year measurements). Ths external data, cordally provded by R Lehmann, allows the estmaton of the between-yearvarablty, of the between-measurement-varablty, and of both combned. Intercomparson study: In 1990/91, an ntercomparson study was conducted to evaluate wthn- and betweenlaboratory-varablty of laboratores from dfferent European countres measurng radon gas concentratons n fve houses wth concentratons typcal of those expected n the then on-gong epdemologcal radon studes (offjn et al., 1992; Kreenbrock et al., 1999). From ths external data, the sx-month measurements from fve detectors placed n each of fve houses conducted by the German laboratory, the Bophyscs Department of the Unversty of Saarland, are utlzed n ths analyss (.e month measurements) to estmate the between-measurement-varablty for the laboratory, whch conducted all measurements of the German case control studes. Statstcal Models to Analyse the Replcate Data Based on the ntercomparson data, the sze of the error from between-measurement-varablty s estmated by applyng log Z ;j ¼ m þ HOUSE þ e ;j ð1þ where Z,j denotes the jth measurement n the th house and HOUSE the effect of the th house, and by computng the SD of the resduals e,j (j ¼ 1, y5, ¼ 1,y5). Based on the bedroom/lvng room measurements, the same error sze s estmated by applyng log Z ;j ¼ m þ HOUSE þ ROOM j þ e ;j ð2þ where Z,j denotes the jth measurement n the th house, HOUSE the effect for the th house, and ROOM j the effect of the bedroom versus the lvng room, and by obtanng the SD of the resduals (j ¼ 0, lvng room, j ¼ 1, bedroom). We also explore whether the floor level dfference explans most of the room effect. Based on the year-by-year data, both the sze of the error from between-measurement-varablty and the sze of the error from between-year-varablty are estmated by applyng log Z ;j;k ¼ m þ HOUSE þ HOUSE YEAR j þ e ;j;k ð3þ where Z,j,k denotes the kth measurement n the jt hyear for the th house, HOUSE the effect of the th house, HOUSE YEAR j the effect of the jt hyearbyhouse,andby dervng the SD of the resduals and the squareroot of the varance estmate of HOUSE YEAR j (k ¼ 1, 2, j ¼ 2, y, 5, ¼ 1, y, 11). An estmate of both errors combned s derved from log Z ;j;k ¼ m þ HOUSE þ e ;j;k ð4þ by the SD of the resduals. By further ncludng a fxed effect of the jth year, YEAR j, n model (3), we test for a potental effectof the years ndependentof the house. Results Identfcaton of Error Components In the followng, we presenta detaled lstof sources of error n radon exposure assessmentwth specal consderaton of ther applcablty for the operatonally defned predctors used n the German analyses, RN 1 and RNCUM(T), and for TWM(T). We propose to dstngush four stages for assessng the predctor resdental radon exposure, plus an addtonal ffth stage f alpha dose s the predctor of nterest: (1) Estmatng the average radon gas concentraton n the th home durng the exposure of the detector, RN (detector). 368 Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5)

5 Two dmensons of measurementerror Hed et al. (2) Usng (1) to estmate the average radon gas concentrat on n t he th home over the year n whch the measurementtook place, RN (year), that s, extrapolatontooneyear. (3) Usng (2) to estmate the radon gas exposure of an ndvdual over a certan tme perod pror to the measurement, RN (resdency), that s, extrapolaton to pror years. (4) Usng (3) for the current home or for all homes nhabted durng a certan tme perod T as operatonally defned predctor for resdental radon exposure, RN(T). (5) Usng (4) as a surrogate for the true alpha dose, D(T). The stages (1 3) descrbe the devaton between the observed predctor and the operatonally defned predctor, stage (4) the devaton between the operatonally defned predctor and exposure, and stage (5) the devaton between exposure and dose. Regardng stage (1), there s (a) the error from betweenmeasurement-varablty, that s the devaton between measurements obtaned repeatedly at the same tme and place. A measurement by alpha track detectors nvolves the exposure of a small box of specfc geometry contanng a thn fol. The emtted alpha partcles leave a small trace (track) on the fol. In order to count these tracks, the fol s etched. The specfc number of counts of a