Overlap Bias in the Case-Crossover Design, With Application to Air Pollution Exposures

Transcription

1 U Boac orkng Papr Sr Ovrlap Ba n h Ca-Croovr Dgn, h Applcaon o Ar Polluon Expour Holly Jan Unvry of ahngon, hjan@u.wahngon.du Lann Shppard Unvry of ahngon, hppard@u.wahngon.du homa Lumly Unvry of ahngon, lumly@u.wahngon.du Suggd Caon Jan, Holly; Shppard, Lann; and Lumly, homa, "Ovrlap Ba n h Ca-Croovr Dgn, h Applcaon o Ar Polluon Expour" January 2004). U Boac orkng Papr Sr. orkng Papr 23. hp://boa.bpr.com/uwboa/papr23 h workng papr hod by h Brkly Elcronc Pr bpr) and may no b commrcally rproducd whou h prmon of h copyrgh holdr. Copyrgh 20 by h auhor

2 I. Inroducon h ca-croovr dgn wa propod a a mhod for mang h aocaon bwn a hor-rm xpour and h rk of a rar vn. Only ca ar rqurd for uch a udy. For ach ca, xpour a h ndx m h haard or a-rk prod pror o h vn) compard o xpour a comparabl conrol, or rfrn m. h dgn can b vwd a a hybrd of h ca-conrol and h croovr dgn: lcon bad on h oucom, mlar o h ca-conrol dgn, bu h ca rv a h/hr own conrol, a n a croovr udy. By makng whn-pron comparon, confoundng du o m-ndpndn facor lmnad and, wh propr lcon of rfrn, m-dpndn confoundr can alo b conrolld. h ca-croovr dgn ha bn wdly ud o udy h ffc of ar polluon xpour on h rk of an advr halh vn. A vary of dffrn rfrn lcon rag hav bn ud n mplmnng h dgn, for xampl 2-8 ). Howvr, h choc of a rfrn chm of parcular mporanc wh ar polluon xpour daa. Rprnav rfrn ar mor dffcul o dfn n an xpour r wh ubanal rucur; ar polluon ofn ha a long rm m rnd, and var wh aon and day of h wk. h facor ar alo ofn aocad wh h oucom of nr, and hnc confoundng a major u. In addon, h dgn rl on h aumpon of a conan drbuon of xpour acro rfrn m. 6;9 Y h choc of rfrn lcon ragy mporan for a mor fundamnal raon. Lumly and Lvy 5 howd ha cran rfrn chm nduc ba n h mang quaon ha ar ypcally ud. call h ba n h mang quaon ovrlap ba ; a vron of h ba nducd from choong non-djon raa o paron h populaon n a machd caconrol udy. 5;0; bgn h papr wh a dcrpon of h ca-croovr dgn, Hod by h Brkly Elcronc Pr

3 ncludng maon and rfrn lcon mhod. dcu h ba aocad wh varaon n h xpour ovr m and confoundng undr cran rfrn lcon rag, and focu n parcular on condraon mporan n choong h rfrn lcon ragy wh ar polluon xpour r. In Scon 4, w gv a drvaon of ovrlap ba, and dcrb h uaon n whch a concrn, a wll a how can b avodd. In Scon 5, w calcula h magnud of h ba for a gvn xpour r, and xplor how h ba dpnd on propr of h xpour r n h abnc of confoundng. conclud wh a dcuon of h mplcaon of h rul on h u of h ca-croovr dgn. 2. Ovrvw of h Mhod h ca-croovr dgn compar xpour a ndx m, whn h xpour rggrng an vn may occur, wh xpour a rfrn m, whn an vn no rggrd. h dgn uabl for ang h rk of an acu-on vn durng a brf prod followng a rann xpour, and for xpour ha ar hough no o hav carryovr ffc. I man advanag ar h fac ha ha achv conrol ovr fxd m-nvaran) confoundr, and, wh lcon of uabl rfrn, conrol for om ffc of m-varyng xpour by dgn, rahr han by acal modlng. Howvr, hr ar many challng n mplmnng h dgn, ncludng, bu no lmd o, drmnaon of h duraon and mng of h haard prod pror o h vn, dfnng h xpour, and lcng rfrn xpour ha ar rprnav of h undrlyng drbuon of xpour. h undrlyng modl for h ca-croovr dgn ypcally h proporonal haard modl for a rar da wh a conan baln haard for ach ndvdual. h modl for h haard 2 hp://boa.bpr.com/uwboa/papr23

