Causal inference in nonexperimental studies typically

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1 Orgnal Artcle Regresson Dscontnuty Desgns n Epdemology Causal Inference Wthout Randomzed Trals Jacob Bor, a,b,c Ellen Moscoe, c Porta Mutevedz, b Mare-Louse Newell, b,d and Tll Bärnghausen b,c Abstract: When patents receve an nterventon based on whether they score below or above some threshold value on a contnuously measured random varable, the nterventon wll be randomly assgned for patents close to the threshold. The regresson dscontnuty desgn explots ths fact to estmate causal treatment effects. In spte of ts recent prolferaton n economcs, the regresson dscontnuty desgn has not been wdely adopted n epdemology. We descrbe regresson dscontnuty, ts mplementaton, and the assumptons requred for causal nference. We show that regresson dscontnuty s generalzable to the survval and nonlnear models that are manstays of epdemologc analyss. We then present an applcaton of regresson dscontnuty to the much-debated epdemologc queston of when to start HIV patents on antretrovral therapy. Usng data from a large South Afrcan cohort ( ), we estmate the causal effect of early versus deferred treatment elgblty on mortalty. Patents whose frst CD4 count was just below the 200 cells/μl CD4 count threshold had a 35% lower hazard of death (hazard rato = 0.65 [95% confdence nterval = ]) than patents presentng wth CD4 counts just above the threshold. We close by dscussng the strengths and lmtatons of regresson dscontnuty desgns for epdemology. (Epdemology 2014;25: ) Submtted 24 July 2013; accepted 07 February From the a Department of Global Health, Boston Unversty School of Publc Health, Boston, MA; b Afrca Centre for Health and Populaton Studes, Somkhele, South Afrca; c Department of Global Health and Populaton, Harvard School of Publc Health, Boston, MA; and d Faculty of Medcne, Unversty of Southampton, Southampton, Unted Kngdom. Ths research was made possble wth fundng from the Wellcome Trust (Afrca Centre for Health and Populaton Studes); Natonal Insttutes of Health grants R01 HD and 1R01MH (T.B., M.L.N.); the Rush Foundaton (J.B., T.B.); Harvard Center for Populaton and Development Studes (J.B.); and US Agency for Internatonal Development (USAID) Cooperatve Agreement AID 674-A (J.B.). The contents are the responsblty of the authors and do not necessarly reflect the vews of any of the funders or the US Government. Supplemental dgtal content s avalable through drect URL ctatons n the HTML and PDF versons of ths artcle ( com). Ths content s not peer-revewed or copy-edted; t s the sole responsblty of the authors. Edtors' note: A commentary on ths artcle appears on page 738. Correspondence: Jacob Bor, 801 Massachusetts Avenue, Boston, MA E-mal: jbor@bu.edu Copyrght 2014 by Lppncott Wllams & Wlkns. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. ISSN: /14/ DOI: /EDE Causal nference n nonexpermental studes typcally requres a strong, untestable assumpton: that no unobserved factors confound the relatonshp between the exposure and the outcome. 1 Volatons of ths assumpton wll lead to based estmaton of causal effects. The regresson dscontnuty desgn s one mportant quas-expermental study desgn n whch ths assumpton s not requred for causal nference. Regresson dscontnuty desgns can be mplemented when the exposure of nterest s assgned at least n part by the value of a contnuously measured random varable and whether that varable les above (or below) some threshold value. Provded that subjects cannot precsely manpulate the value of ths varable, assgnment of the exposure s as good as random for observatons close to the threshold, and vald causal effects can be dentfed. 2 The regresson dscontnuty desgn frst appeared n the educatonal psychology lterature n 1960, 3 5 was further developed n the 1970s and 1980s, 6,7 and has become well establshed n economcs over the last 2 decades. 2,8,9 In recent years, a number of clncal and populaton health studes have been publshed n economcs journals usng regresson dscontnuty desgns These studes have used regresson dscontnuty to estmate the health effects of clncal care, 10,11 health behavors, 12,13 socal determnants, 14,15 and envronmental exposures 16 questons of nterest to epdemologsts. Yet regresson dscontnuty has not been wdely adopted n epdemology. To date, no emprcal regresson dscontnuty studes have been publshed n leadng epdemology journals, and, when economcs journals are excluded, just 8 such studes appear n PubMed. 