randomly chosen area of the fol s obtaned manually or by a computer program, and the number of counts per unt s then calculated. The exposure of the detector to radon s derved from the track densty on the fol by takng nto account the average background track densty on smlar fols and the senstvty of the fol to radon exposure, determned by calbraton Thus, ths error componentncludes the error from background track densty (number of tracks observed on a detector notexposed to radon), the error from mscountng the number of tracks, the error from varatons n track countng effcency, t he error from calbraton, and the error from underestmatng hgh exposure, when the tracks are so close together to cause dffculty n dstngushng them after etchng. Further, there are (b) the error from between-laboratoryvarablty (not applcable for the German studes, snce all measurements were conducted by the same laboratory), (c) the error from between-detector-placement-varablty due to the varaton of the radon concentratons dependng on the placementn the room, (d) the error from between-roomvarablty due to the fact that radon concentratons n the rooms wthout measurements dffer from the radon concentraton n the lvng room, whch was used as proxy for the concentratons n the other rooms (except the bedroom). Regardng stage (2), there s (a) the error from betweenseason-varablty due to seasonal varaton of the radon concentraton and applcable, f a measurement of less than a year s used to estmate the 1-year-average (not applcable for the German studes, where only 1-year-measurements were used). If seasonal correcton s appled, (2a*) an error from statstcal uncertanty n estmatng the correcton factor remans and (2a**) an error from assgnng a group-matched correcton factor s ntroduced. (One factor s assgned to all ndvduals wth a certan sesaonal pattern.) Regardng stage (3), there s (a) the error from betweenyear-varablty from radon concentratons year-by-year varaton due to dfferences n the weather and the habts of the occupants, and (b) the error from between-subphasevarablty. We defne the perod of tme that a house remans wthout radon-relevant changes as a subphase due to the fact that the radon concentraton durng the measurement dffers from the concentratons before radon-relevant alteratons to the home (Gunby et al., 1993). If the operatonally defned predctor takes nto account homes other than the current home (RNCUM(T) or TWM(T)), there s (c) the error from between-owner-varablty arsng from the dfferent ventlaton habt of the current owners of the study subject s prevous homes, whch leads to condtons n the home durng the measurement dfferent from the condtons durng the resdency of the study subject. If correcton from nformaton on the change of the average radon concentraton by certan house alteratons or ventlaton habt s performed (Gerken etal., 2000), an error from the statstcal uncertanty of estmatng the correcton factor, (3b*) or (3c*), and an error from assgnng a group-matched correcton factor, (3b**) or (3c**), s ntroduced. (A constant multplcatve effect on the radon concentraton s assgned to all houses wth a certan pattern n house characterstcs or ventlaton dfferences.) Regardng stage (4), there s (a) the error from the dfferences n the ventlaton habt dependng on room and daytme. Ths error s due to the fact that the detectors measure average radon concentraton for the full day, but the bedroom s occuped durng the nght and the other rooms durng the day. If the bedroom s ventlated more durng the day than at nght, the measured bedroom concentraton underestmates the concentraton durng the bedroom s occupancy; f a partcpant sleeps wth wndow open and the wndow s closed durng the day, the measured radon concentraton overestmate the concentraton durng the occupancy. Ths nduces a random error, f tcan be assumed that there s no systematc pattern n the day nght cycle of ventlatng the rooms across all study partcpants (that s some partcpants sleep wth wndow open, some wth wndow closed). Further, there s (b) the error from between-envronmentvarablty due to the fact that the radon concentraton n Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5) 369