4 ra of pron a m gvn m-varyng covara x λ x ; ) λ xp x) 6 for a logc rgron formulaon applcabl o mor common oucom). Ovr a hor m prod, h conan baln haard aumpon λ ) ofn raonabl, and quvaln o aumng mooh aonal ffc n a m r analy. Aumng an appropra rfrn amplng chm, an unbad ma h ffc of hor-rm xpour can b oband ung condonal logc rgron. h mlar o h analy of machd ca-conrol ud; hr, h xpour a ach ndx m par of a machd of xpour conng of xpour for h am ubjc a h/hr rfrn m. Condr h uaon n whch w hav a hard xpour r,, dfnd a m,2,, common o all,2,,n ubjc. Dno h ndx m for ubjc by, h xpour a h ndx m by, and l rprn h rfrn wndow for ubjc ncludng h ndx m and all rfrn m). h condonal logc rgron mang quaon ar n n U ). ) For ach ubjc, h mang quaon h dffrnc bwn xpour a h ndx m and a wghd avrag of xpour a all m n h rfrn wndow; hnc, h ma of oband rprn h chang n h rk of an vn aocad wh a hor-rm un ncra n xpour. wll dcu h u of h condonal logc rgron mang quaon n Scon 4; only for om rfrn lcon rag ha h quaon hav man ro. 3 Hod by h Brkly Elcronc Pr

5 ypcally, xpour daa for ar polluon ud com from cnrally locad monor of ambn polluon, and hu xpour common o all ndvdual. Conqunly, rarch on h ca-croovr dgn n h ar polluon xpour conx hould focu on propr of h ffc ma condonal on a known and hard xpour r. wan o know ha h ma and h corrpondng mang quaon ar unbad for a gvn xpour r; knowng ha hy ar unbad whn avragd ovr all pobl xpour r of ll valu whn hr a ngl xpour. In h papr, w rrc our anon o modl ha condon on xpour. 3. Rfrn Slcon Srag In a ypcal ar polluon m r udy, daly vn coun.g. moraly or hopal admon) ar rgrd on h hard xpour r ung a Poon rgron modl 2 for a rvw of h approach). Srong confoundng ffc of aon and wahr mu b conrolld acally n h modl. I nrucv o no ha h ca-croovr dgn quvaln o a Poon rgron analy xcp ha confoundng conrolld for by dgn by machng) nad of n h rgron modl. 5 Rrcng rfrn o h am day of wk and aon a h ndx m conrol for h confoundng ffc by dgn. Dnc from confoundng anohr concrn rgardng m rnd n h xpour r. If hr a long-rm m rnd, choong rfrn only pror o h ndx day may lad o ba. Navd 6 propod ha h m rnd ba could b lmnad by choong rfrn boh bfor and afr h ndx m, a ragy calld bdrconal rfrn lcon. chncally, bdrconal amplng only vald whn ca ar ll a rk afr h vn, an aumpon ha 4 hp://boa.bpr.com/uwboa/papr23

6 cranly volad whn h vn dah. Howvr, Lumly and Lvy 5 howd ha, wh a rar vn, h ba du o amplng rfrn afr h a-rk prod vry mall. Mo mporanly, h ba from amplng rfrn oud of h m a rk mallr han h ba aocad wh h alrnav of only amplng rfrn pror o h ndx m n h prnc of a m rnd. Bcau of concrn ovr xpour rnd, bdrconal rfrn lcon rag hav bn almo unvrally adopd among rarchr prformng ca-croovr analy of ar polluon daa. Fgur dplay an array of pobl rfrn amplng rag ud n ar polluon ud. Fgur a how h oal hory undrconal rfrn ampl, n whch rfrn ar all day n h xpour r pror o h ndx day. Bcau rfrn can b long bfor h ndx day, h dgn wll b ubjc o ba du o non-aonary of h xpour, vn n h abnc of confoundng by m rnd, aon, and/or day of h wk. h ragy could b mprovd by rrcng rfrn o b clor o h ndx m, uch a no mor han 30 day pror o h ndx m. Fgur b hrough d gv xampl of rrcd undrconal rfrn amplng dgn. All of h approach fx h locaon of h rfrn o b pror o and rlav o h ndx m. Fgur d how rfrn placd 7, 4, and 2 day pror o h ndx day n ordr o conrol for day-of-wk confoundng. Fgur hrough g how xampl of bdrconal rfrn amplng. Fgur how h full raum bdrconal rfrn ampl 6 whr h rfrn ar all day n h xpour r ohr han h ndx day. h h dgn, aon mu b modld or h ar polluon ffc ma wll b confoundd. A popular alrnav o modlng rrcon n m ung 5 Hod by h Brkly Elcronc Pr