17 Ths paper serves as an ntroducton to regresson dscontnuty for applcaton n epdemology. We descrbe the regresson dscontnuty approach, the assumptons that enable dentfcaton of causal effects, and methods of mplementaton. To date, regresson dscontnuty studes have prmarly used lnear regresson models for contnuous outcomes. 2 We show that the desgn s generalzable to bnary, count, and tme-to-event outcomes, and to the models that epdemologsts commonly use to analyze them. We then present an applcaton of regresson dscontnuty to answer a much-debated queston: when to start treatng HIV patents wth antretrovral therapy (ART). 18 We close by dscussng the benefts and lmtatons of regresson dscontnuty n comparson wth other study desgns and suggest some addtonal applcatons. Epdemology Volume 25, Number 5, September

2 Bor et al Epdemology Volume 25, Number 5, September 2014 REGRESSION DISCONTINUITY DESIGNS: THEORY AND PRACTICE When an exposure or treatment s determned by a threshold rule, the regresson dscontnuty desgn can be used to estmate causal effects. Threshold rules are common n medcne. Patents are often assgned to a therapeutc regmen f they are dentfed as hgh rsk wth respect to a contnuous bomarker such as cholesterol, blood glucose, or brth weght. 10 As wth most measures n nature, the contnuous measures that determne treatment elgblty are subject to random varablty due to measurement error, samplng varablty, and chance factors that affect bomarkers such as ambent temperature. Random varablty mples that patents who score mmedately above and below the threshold wll be smlar, n expectaton, on all observed and unobserved pretreatment characterstcs, just as n a randomzed controlled tral (RCT). Causal effects can be estmated by comparng outcomes n these patents. Threshold rules also appear n nonclncal settngs. Elgblty for a program may depend on beng born after a certan date, 19 resdng n a suffcently poor county, 14 or on one sde of an admnstratve boundary. 20 Indeed, the assgnment varable could be any contnuous pretreatment measure ncludng the outcome varable measured at baselne 4 or another measure of rsk 7,21 ; a baselne covarate that s loosely correlated wth the outcome 14,15,19 ; or even a random number, n whch case regresson dscontnuty s dentcal to an RCT. 2 In ths paper, we use as a runnng example the clncal measurement of CD4 counts (cells/μl of blood), whch are used to determne elgblty for ART. Measured CD4 counts contan substantal random varablty. 22,23 For true CD4 counts close to the threshold, these sources of varablty wll randomly allocate HIV patents to measured CD4 counts above or below the threshold and hence to dfferent probabltes of ART ntaton. Causal Inference n Regresson Dscontnuty Desgns We provde a bref ntroducton to regresson dscontnuty as a method of causal nference, usng the potental-outcomes framework. 24 Detaled dscussons have been publshed elsewhere. 2,6,8,25 We assume a bnary treatment, although results can be generalzed to contnuously valued exposures. 14 By defnton, causal nference requres comparson of outcomes for the same patents (or other unt of analyss) n 2 states of the world: f treated, Y () 1, and f not treated, Y ( 0 ). Only one of these potental outcomes s ever observed: Y =Y (1) f T =1 or Y =Y (0) f T =0, where T = {,} 01 s the treatment ndcator, as assgned. The challenge faced by nonexpermental studes s that f there are unobserved confounders of the relatonshp between T and Y, then the potental outcomes wll be correlated wth treatment assgnment and effect estmates wll be based. Regresson dscontnuty desgns are feasble when the probablty of treatment assgnment changes dscontnuously at some threshold value, c, of a contnuous assgnment varable, Z : lm Pr( T = 1 Z = z) lm Pr( T = 1 Z = z). If the Z c Z c probablty of treatment assgnment changes from 0 to 1 at the threshold, then treatment assgnment s a determnstc functon of Z : T = 1[ Z < c], where 1[] s the ndcator functon; ths s known as sharp regresson dscontnuty (SRD). When the probablty of treatment changes at the threshold, but not from 0 to 1, ths s known as fuzzy regresson dscontnuty (FRD). 