6 Hed et al. Two dmensons of measurementerror resdental envronments other than the prncpal home are usually notmeasured and assumed to be as hgh, on average, as the prncpal home. (Ths error s lessened n the German studes by ncludng only subjects wth home occupancy of at least 25%.) Note that t s resdental radon exposure that s consdered here, whch does not nclude the exposure at the workplace. There are (c) the error from betweenhome-varablty (for RN 1 ) from usng the radon concentraton n the current home as proxy for the average radon concentraton n all homes nhabted durng the relevanttme perod, (d) the error from false recall of the resdency tme (for (RNCUM(T) or TWM(T)), (e) the error from false recall of the relatve bedroom occupancy, (f) the error from false recall of the absolute house occupancy (for (RNCUM(T)), (g) the error from ms-specfyng the relevant exposure-wndow (for (RNCUM(T) or TWM(T)) due to the fact that a tme perod other than T may be relevant for the lung cancer geness at ndex date, and (h) the error from gnorng the absolute house occupancy (for RN 1 and TWM(T)). Regardng stage (5), there are (a) the uncertanty n determnng the equlbrum factor and (b) the error from between-person-varablty due to the fact that the lung doses of persons wth the same radon and radon progeny exposure vary due to respratory dfferences. Classfcaton of Error Components: Classcal versus Berkson In Table 1, the error components correspondng to each of the fve stages are summarzed ndcatng the dependence on the operatonally defned predctor, the applcablty to the German radon studes and the classfcaton nto Berkson or classcal type. We used dfferent arguments to classfy errors as classcal error or Berkson error: Table 1. Components of error n assessng resdental radon exposure or alpha dose ndcatng the applcablty dependng on the operatonally defned predctor, RN 1, TWM(T), RNCUM(T) byx. Error componenttype Applcable for RN 1 TWM(T) RNCUM(T) (1) Estmatng average radon gas concentraton n home durng exposure of detector (a) Error from between-measurement varablty Classcal x x x (b) Error from between-laboratory-varablty Classcal x (notg) x (notg) x (notg) (c) Error from between-placement-varablty Classcal x x x (d) Error from between-room-varablty Classcal x x x (2) Usng (1) to estmate average radon gas concentraton n home over the year of the measurement (a) Error from between-season-varablty Classcal x (notg) x (notg) x (notg) (a*) Error from ms-specfyng the correcton factor (samplng error) Classcal x x x (a**) Error from applyng correcton factor for seasonal varaton Berkson x x x (3) Usng (2) to estmate radon gas exposure of an ndvdual over a certan perod of years pror to measurement (a) Error from between-year-varablty Classcal x x x (b) Error from between-subphase-varablty Classcal x (notg) x (notg) x (notg) (b*) Error from ms-specfyng the correcton factor (samplng error) Classcal x x x (b**) Error from applyng correcton factor for house phases and Berkson x x x (c) error from between-owner-varablty Classcal F x (notg) x (notg) (c*) Error from ms-specfyng the correcton factor (samplng error) Berkson x x x (c**) error from applyng correcton factor for prevous owners Classcal x x x (4) Usng (3) as operatonally defned predctor for resdental radon exposure (a) Error from dfferences n room and daytme dependent ventlaton habt Classcal x x x (b) Error from between-envronment-varablty Classcal x (less G) x x (less G) (c) Error from between-home-varablty Classcal x F F (d) Error n recall of resdency tme Classcal F x x (e) Error from false recall of relatve occupancy of bedroom Classcal x x x (f) Error from false recall of absolute occupancy of home Classcal F F x (g) Error from ms-specfyng the relevant exposure-wndow Classcal F x x (h) Error from gnorng the absolute occupancy tme of home Berkson x x F (5) Usng (4) as proxy for alpha dose (a) Error n the equlbrum factor Berkson x x x (b) Error from between-person-varablty Berkson x x x In parentheses, not G or less G denote that ths component s not applcable or reduced n the German radon studes, small ndcates that ths error s smaller than the others n the same row. 370 Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5)