7 ymmrc bdrconal amplng. Rfrn ar lcd a fxd nrval bfor and afr h ndx m.g. 2;3;8;3 for applcaon). h rfrn lcon ragy conrol for ba du o m rnd, aon, and day of h wk, f lag ar mulpl of vn. 4 Fgur f how h ymmrc bdrconal dgn wh wo rfrn 7 day bfor and afr h ndx m, and panl g how a ragy wh four rfrn 7 and 4 day bfor and afr h ndx m. h man problm w focu on wh rpc o h rfrn chm n Fgur a-d, and f-g ovrlap ba. h dgn ar all xampl of non-localabl rfrn wndow rag. h h dgn, an unbad mang quaon ha rrcd o h rfrn wndow do no x. hu, h condonal logc rgron mang quaon, whch ar rrcd o xpour whn h rfrn wndow, ar bad. call h ba ovrlap ba, and dcu magnud n Scon 5. In conra, w dfn localabl rfrn dgn a ho for whch hr x an unbad mang quaon rrcd o h rfrn wndow. Localably a drabl propry nc allow u o oban unbad ma by makng comparon whn rfrn wndow. Snc h ar uually rlavly hor prod of m, w ar abl o conrol for confoundng n h way. h m rafd 5 and h m-ymmrc bdrconal 6 dgn ar boh localabl dgn. h m rafd rfrn ragy dvd m no djon raa, u h ndx m o drmn whch raum lcd for a gvn ca, and lc all or a ampl of h rmanng m n h raum a rfrn. h ragy manan conrol ovr confoundr by dgn 6 hp://boa.bpr.com/uwboa/papr23

8 nc day whn ach raum ar machd on mporan confoundr. A common rafcaon n h ar polluon ng o lc rfrn ha fall on h am day of h wk and n h am monh and yar a h ndx day. h conrol for confoundng du o aon and day of h wk. In addon, hr no ba du o m rnd nc hr no parn n h placmn of rfrn rlav o h ndx m. h full raum bdrconal rfrn ragy a pcal ca of h m rafd dgn whr hr only on raum; wh h rafcaon, howvr, confoundr uch a aon canno b aumd o b conan whn raum. h m-ymmrc bdrconal rfrn lcon, on rfrn randomly chon from h par of day a fxd lag pr- and po-vn; f only on avalabl.g. a h bgnnng and nd of a m r), rv a h rfrn. Confoundng conrolld by dgn whn an appropra lag lcd. h m-ymmrc bdrconal dgn no a m rafd dgn, bu a lgh modfcaon of. call h h adjud m-ymmrc bdrconal rfrn ragy. For vn a h bgnnng or nd of h m r, h rfrn ragy ll randomly lc from h par of day pr- and po-vn. Howvr, whn h rfrn chon oud h r, h ca xcludd. h modfd dgn can alo b hough of a randomly choong bwn wo ovrlappng of djon raa, whr ach bad on a dffrn m rafcaon. For ach day xcp ho a h nd of h r) hr ar wo pobl raa.g. f h lag on day, a ca on day wo ha day on and wo a on pobl raum, and day wo and hr a anohr pobl raum). On raum randomly lcd for ach ca, and ca ha fall oud h lcd raum ar xcludd. 7 Hod by h Brkly Elcronc Pr