5,6 The key nsght that motvates regresson dscontnuty s that, n a small neghborhood around c, as that range goes toward 0, treatment assgnment s gnorable, that s, ndependent of the potental outcomes, just as n randomzed experments: lm ɛ 0 Y (0),Y (1) T c ɛ < Z < c + ɛ. Ths follows from the 2 dentfyng assumptons of regresson dscontnuty: frst, that Z s contnuous at c; and second, that the relatonshp between Z and the potental outcomes Y( 0), Y() 1 s contnuous at c. Under these assumptons, the condtonal dstrbuton f ( Y( 0 ) Z) s dentcal as Z approaches c from above and below, and smlarly for f ( Y() 1 Z). Equvalently, all potental confounders are balanced n a small area around the cutoff. Although contnuty at the cutoff may seem lke a strong assumpton, n fact t follows drectly f there s random nose n Z (e, f t s a random varable or f t s measured wth error), and patents are unable to manpulate the precse value of Z. 2,26 If Z s not measured wth error (eg, date of brth), f Z s noncontnuous (eg, ordnal), or f there s a phase-n regon around the cutoff, then regresson dscontnuty desgns can be mplemented under a more strngent but often plausble assumpton that there are no other reasons for a dscontnuty n potental outcomes at the threshold other than treatment assgnment. Most regresson dscontnuty applcatons have been concerned wth estmatng dfferences n means at the threshold, E[ Y() 1 Z = c] E[ Y( 0) Z = c], an average causal effect (ACE). If treatment assgnment s determnstc (e, a "sharp" dscontnuty), then patents are assgned to the treatment wth certanty f they fall below the threshold and to the control condton f they fall above the threshold: that s, E[ Y Z] = EY [ () 1 Z] when Z < c, and E[ Y Z] = EY [ ( 0 ) Z] when Z c. Fgure 1 shows the contnuous condtonal expectaton functons for the potental outcomes, E[ Y( 0 ) Z] and E[ Y() 1 Z]. The sold lnes show the observed data, E[ Y Z] ; the dotted lnes show the regons of the potental outcome condtonal expectaton functons that are not observed. At the threshold, both E[ Y() 1 Z] and E[ Y( 0 ) Z] are dentfed by lmts n the observed data. Thus, the sharp regresson dscontnuty desgn dentfes the average causal effect at the threshold: ACE = SRD lm EY [ Z = z ] lm EY [ Z = z ] (1) { z c} { z c} Often, treatments are not assgned determnstcally but probablstcally (e, a "fuzzy" dscontnuty). Ths would occur f, for example, clncans prescrbed a therapy to patents based n part on a threshold rule and n part on ther clncal judgment. Such s the case wth ART for HIV: patents are elgble ether f ther CD4 count falls below a Lppncott Wllams & Wlkns

3 Epdemology Volume 25, Number 5, September 2014 Regresson Dscontnuty Desgns n Epdemology threshold value or f they exhbt clncal symptoms that sgnal the severty of ther dsease. In the fuzzy regresson dscontnuty desgn, Equaton 1 s now the ntent-to-treat (ITT FRD ) effect, that s, the effect of the patent presentng just below the threshold. ITT FRD measures the effect of treatment elgblty, as determned by the threshold rule, and s often of nterest n ts own rght. In partcular, ITT FRD can be nterpreted as the effect of rasng the threshold on outcomes for the full populaton of patents close to the threshold. In addton, clncans may be nterested n the effect of therapy tself on those nduced to take up the treatment because of the threshold rule (so-called complers). To obtan ths compler average causal effect (CACE FRD ), t s necessary to scale ITT FRD by the dfference n the probablty of treatment at the cutoff (e, the Wald nstrumental varables estmator, Equaton 2). Fuzzy regresson dscontnuty can be thought of as an nstrumentalvarables approach, where 1[ Z < c Z c] s the nstrument. lm lm CACERDD = EY [ = ] [ = ] { z c} Z z EY { z c} Z z lm PT ( { z c} = 1 Z = z) lm PT ( = = 1 Z z) { z c} When the denomnator of Equaton 2 s equal to 1, we are n the sharp regresson dscontnuty case, and ITTFRD = CACEFRD = ACESRD; when t s 0, there s no dscontnuty, and the causal effect s not dentfed. In our example, CACE FRD measures the casual effect of rapd (vs. deferred) ART ntaton only for those nduced to ntate because they had an elgble CD4 count; ths effect may dffer from the (unobserved) treatment effects for patents that would have ntated ART regardless of CD4 count, for example, because of clncal symptoms (so-called always-takers), or patents who would not have ntated ART even f elgble (so-called never-takers). 