7 Two dmensons of measurementerror Hed et al. Classcal I: Repeated observatons, gven all other error components were nonexstent, would yeld dfferent values and vary about the true value: Appled for the components 1a, e, 2a, 3c, 4a, d g. If the radon gas concentraton measurements were repeated under dentcal condtons (1a), f the measurementwas repeated n a newly specfed house (1e), f the tme perod of the detector exposure covered a dfferent perod of the year (2a), f the measurement was repeated n the same house wth agan a dfferent owner (3c), f the measurement was repeated wth dfferentday nghtvaraton n ventlaton of the rooms (4a), f the partcpant was ntervewed agan (4d f), f the observaton was repeated wth a dfferent exposure-wndow (4g), the new observaton would dffer from the orgnal. Classcal II: One measurements used as proxy for the average (Repeated observatons would vary about the average): Appled for the components 1d, 3a, b, 4b, c. The measurementn the lvng room s a proxy for all rooms (1d); the measurement durng 1 year s a proxy for the average over all relevant years (3a); the measurementof the currentsubphase s a proxy for the average over all subphases (3b); the measurement n the current prncpal home s a proxy for the average of all currently occuped homes (4b) or for the average of all prncpal homes nhabted durng the relevant tme perod (4c). ndvduals wth the same changes n ventlaton habt between current house owner and study partcpant. Evdence of Multplcatve Error Structure Informaton on the error from between-measurement-varablty under epdemologcal condtons s provded by the bedroom and lvng room measurements of the German case control studes. lottng the measurements versus ther mean wthn house (Fgure 3) shows the trumpet on the orgnal scale and the tube on the log-scale ndcatng a multplcatve structure of ths error component (compare Fgure 2). The analogous graph of the year-by-year data (Fgure 4) presents a smlar pcture of a rather multplcatve structure of the error from between-year-varablty. However, the smaller number of measurements can clearly be vewed, and a glance at the unt of the axes labellng, 1000 Bq/m 3 nstead of 1 Bq/m 3 n Fgure 3, shows that the radon concentratons encountered n these houses do not reflect the epdemologcal stuaton. The ntercomparson data provdes, agan, nformaton on the error from betweenmeasurement- varablty. Fgure 5 dsplays the orgnal data Classcal III: Uncertanty n the estmaton of a correcton factor (samplng error): Resamplng, that s, the repetton of the observaton wth a dfferent sample of partcpants, would yeld dfferentcorrecton factors. Appled for 2a *,3b *,c *. Berkson I: A group s observaton s assgned to each ndvdual n the group, but the ndvdual s values dffer wthn each group. Appled for 4h, 5a, b. A certan level of RN 1 or TWA(T) sassgnedtoa group of persons regardless of ther absolute home occupancy; a certan level of radon gas exposure s assgned to a group of persons regardless of the specfc equlbrum factor n ther envronment (5a) or of the persons specfc respratory characterstcs (5b), whch may cause dfferng exposure to radon progeny (for 5a) and dfferng lung dose (5a,b). Berkson II: A correcton factor derved for a group of ndvduals wth certan characterstcs n common s assgned to all ndvduals of ths group: Appled for 2a **,3b **,c **. A certan factor s assgned to all ndvduals wth the same seasonal pattern (2a ** ), to all ndvduals wth the same radon-relevant house alteratons (3b ** ), or to all Fgure 3. Bedroom/lvng room measurements: Radon concentratons for controls of the German East study wth both rooms at the same floor versus the mean of the two measurements (a) on the orgnal scale and (b) on the log-scale. Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5) 371