9 hough h m rafd and m-ymmrc bdrconal dgn ar boh localabl, hy dffr n on mporan rpc. furhr paron h localabl dgn no wo group calld gnorabl and non-gnorabl dgn o hghlgh h dffrnc. h an gnorabl dgn, uch a h m rafd dgn, h rfrn amplng chm can b gnord n conducng h analy, and h condonal logc rgron mang quaon ar unbad. Howvr, wh a non-gnorabl dgn, uch a h m-ymmrc bdrconal dgn, h rfrn amplng chm canno b gnord. h lklhood of h daa dpnd on h amplng chm, and mu b ud for an unbad analy. Alrnavly, condonal logc rgron could b ud f an appropra off wr pcfd. S Scon 4 for a drvaon of h lklhood of h daa undr h dffrn yp of rfrn rag.) hu, h dncon bwn gnorabl and non-gnorabl dgn rfr o whhr andard condonal logc rgron can b ud for an unbad analy. 4. Ovrlap Ba h u of condonal logc rgron for analy of h ca-croovr dgn ha bn movad by h analogy o machd ca-conrol dgn, whr condonal logc rgron maxm h ru condonal) log-lklhood of h daa. h ffc ma,, found by ng h condonal logc rgron cor quaon ) qual o ro. h localabl and gnorabl dgn, uch a h m rafd dgn, quaon ) h drvav of h ru condonal log-lklhood. drv h lklhood a follow. Snc h ndx m ar random condonal on h rfrn wndow, w can u h lklhood of h ndx m condonal on h rfrn wndow o ma. Condr agan h uaon n whch w 8 hp://boa.bpr.com/uwboa/papr23

10 hav an xpour r,, dfnd a m,2,, common o all,2,,n ubjc. Dno h ndx m for ubjc by, h xpour a h ndx m by, and l rprn h rfrn wndow for ubjc ncludng h ndx m and all rfrn m). L Y b an ndcaor of whhr ubjc ' ndx m wa on day. aum ha vn follow a proporonal haard modl n whch h lklhood of an vn occurrng a m for ubjc λ. condon on h xpour r, h rfrn wndow, and aum ha vn ar rar. no ha ubjc whou vn do no conrbu nformaon. h lklhood P n, Y, ) n P,, Y ) P, Y, ) n λ n λ. 2) h λ cancl, o w can u ca only o ma, and nd no ma λ. no ha h lklhood dpnd only on xpour a m whn h rfrn wndow. In addon, w fnd ha h drvav of h logarhm of 2) xacly h condonal logc rgron mang quaon ), and hu h condonal logc rgron mang quaon hav man ro f localabl, gnorabl rfrn ar ud. If, howvr, non-localabl rfrn wndow ar ud, h condonal logc rgron mang quaon ar no h drvav of h log-lklhood of h daa. In ordr o h, 9 Hod by h Brkly Elcronc Pr

11 w fr no ha h non-localabl dgn ar xacly ho for whch h locaon of h ndx m whn h rfrn wndow ar fxd.., wh ymmrc bdrconal rfrn, h ndx m h m n h cnr of h rfrn wndow). Hnc, wh non-localabl rfrn, h lklhood of h ndx m condonal on h rfrn wndow n P P,, ) n Y n,, Y ) P, Y, ) I[ ], whr I f A ru, and ro ohrw. h lklhood unnformav, nc knowng a [ A] rfrn wndow gv h aocad ndx m xacly. h non-localabl dgn, h random varabl ar acually h rfrn wndow, whch fall around ach ndx m wh probably λ. h appropra lklhood o u ha of h rfrn wndow, uncondonal on h ndx m. Agan, w condon on hr bng on vn for ach ca. h margnal lklhood of h rfrn wndow P n n w, P w, Y ) Y ) P w, Y ) w n n λ w λ w. 3) No ha h lklhood dpnd on xpour a all m n h xpour r. hu, h drvav of 3) no h condonal logc rgron mang quaon ), whch dpnd 0 hp://boa.bpr.com/uwboa/papr23

12 only on xpour whn h rfrn wndow. I can b hown ha ) do no hav man ro for non-localabl rfrn wndow amplng chm Scon 5). Hnc, hr ovrlap ba aocad wh h u of h condonal logc rgron mang quaon undr non-localabl rfrn lcon. wan o mpha h dffrnc n h lklhood for h non-localabl and localabl, gnorabl rfrn chm. No ha hr ar wo random varabl n ach rfrn chm: h ndx m,, and h rfrn wndow,. h a non-localabl rfrn wndow chm, and ar mpl funcon of ach ohr, and hnc h lklhood of on condonal on h ohr unnformav. hu, w u h margnal lklhood of h rfrn wndow o ma. A margnal lklhood could alo b calculad for h localabl, gnorabl dgn nad of h condonal lklhood), bu would no yld a uful ma of. h ma would b nrly confoundd, du o varabl uch a aon whch ar aocad boh wh ndx m and hnc h rfrn wndow lcd) and wh xpour. alo xamn h lklhood of h daa undr h m-ymmrc bdrconal dgn, an xampl of a localabl, bu non-gnorabl dgn. h h ragy, h rfrn wndow for a ubjc wh an ndx m a { δ, } wh probably 0.5, and {, + δ } wh probably 0.5, for om lag δ. 6 can agan u h condonal lklhood o ma. h condonal lklhood can b wrn a n P,, Y ) n P P w w,, Y ) P,,, ) P, Y Y Y ) ) Hod by h Brkly Elcronc Pr