27 Addtonally, dentfcaton of CACE FRD requres the assumptons of monotoncty (e, that no patents who would have taken up ART f nelgble would FIGURE 1. Sharp regresson dscontnuty desgn. Ths fgure shows the condtonal expectaton functons for each of the potental outcomes E[ Y( 1 ) Z and E[ Y( 0 ) Z. The sold lnes show the condtonal expectaton functon of the observed data, E[ Y Z. (2) refuse ART f elgble and vce versa) and of excludablty (e, that Z < c may affect Y only through T ). ITT effects have been popular n epdemology because they do not requre these assumptons. 28 In both sharp regresson dscontnuty and fuzzy regresson dscontnuty desgns, causal treatment effects are dentfed at the threshold. If treatment effects are constant or ndependent of Z, then ITT FRD (and equvalently ACE SRD ) s equal to the populaton average treatment effect dentfed n an RCT. (In fact, an RCT can be thought of as a dscontnuty desgn n whch Z s a random number.) If treatment effects are heterogeneous n Z (e, E[Y (0) Z ] and E[Y (1) Z ] are not parallel, as n Fgure 1), then the regresson dscontnuty estmand should be nterpreted as a local treatment effect at Z = c. Ths local effect s more generalzable than t may frst appear. Due to random nose n measurements of Z, observatons wth Z = c are drawn from a dstrbuton of true Z *. Thus, treatment effects dentfed at a sngle value of the measured Z can be thought of as a weghted average across a wder range of true Z *, wth the weghts proportonal * to Pr( Z = c Z = z). Furthermore, even f effects are heterogeneous across the full range of Z, they may be approxmately constant (on the approprate scale) for a wde range of values around the threshold; the assumpton of constant proportonal or addtve effects s often nvoked n epdemologc studes (e.g. nonsaturated regresson models). The presence of effect heterogenety close to the threshold can be tested by assessng whether the slope of E[ Y Z] changes at c. We cauton, however, that local effects may not be generalzable to populatons far from the threshold (eappendx, com/ede/a808). An alternatve to local dentfcaton at the threshold mght be to estmate a global average causal effect by extrapolatng the condtonal expectaton functons across the entre range of Z ; however, ths requres much stronger assumptons to dentfy causal effects n partcular, that the functonal forms of E[ Y() 1 Z] and E[ Y( 0 ) Z] are known across the full range of Z. 6,21,29 Consstent estmaton of the lmts n Equatons 1 and 2 does not depend on knowledge of the functonal form of the condtonal expectaton functons, so long as one s wllng to shrnk the bandwdth as the sample sze ncreases. 25 Estmaton n Regresson Dscontnuty Desgns The task for estmaton n regresson dscontnuty desgns s to estmate the lmts n Equatons 1 and 2: lm { } EY [ ] z c Z = z and lm { } EY [ ] z c Z = z. One approach mght be to compare means n a range of Z above and below the threshold. However, f the slope of E[Y Z ] s non-zero on ether sde of the threshold, then these averages wll be based estmates of the true averages at the lmt, as Z c. Estmatng local lnear (or cubc) regresson models substantally mtgates ths problem. 30 In practce, ACE SRD and ITT FRD estmates are typcally formed by fttng parametrc functons of E[ Y Z, Z c] and E[ Y Z, Z < c] for a range 2014 Lppncott Wllams & Wlkns 731

4 Bor et al Epdemology Volume 25, Number 5, September 2014 of data around the threshold and takng the dfference n the predctons at Z = c. It s customary to ft models of the form EY [ Z ] = β + β + β 1[ Z < c]+ β * 1 [ Z < c] (3) where β 1 s the slope of the lne below the threshold, β1+ β3s the slope of the lne above the threshold, and β 2 s the dfference at the cutoff. 8 The nteracton term allows for the possblty that treatment effects are heterogeneous. Unless the correct functonal forms for E[ Y( 0 ) Z and E[ Y( 1 ) Z are known, the fnte sample estmate always runs the rsk of beng based. However, ths problem s consderably reduced by estmatng the model usng a smaller bandwdth (e, a narrower wndow of data ( c h, c+ h) around the cutoff) and by assessng the robustness of the results to the ncluson of hgher order polynomal terms for Z. CACE FRD s estmated by dvdng the dfference n E[ Y Z] at the threshold by the smlarly formed estmate of the dfference n E[ T Z] at the threshold. In regresson dscontnuty studes, unbased vsual presentaton of the data s