8 Hed et al. Two dmensons of measurementerror houses (houses 3 and 4) hntng at a multplcatve error structure. Error Sze The converson table of several measures of multplcatve error szes, Table 2, shows that the two defntons of the CV yeld smlar results for small errors, but that the dfference ncreases rapdly for errors larger than 0.3. The SD of the log of the error s close to the CV defned as SD dvded by the mean wth the dfference ncreasng, agan, wth error sze. Table 3 relates the SD of the log of the error to the percentage of the error varance compared to the observed radon exposure varance (on the log-scale for multplcatve error). These table entres are data-dependent, that s, for a gven percentage, the correspondng SD depends upon the study data and s here gven for the German West study. (Smlar results are obtaned from the East study.) Table 4 shows the range of radon exposure values that s lkely to be observed gven a certan true radon exposure level and a certan classcal error. For example for an error of 0.4, measurements from 22 to 111 Bq/m 3 can be expected to be observed gven a true radon exposure of 50 Bq/m 3. Table 2. Several measures of sze for multplcatve error. SD of log of error Varance of log of error GSD of error CV as SD/mean (%) CV as SD/GM (%) Fgure 4. Year-by-year data: Radon concentratons ( corrected for between-measurement error by takng the mean of the two measurements n the same year and home) versus the mean of these measurements n one house (a) on the orgnal scale and (b) on the log-scale GSD geometrc standard devaton; GM geometrc mean; and CV the coeffcentof varaton. Table 3. Error sze for German case control studes Error sze as % of observed radon exposure varance SD of log of multplcatve error n log Bq/m 3 SD of addtve error n 100 Bq/m 3 Fgure 5. Intercomparson data: radon concentratons by house. by house. (The analogous graph to Fgures 3 and 4 s not dsplayed, snce t would be un-nformatve due to sparse data.) It shows that the between-measurement-varablty s small, but slghtly larger for the two most hghly exposed Table entres are the SD of the error n resdental radon exposure so that the error varance meets a gven percentage of the radon exposure varance (on log-scale for multplcatve error, on orgnal scale for addtve error) observed among the West study controls (smlar for East). 372 Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5)

9 Two dmensons of measurementerror Hed et al. Table 4. Error sze: range of 95% of the observed radon exposures (on orgnal scale) gven a certan true radon exposure and error sze. Multplcatve error Addtve error True radon exposure Error varance as % n Bq/m 3 of observed radon exposure varance SD of log error n log Bq/m 3 Range of 95% of observed radon exposure varance n Bq/m 3 SD of error n Range of 95% 100 Bq/m 3 of observed radon exposure n Bq/m to to to to The error sze s gven as the SD of the error (on log-scale for multplcatve error, on orgnal scale for addtve error). Further, the error varance sgvenas the percentage of the radon exposure varance observed among the German West study controls (on the log-scale for multplcatve error, on the orgnal scale for addtve error). Table 5. Error sze estmates from replcate data Data ANOVA model Error sze Err(meas) Err(year) Both Intercomparson data a log Z ;j ¼ m þ HOME þ e ;j 0.10 F F Bedroom/lvng room (West) b log Z ;j ¼ m þ HOME þ ROOM j þ e ;j 0.28 F F Bedroom/lvng room (East) b log Z ;j ¼ m þ HOME þ ROOM j þ e ;j 0.33 F F Year-by-year data c,d log Z ;j;k ¼ m þ HOME þ HOME YEAR j þ e ;j;k 0.07 (0.07) 0.58 (0.51) F log Z ;j;k ¼ m þ HOME þ e ;j;k F F 0.55 (0.49) For each data set, the ANOVA model and the estmated error sze (SD of the log of the error) s gven for the error from between-measurement-varablty, Err(meas), for the error from between-year-varablty, Err(year), and for the combnaton of both. a Z,j s the jth radon concentraton measure n the t hhouse(j ¼ 1,..., 5, ¼ 1,..., 11). b Z,j s the jth radon concentraton measure n the t hhouse(j ¼ 1, lvng room; j ¼ 2; bedroom; ¼ 1,..., 4000). c Z,j,k s the kth radon concentraton measure durng the jt hyearntheth house (k ¼ 1,2, j ¼ 1,...5, ¼ 1,..., 11). d The error szes n parentheses are computed wthout house 11. The results of the ANOVA of the replcate data are summarzed n Table 5. It can be seen that the estmated sze of the error from between-measurement-varablty s 0.07 (year-by-year data, ANOVA model (3), 0.10 (Intercomparson data, ANOVA model (1)), 0.28, or 0.33 (bedroom/ lvng room measurements, ANOVA model (2)) dependng on the analysed data set. In the case of the bedroom/lvng room measurements, we need to ascertan that the room dfference s suffcently controlled for. The measured radon concentratons n the bedrooms are overall about 10% (30%) hgher than those n the lvng rooms n the West (East) study. Addng