13 2 n Y P Y w P Y P Y w P ), ),, ), ),, n n }, {, ) ) ) }, {, ) ) ) δ λ δ π λ π λ π δ λ δ π λ π λ π δ δ 4) whr ),, ) Y w P π. no ha h lklhood dpnd only on m whn h rfrn wndow, bu ha h form of h lklhood dpnd on h rfrn amplng chm. h m-ymmrc bdrconal rfrn lcon, ) 0.5 j π f j n h mddl of h xpour r, bu ndx m a h bgnnng or nd of h xpour r hav only on pobl rfrn wndow. If ) ) k j π π for all k j,, h lklhood would rduc o h condonal logc rgron lklhood 2).) h lklhood, 4), mu b ud n ordr o oban an unbad ma of. Y, apparn from quaon 4) ha f w u an off rm of log2 for day a h bgnnng and nd of h r n a condonal logc rgron analy, w wll g h am ma of a f w had ud 4) o ma, nc 0 log2)+ 5.. A fnal obrvaon concrn h adjud m-ymmrc bdrconal dgn, a localabl and gnorabl dgn. h h ragy, f w randomly lc a rfrn wndow a h bgnnng or nd of h r ha don x, w drop h ca from h analy. h quvaln o wghng ca a h bgnnng and nd of h r by 0.5, whch would nur ha h wgh π) n 4) cancl. hu, wh adjud m-ymmrc bdrconal rfrn lcon, hp://boa.bpr.com/uwboa/papr23

14 3 h lklhood of h daa xacly h condonal logc rgron lklhood, and ) can b ud o oban an unbad ma of. 5. Magnud of h Ovrlap Ba Lumly and Lvy 5 wr h fr o pon ou h xnc of ovrlap ba wh non-localabl rfrn wndow lcon. h boh non-localabl and localabl bu non-gnorabl dgn, h condonal logc rgron mang quaon do no hav man ro. h xpcd valu of h mang quaon for on ubjc, rgardl of h rfrn lcon ragy, v u u v u U E )). 5) h xpcaon h um ovr all pobl m of h mang quaon wghd by h margnal drbuon of ndx m, Y P / ),. No hr ha w do no wan o condon on h rfrn wndow, nc w wan o compu h xpcaon ovr all ndx m. If w rwr 5) wh and, w oban ) U E ) )) ν, whr / ν ar wgh, h rfrn wndow o whch blong, and ) / u v u u v a rfrn wndow-wghd avrag of xpour. Hod by h Brkly Elcronc Pr

15 For h m rafd dgn, w can rwr h um ovr all m a a um ovr raa,, S and m whn raa, nc m parond no a compl of djon raa. hu, w can wr 5) a E U S u )) u 0, l u u l u nc h rm n parnh ro. h ohr rfrn lcon rag, m no parond no djon, and hr no mplfcaon ha rduc h xpcaon o ro. Howvr, xpron 5) gv u h magnud of h ovrlap ba n h condonal logc rgron mang quaon undr any rfrn ragy for a gvn xpour r. Examnaon of h xpron rval ha, whn 0, ba occur only f h rfrn ragy dffr among m n h xpour r. h occur, for xampl, n h ymmrc bdrconal dgn, n whch ca occurrng a h bgnnng of h xpour r do no hav pr-vn rfrn, and ca a h nd do no hav po-vn rfrn.) Baon and Schwar 5 ugg ubracng off h ba, bu h only accoun for h ovrlap ba a 0. alo no ha an unbad mang quaon could b oband by ubracng off h ba, 5), from h condonal logc rgron mang quaon, ). Howvr, h unbad mang quaon no local; dpnd on xpour a all m n h xpour r. would lk o xamn h ovrlap ba on h cal, rahr han h mang quaon cal, o w u h approxmaon 4 hp://boa.bpr.com/uwboa/papr23