essental. In partcular, the researcher should plot E[ Y Z] and E[ T Z] to show the dscontnuty n the outcome and n treatment assgnment. Researchers should also provde vsual evdence n support of the key dentfyng assumpton (e, contnuty of f ( Y( 0 ) Z = z) and f ( Y() 1 Z = z) n Z ), whch results f there s random nose n measurements of Z. Ths assumpton has two mportant mplcatons that can be tested n the data. The frst s that the densty of the data should be contnuous around the threshold; ths would be volated f patents (or provders) could precsely manpulate Z. 31 The second mplcaton s that baselne covarates should be balanced (e, contnuous) at the threshold. As n RCTs, evdence of balance on baselne observables provdes confdence that patents assgned to treatment and control condtons are exchangeable. Regresson Dscontnuty wth Nonlnear and Censored Regresson Models Regresson dscontnuty studes have typcally used lnear regresson models, popular among economsts. 2,8 There are very few examples of regresson dscontnuty desgns appled to the bnary, count, and survval models most often used by epdemologsts. 21,32,33 The extenson of regresson dscontnuty to nonlnear models s straghtforward for ACE SRD and ITT FRD. Contnuty n the condtonal expectaton functons, E[ Y( 0 ) Z and E[ Y() 1 Z, s suffcent for dentfcaton of regresson parameters across the class of generalzed lnear models, whch relate the condtonal expectaton (mean, probablty, rate) to a lnear model va a contnuous lnk functon (such as the log or logt). 34 More generally, contnuty n the densty functons f [ Y( 0 ) Z and f [ Y() 1 Z mples that regresson dscontnuty can be appled to other estmators that do not rely solely on the mean, such as margnal effects (rsk or rate dfferences) n multplcatve models and quantle regresson estmators. 35 For applcatons to survval analyss, Equaton 3 can be adapted to parametrc and semparametrc regresson models that specfy the hazard, cumulatve hazard or survvorshp as a functon of the assgnment varable and tme. A common feature of tme-to-event data s that some duratons are censored, that s, the falure tme exceeds the censorng tme Y > C. The usual assumpton nvoked n survval analyss s that the censorng tmes are nonnformatve, that s, ndependent of falure tmes. For ths to hold n regresson dscontnuty desgns, contnuty n the dstrbuton of censorng tmes s requred. The nablty of agents to manpulate the assgnment varable ensures contnuty as long as censorng s not a result of treatment assgnment. Ths excluson s not so nnocuous because treatment assgnment may nfluence retenton n clncal care and hence the avalablty of follow-up data. However, ths cauton apples to longtudnal data collecton n general. Valdty s enhanced when follow-up data are collected separately from routne montorng of treated patents. In fuzzy regresson dscontnuty desgns wth nonlnear models, ITT FRD s often of nterest and easly estmated. For analysts nterested n the effect of the treatment among complers, rather than the effect of treatment elgblty, CACE FRD can be estmated on the rsk dfference scale usng the smple Wald estmator evaluated at the threshold. Ths lnear estmator s unbased for nonlnear models wthout covarates and s dentcal to the addtve structural mean model. 36,37 Compler causal relatve rsks (CCRR FRD ) can be estmated n multplcatve structural mean models Instrumental varables technques that account for censorng n survval analyss are under development. 40 A smple approach s to use predcted survval probabltes for the numerator n the Wald estmator 40 ; under some assumptons, predcted hazards could also be estmated and plugged n (eappendx, EDE/A808). We note that the null hypothess CACE FRD = 0 s equvalent to ITT FRD = 0 and the varance of CACE FRD s strctly larger than the varance of ITT FRD ; f a result s not statstcally sgnfcant n the ITT framework, t wll not be sgnfcant after scalng by take-up. AN APPLICATION OF THE REGRESSION DISCONTINUITY DESIGN: WHEN TO START ANTIRETROVIRAL THERAPY FOR HIV To llustrate the potental for regresson dscontnuty n epdemology, we present a real-lfe applcaton to a muchdebated queston: when n the course of HIV dsease progresson to start lfe-prolongng ART. We assessed the causal effect of early versus delayed ART elgblty on survval usng data from a large cohort of HIV-nfected patents n rural South Afrca. Our applcaton explots the threshold rule used to determne ART elgblty durng the study perod Our analyss contrbutes causal evdence to a queston on whch expermental evdence s lmted. In an RCT Lppncott Wllams & Wlkns