a fxed effect of the floor dfference between the rooms dd not nfluence the error sze estmate, but reduced the effect of the bedroom to 5% (20%) n the West (East) study. Repeatng the applcaton of ANOVA model (2) for the sample reduced to only those ndvduals wth bedroom and lvng room on the same floor yelds smlar results. The sze of the error caused by between-yearvarablty s estmated as 0.58 (year-by-year data, ANOVA model (3)). The estmate of the sze of both error components combned, 0.55, s smaller than the sum of the error szes (year-by-year data, ANOVA model (4)), hntng at a correlaton of the two error components. Includng a fxed effectof the year and graphcal nspecton (notshown) shows no effect(-value ¼ 0.2) of the years and certanly no trend over the years. The results wthout house 11, the most nfluental, reported n parentheses n Table 5, show smaller estmates. Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5) 373

10 Hed et al. Two dmensons of measurementerror Dscusson The mportance of hgh-qualty exposure assessment and the need to mnmze errors n the exposure s well acknowledged n epdemology. Whereas the statstcal methodology for dealng wth such errors n the estmaton of dsease rsk has been around for already 20 years (Carroll etal., 1984; Rosner et al., 1989), the ncreasng awareness of ther practcal mplcatons n epdemology called for a renassance of measurementerror (Mchels, 2001). Frequently, the dscussons of publshed epdemologcal studes nclude a note-of-cauton regardng the nvolvement of measurement error n the exposures hntng at a potental attenuaton of rsk estmates. However, there s often a lack of clear dfferentaton between classcal and Berkson error despte ther greatly dfferent mpact. Fgure 6 shows several theoretcally observed dose response curves. The x-axs shows the exposure F here mmckng the German radon study stuaton (range of Bq/m 3 ); however, ths quantty can be any other epdemologcal contnuous exposure. The y-axs shows the RR that would be observed, f the exposure was measured wth a certan error and the RR was not corrected for t. Each curve shows the ncrease of the RR under the logstc model for ncreasng exposure assumng varous error models. The four curves correspond to four dfferent error models, the addtve classcal error, the multplcatve classcal error, the addtve Berkson error, or the multplcatve Berkson error. The curve labelled none ¼ add Berk shows the true dose response curve wth an RR of 1.12 per 100 unts of the exposure, that s, the RR under a normal logstc model wthout errors n the exposure. The curves are drawn based on the expected exposure gven the observed exposure and gven a certan error model (followng the reasonng of the regresson calbraton method). Ths fgure clearly ndcates that classcal error attenuates the dose response curve, n the case of multplcatve error even nducng a spurous curvature, that addtve Berkson error has no effect Fgure 6. Theoretcally observed dose response when the dose s measured wth addtve (SD ¼ 50) or multplcatve (SD on logscale ¼ 0.4) error of the classcal or Berkson type and a true RR of 1.12 (see none ). on the rsk estmate and that multplcatve Berkson error, f any, slghtly ntensfes the dose response relatonshp. However, the fact that the mpact of the Berkson error on the rsk estmate (pont estmate) s neglgble does not mean that the Berkson error can be gnored, snce the estmate s precson may suffer severely. In the statstcal lterature, new error correcton methodology for error models combnng both classcal- and Berksontype have recently been developed and appled (Reeves et al., 1998; Schafer etal., 2001; Hed, 2002). For the correct applcaton of these methods, the foremost prerequste s the metculous dentfcaton of all sources of error n the exposure assessment, ther correct classfcaton (classcal versus Berkson) and collecton of nformaton on error structure (addtve versus multplcatve) and sze, whch we provded for resdental radon exposure assessment va ar measurements wth partcular reference to the German lung cancer studes. Random error n the exposure assessment from the physcal process of measurng radon gas concentratons were prevously studed n great detal (Wrxon et al., 1988; Hardcastle and Mles, 1996), but for the