16 E )) ˆ U. 6) δ E U )) δ h dnomnaor can b found by dffrnang ) wh rpc o and akng h xpcaon ovr all pobl m a n 5). u h approxmaon n 6) o how, for a gvn xpour r, h larg ampl ba n h ffc ma, ˆ. In ordr o xplor ba n h ma of undr varou rfrn lcon rag, w mulad hr dffrn yp of xpour r. h alo allow u o drmn how h ba dpnd on propr of h xpour r. No ha, bcau w condr h xpour r o b fxd rahr han random, propr of h xpour r, uch a auocorrlaon, hould b hough of a mahmacal dcrpon of h r, rahr han a paramr of h drbuon of xpour. mulad xpour o mmc PM 0 parcula mar l han 0 µm n damr) n Sal ovr a on yar prod. hr lognormal xpour wr gnrad, wh. No mporal rucur Fgur 2); 2. Sral corrlaon on adjacn day ρ 0.6; Fgur 3); and 3. Sral corrlaon and a m-dpndn man and varanc Fgur 4 and 5). All lognormal xpour had a man of 3.6 and a varanc of 0.2. Man and varanc rucur addd ncludd a dcrang mporal rnd, day of h wk ffc, aonaly modld wh n and con rm wh prod of 2,,,,and of a yar), and aonal varanc modld wh n and con rm wh prod of and of a yar). Lognormal xpour wr hn 2 xponnad o yld mulad xpour,. 5 Hod by h Brkly Elcronc Pr

17 how, n Fgur 2, 3, 4, and 5, h larg ampl ba for hr randomly chon ralaon of ach yp of xpour r a a funcon of. For ach yp of xpour, h am hr r ar hown acro all rfrn rag. h ba hown for ach of h rfrn wndow rag n Fgur, a wll a for h m rafd and m-ymmrc bdrconal dgn. don how h ba for h full raum bdrconal dgn, howvr, nc h a pcal ca of h m rafd approach, and hnc h ba ro for all. No ha, wh h m-ymmrc bdrconal dgn, w how h ba avragd ovr all ralaon of h rfrn ragy; for a gvn ralaon, h ba would b largr. For h r wh mvaryng man and varanc, w how h ba on wo dffrn cal: a largr wndow n whch var from -0.2 o 0.2, and a mallr wndow n whch var from -0.0 o 0.0 Fgur 4 and 5), whl for h unrucurd Fgur 2) and auocorrlad Fgur 3) r w only how ba on h mallr cal. h mallr cal how ffc ma on h ordr of ho ypcal n ar polluon ud, nc a of corrpond o a rlav rk of. for on nrquarl rang chang n xpour whr IQR 20). I hould b mphad ha h ba hown n Fgur 2, 3, 4 and 5 ovrlap ba only. h fgur do no nclud ba du o confoundng, or addr h u of whch rfrn rag b dal wh h problm. Fgur 2, 3, 4, and 5 ugg a fw pon rgardng ovrlap ba. A xpcd, hr no ovrlap ba for h m rafd and full raum bdrconal dgn. In om ca, hr ubanal ba for h undrconal dgn, bu h ba wh h varou ymmrc bdrconal dgn mall. h ba n h ffc ma gnrally mall, pcally for mall, bu nd o ncra wh mor rucur n h xpour r. Howvr, h rul dpnd on h parcular ralaon of h xpour r. For om xpour, uch a h hr 6 hp://boa.bpr.com/uwboa/papr23

18 hown wh m varyng man and varanc, hr ubanal ba n h ffc ma wh undrconal rfrn, vn for mall. I pobl ha, for om xpour, h ba may b largr han lf. In addon, for om xpour, hr may b ba vn whn 0. Fnally, for om xpour, hr ba apparn wh cran rfrn lcon rag, bu no wh ohr. Hnc, clar ha h magnud and drcon of ovrlap ba canno b prdcd bfor h xpour known. I m logcal ha on hould choo a rfrn lcon ragy ha avod h ba nrly. 6. Dcuon and Concluon h ca-croovr dgn wll ud o h udy of h acu halh ffc of ar polluon, bu propr lcon of rfrn parcularly mporan. h conrol xpour mu b rprnav of h xpour ha gnrad h oucom, and alo mu b chon o conrol for confoundng or confoundng ffc mu b ohrw ncludd n h modl). h pon w hav mphad hr, howvr, ha an analy mhod mu b pard wh an appropra rfrn lcon ragy n ordr o nur unbad mang quaon. hav prnd a nw axonomy of rfrn lcon rag. Mo commonly ud rfrn dgn.g. h ymmrc bdrconal dgn) ar non-localabl, and ar ubjc o ovrlap ba. hn h localabl dgn ar h non-gnorabl dgn, uch a h mymmrc bdrconal dgn. h rag rqur analy ung h ru lklhood of h daa or pcfyng an appropra off n h condonal logc rgron analy). If condonal logc rgron mang quaon ar ud whou an off), ovrlap ba wll 7 Hod by h Brkly Elcronc Pr