errors n the epdemologcal settng, only a very crude lst of such errors was descrbed so far (Ba verstam and Swedjemark, 1991, Lubn et al., 1995). A detaled revaldaton s of mmedate nterest n the lght of the on-gong mplementaton of a new assessmentprocedure of resdental radon exposure by measurng polonum n glass objects (e.g. Lagarde et al., 2002). Further, we showed the usefulness of the concepts predctor of nterest and operatonally defned predctor, ther mpact on the applcablty of error components, on sze and predomnant type of the error. We found that external resdental radon exposure as lung cancer predctor nvolves almostno Berkson error component, whereas the predctor lung dose ntroduces a Berkson error. Further, t became clear that, however the choce of the operatonally defned predctor, there s a trade-off to be made: The TWM(T) radon concentraton n all relevant homes, rather than the concentraton n the current home, s closer to the predctor of nterest, but more dffcult to measure. Next to the dfferentaton between classcal- and Berksontype error, t s the error sze that s generally very nfluental on the mpact of the measurement error on rsk estmaton. The computaton formulas (Methods Secton) and the converson table (Table 2) should gude the reader through the varous measures of error sze. Addtonally, the sze of the error referred to as percentage of the error varance compared to the observed exposure varance (Table 3) s partcularly valuable to put the error sze n the rght perspectve when consderng real data. For classcal error, ths percentage descrbes the proporton of the observed radon exposure varance that s explaned by the error and whch would dsappear, f the varable was measured wthout error. For Berkson error, t s the proporton, by whch the 374 Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5)

11 Two dmensons of measurementerror Hed et al. true exposure varance exceeds the observed exposure varance and by whch the observed exposure varance would ncrease, f the error was elmnated. For example, a classcal multplcatve error wth SD (on the log-scale) of 0.4, whch was suggested as reasonable for the German studes (Hed., 2002), explans 50% of the observed radon exposure varance (on the log-scale); a Berkson error of ths sze would ndcate that the true exposure varance s 1.5 tmes as large as the one observed. For a classcal error of 100%, all the observed exposure varance would be due to error, the true exposure varance would be zero, whch s the reason for the classcal error notexceedng 100%. The 300% Berkson error ndcates that the true exposure varance trebles the observed exposure varance. Ths proporton s thus an ndcator for what the error does to the exposure varance, but even more, for classcal error, ts a measure for the mpactof the error on rsk estmates across data sets. For example, a multplcatve classcal error wth SD (on the log-scale) of 0.48, as estmated for the Englsh radon study (Darby et al., 1998), explans about20% of the observed radon exposure varance n ths study, but would explan over 65% n the German studes. The mpact of such an error s, hence, more severe n the German studes. Note that the Berkson error s mpact does not depend on the exposure varance and s thus the same across data sets assumng the same underlyng error sze and rsk (Hed etal., 2002), whch s of nterestn metaanalyses. A real example of the error sze of two error components, the errors from between-measurement-varablty and from between-year-varablty, s gven by three data sets wth repeated measurements: The sze of the error from betweenmeasurement-varablty s 0.07 (year-by-year data), 0.10 (ntercomparson data), and 0.3 (bedroom/lvng measurements) dependng on the analysed data set. These dfferences ndcate the mportance of havng a crtcal look at the replcate data. In the year-by-year data, the houses wth very hgh radon concentratons were not representatve for the epdemologcal stuaton. The ntercomparson data, whle overcomng ths problem, are rather sparse and the laboratory personnel were aware that ther results were beng evaluated. The value of 0.1 obtaned n a controlled exercse wthalmt numberofmeasurementscanthusbevewedasa mnmum error sze for the epdemologcal studes, where over 10,000 measurements were