19 b ncorporad. h localabl, gnorabl dgn, uch a h m rafd dgn, hr no ovrlap ba aocad wh a andard condonal logc rgron analy. Our ud of ovrlap ba ndca ha uually mall, a wa uggd by Lumly and Lvy. 5 Howvr, w hav alo n ha h ba qu unprdcabl n magnud and drcon; dpnd on h parcular hard xpour r. h ba wll b xacrbad by modl hoppng on rfrn r, nc h magnud and drcon of h ba dpnd on h parcular ralaon of h xpour r. hav hown ha pobl o hav ba vn for mall, and, ndd, vn whn 0. hl pobl o ma h xpcd ba a a funcon of for any pcfc xpour r, vry dffcul o prdc n advanc h xn of h problm. hrfor, m mo prudn for rarchr o choo a rfrn lcon ragy ha avod ovrlap ba nrly. wo of h man rngh of h ca-croovr dgn n h ar polluon conx ar ha conrol fxd confoundr and allow for conrol ovr m-dpndn confoundr by dgn. Choong rfrn ha ar rrcd o h am day of h wk and aon a h ndx m wll conrol for h ffc. hough w hav no xamnd h dgr o whch varou rfrn lcon rag conrol ba du o confoundng, vral pon ar clar. Rfrn ha ar clor o h ndx m wll achv grar conrol of aonal confoundng. Howvr, amplng rfrn clor o h ndx m wll alo rul n a lo of powr du o l hrogny n h xpour r, and a mallr numbr of rfrn wll alo rul n lowr powr. 8 hp://boa.bpr.com/uwboa/papr23

20 Acknowldgmn: h work wa uppord by h EPA Norhw Cnr for Parcula Mar and Halh gran numbr R h conn of h arcl do no ncarly rflc h vw and polc of h EPA. Rfrnc ) Maclur M. h Ca-Croovr Dgn - A Mhod for Sudyng rann Effc on h Rk of Acu Evn. Amrcan Journal of Epdmology 99; 332): ) Kwon HJ, Cho SH, Nybrg F, Prhagn G. Effc of Ambn Ar Polluon on Daly Moraly n a Cohor of Pan wh Congv Har Falur. Epdmology 200; 2: ) L J, Schwar J. Ranaly of h ffc of ar polluon on daly moraly n Soul, Kora: A ca-croovr dgn. Envronmnal Halh Prpcv 999; 07: ) Lvy D, Shppard L, Chckoway H, Kaufman J, Lumly, Kong J al. A cacroovr analy of parcula mar ar polluon and ou-of-hopal prmary cardac arr. Epdmology 200; 2: ) Lumly, Lvy D. Ba n h ca-croovr dgn: mplcaon for ud of ar polluon. Envronmrc 2000; : ) Navd. Bdrconal ca-croovr dgn for xpour wh m rnd. Bomrc 998; 54: ) Pr A, Dockry D, Mullr JE, Mlman MA. Incrad parcula ar polluon and h rggrng of myocardal nfarcon. Crculaon 200; 03: ) Sunyr J, Schwar J, oba A, Macfarlan D, Garca J, Ano JM. Pan wh chronc obrucv pulmonary da ar a ncrad rk of dah aocad wh urban parcl ar polluon: A ca-croovr analy. Amrcan Journal of Epdmology 2000; 5: Hod by h Brkly Elcronc Pr

21 9) Grnland S. Confoundng and xpour rnd n ca-croovr and ca m-conrol dgn. Epdmology 996; 7: ) Aun H, Flandr D, Rohman KJ. Ba arng n ca-conrol ud from lcon of conrol from ovrlappng group. Inrnaonal Journal of Epdmology 989; 8: ) Robn J, Pk M. h valdy of ca-conrol ud wh nonrandom lcon of conrol. Epdmology 990; : ) Domnc F, Shppard L. Halh ffc of ar polluon: a acal rvw. Inrnaonal Sacal Rvw 2003 n pr). 3) Na LM, Schwar J, Dockry D. A ca-croovr analy of ar polluon and moraly n Phladlpha. Envronmnal Halh Prpcv 999; 07: ) Baon F, Schwar J. Conrol for aonal varaon and m rnd n ca croovr ud of acu ffc of nvronmnal xpour. Epdmology 999; 0: ) Baon F, Schwar J. Slcon ba and confoundng n ca-croovr analy of nvronmnal m-r daa. Epdmology 200; 2: ) Navd, nhandl E. Rk amplng for ca-croovr dgn. Epdmology 2002; 3): hp://boa.bpr.com/uwboa/papr23

22 Fgur Rfrn lcon rag 2 Hod by h Brkly Elcronc Pr

23 Fgur 2 Larg ampl ba a a funcon of ba for hr ralaon of an xpour r wh random no, for gh dffrn rfrn lcon rag: oal hory undrconal, undrconal 2, undrconal 3,4,5, undrconal 7,4,2, ymmrc bdrconal 7, ymmrc bdrconal 7,4, m-ymmrc bdrconal 7, and m rafd. For ach rfrn ragy, w how h ba for n -0.0, 0.0). Undrconal Un. 2 Larg Sampl Ba Larg Sampl Ba ba ba Un. 3,4,5 Un. 7,4,2 Larg Sampl Ba Larg Sampl Ba ba ba Sym. B 7 Sym. B 7,4 Larg Sampl Ba Larg Sampl Ba ba ba Sm-Symm. m Sra. Larg Sampl Ba Larg Sampl Ba ba ba 22 hp://boa.bpr.com/uwboa/papr23

24 Fgur 3 Larg ampl ba a a funcon of ba for hr ralaon of an xpour r wh auocorrlaon 0.6, for gh dffrn rfrn lcon rag: oal hory undrconal, undrconal 2, undrconal 3,4,5, undrconal 7,4,2, ymmrc bdrconal 7, ymmrc bdrconal 7,4, m-ymmrc bdrconal 7, and m rafd. For ach rfrn ragy, w how h ba for n -0.0, 0.0). Undrconal Un. 2 Larg Sampl Ba Larg Sampl Ba ba ba Un. 3,4,5 Un. 7,4,2 Larg Sampl Ba Larg Sampl Ba ba ba Sym. B 7 Sym. B 7,4 Larg Sampl Ba Larg Sampl Ba ba ba Sm-Symm. m Sra. Larg Sampl Ba Larg Sampl Ba ba ba 23 Hod by h Brkly Elcronc Pr

25 Fgur 4 Larg ampl ba a a funcon of ba for hr ralaon of an xpour r wh m varyng man and varanc, for gh dffrn rfrn lcon rag: oal hory undrconal, undrconal 2, undrconal 3,4,5, undrconal 7,4,2, ymmrc bdrconal 7, ymmrc bdrconal 7,4, m-ymmrc bdrconal 7, and m rafd. For ach rfrn ragy, w how h ba for n -0.2, 0.2). Undrconal Un. 2 Larg Sampl Ba Larg Sampl Ba ba ba Un. 3,4,5 Un. 7,4,2 Larg Sampl Ba Larg Sampl Ba ba ba Sym. B 7 Sym. B 7,4 Larg Sampl Ba Larg Sampl Ba ba ba Sm-Symm. m Sra. Larg Sampl Ba ba Larg Sampl Ba ba 24 hp://boa.bpr.com/uwboa/papr23

26 Fgur 5 Larg ampl ba a a funcon of ba for hr ralaon of an xpour r wh m varyng man and varanc, for gh dffrn rfrn lcon rag: oal hory undrconal, undrconal 2, undrconal 3,4,5, undrconal 7,4,2, ymmrc bdrconal 7, ymmrc bdrconal 7,4, m-ymmrc bdrconal 7, and m rafd. For ach rfrn ragy, w how h ba for n -0.0, 0.0). Undrconal Un. 2 Larg Sampl Ba Larg Sampl Ba ba ba Un. 3,4,5 Un. 7,4,2 Larg Sampl Ba Larg Sampl Ba ba ba Sym. B 7 Sym. B 7,4 Larg Sampl Ba Larg Sampl Ba ba ba Sm-Symm. m Sra. Larg Sampl Ba Larg Sampl Ba ba ba 25 Hod by h Brkly Elcronc Pr

27 26 hp://boa.bpr.com/uwboa/papr23

28 27 Hod by h Brkly Elcronc Pr