conducted by ths laboratory. The bedroom/lvng room measurements have the advantage of beng avalable for nearly all study partcpants and of beng conducted under epdemologcal feld condtons. However, we showed that there are dfferences between the radon concentratons n the rooms, whch were beyond that due to dfferent floor levels and whch we mght not have been able to completely correct for n the error sze estmaton. The error from between-year-varablty, estmated as 0.58 (year-by-year data) s qute large compared to the between-measurement-error. However, agan, the fact that these data are unrepresentatve calls for cauton. It should be noted that the estmated error varance of the combnaton of both between-year-varablty and betweenmeasurement-varablty s smaller than the sum of both, whch may well be due to correlatons between the components. It s therefore of no use to estmate each error component s sze separately and to take ther sum as the total error sze. The multplcatve error model n the assessment of resdental radon exposure s already establshed (Gunby etal., 1993; Lagarde etal, 1997; Darby etal., 1998). We fnd that the replcate measurements provde further evdence of the multplcatvty by graphcally vewng the mean versus sngle measurements (Fgures 3 and 4). Conclusons We conclude that, generally n epdemology, clear dfferenton between classcal and Berkson error components s essental n the assessment of error sources and for establshng an error model, a factwhch we beleve s notfully acknowledged. Ths dfferentaton s crucal due to the dfferent mpact of these two error types. The classcal error can nduce severe bas on the rsk estmate; multplcatve classcal error can even dstort the dose response curve. Ths bas can be reduced by usng the mean of multple measurements n the analyss requrng nternal replcates for each ndvdual, or tcan be corrected for by usng the nformaton from (nternal or external) replcate measurements for a subgroup. Also, the spurously narrow confdence ntervals for the rsk estmates n the presence of classcal error n the exposure, whch are yelded wthout error correcton, can be corrected. The analyss of our replcate data, ther usefulness, and the struggle wth ther lmtatons motvate our recommendaton for more nternal repeated measurements n future epdemologcal studes (e.g., n radon studes, more than one detector per room and repeated measurements over a seres of years). At frst glance, the Berkson error s less problematc, snce t does not nduce notable bas on the rsk estmates. However, t weakens the precson of the estmates, whch s often more dffcult to correct for than n the classcal stuaton due to the problem of graspng the extent of the Berkson error. For example, the lung dose s hard to measure and such a measurementwould rentroduce classcal error. Justreplcatng measurements does not help n the Berkson case. Smplfed, classcal error s rather related to the measurement process, whereas Berkson error s often a matter of defnng the exposure: Usng fxed montors (e.g. usng the dstance of a home to the next power staton as predctor nstead of ndvdual measurements), usng measurements n the envronment(e.g. resdental radon exposure nstead of lung dose), or usng a person s afflaton to a group n order to use the Journal of Exposure Analyss and Envronmental Epdemology (2004) 14(5) 375

12 Hed et al. Two dmensons of measurementerror exposure assgned to ths group (e.g. usng job-exposure matrces) nstead of personal montors s a queston of how to defne the exposure; t nduces Berkson error. The general statement that well-behaved (random, nondfferental, homoscedastc) errors attenuate regresson coeffcents apples only to the classcal error. It should be kept n mnd, that consderng more precse (or more relevant) exposures and thus nducng more potental sources of error does not necessarly ncrease the bas of the rsk estmates (compare to Lubn et al., 1995). For example, extendng the defnton of the predctor from resdental radon exposure to lung dose nduces Berkson error and does not attenuate the rsk estmate. To assume the sum of both error type s szes as known and to vary the percentage of the Berkson error s no opton n such stuatons (see Mallck et al., 2002). 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