5 Epdemology Volume 25, Number 5, September 2014 Regresson Dscontnuty Desgns n Epdemology n Hat, Severe et al 41 found a 75% reducton n mortalty among HIV patents who ntated treatment when ther CD4 counts were between 200 and 350 cells/μl, rather than watng for ther CD4 counts to fall below 200 cells/μl. Cohen et al 42 found a 41% decrease n clncal events among patents who began treatment between 350 and 550 cells/μl compared wth those who delayed therapy untl ther CD4 count went below 250 cells/μl; however, the study dd not have suffcent power to detect dfferences n survval. No RCT has evaluated the effect of early versus delayed therapy on survval n sub-saharan Afrca where most people recevng ART lve. Several large clncal cohort studes have reported hgher mortalty for patents who ntated ART at lower CD4 counts ; however, these studes are lmted by the potental for bas due to unobserved confounders that determne treatment-seekng behavor and by the excluson of patents who never ntated ART. CD4 counts at enrollment n care were obtaned for all patents n the Hlabsa HIV Treatment and Care Programme. Dates of ART ntaton were obtaned for those who ntated therapy. 48 Patents were elgble for ART f ther CD4 count was less than 200 cells/μl or f they had stage IV AIDSdefnng llness, as per natonal gudelnes. 49 Dates of death were obtaned from the Afrca Centre for Health and Populaton Studes, whch mantans a demographc survellance system n the clncal catchment area. 50 Survval data were lnked to clncal records by natonal ID number, full name, age, and sex. 51 The study populaton ncluded all patents who had a frst CD4 count between 1 January 2007 and 11 August 2011 regardless of whether they later ntated ART and who were under survellance at that tme. Patents wth frst CD4 counts greater than 350 cells/μl were excluded. Patents were followed from the date of ther frst CD4 count to ther date of death or the date when ther vtal status was last observed n the populaton survellance system. Out of 4391 patents who sought care, 2874 ntated ART and 820 ded durng 13,139 person-years of follow up. Stata 11 was used for all statstcal analyss (StataCorp, College Staton, TX). Fgure 2 shows the dstrbuton of baselne CD4 counts among patents n the study sample. Causal nference would be jeopardzed f health workers or patents manpulated CD4 counts, for example, n an effort to access treatment earler. We found no evdence of bunchng at the threshold, as would result from manpulaton. Further analyss revealed balance n varables observed at baselne (age and sex) at the cutoff (not shown). Fgure 3 dsplays the cumulatve probablty of ART ntaton wthn 3 and 12 months of a patent s frst CD4 count. The probablty of rapd ART ntaton (wthn 3 months) was hgher for patents presentng below 200 cells/μl; ths dscontnuty perssted at 1 year. We frst examned the effect of treatment elgblty (CD4 < 200 cells/μl) on mortalty n an ITT analyss. The Table presents the results of hazard regresson models, wth the loghazard replacng E [ Y Z ] n Equaton 3. We present estmates Frequency Frst CD4 count (cells/µl) FIGURE 2. Dstrbuton of frst CD4 counts n the HIV treatment and care program. Cumulatve probablty of ART ntaton Frst CD4 count (cells/µl) 3 months 12 months FIGURE 3. Frst CD4 count and ART ntaton. Kaplan Meer estmates of the probablty that a patent ntated ART wthn 3 and 12 months of frst CD4 count n the HIV treatment and care program. lmtng the data to several ranges (bandwdths) around the cutoff. Smaller bandwdths reduce the potental for bas from usng a lnear functon to approxmate the relatonshp between frst CD4 count and log-mortalty rates; however, ths reducton n possble bas s attaned at the expense of precson. Fgure 4 dsplays ftted values from model 2a, supermposed over hazards predcted for CD4 count bns of wdth 10 cells. In general, mortalty was lower for patents presentng wth hgher ntal CD4 counts (Table, Fgure 4). However, there was a dscontnuty at 200 cells/μl: patents presentng just below the threshold had a 35% lower hazard of death than those presentng just above the threshold (ITTFRD HR = 065. ; model 2a n the Table). Ths result was robust to varyng specfcatons of the hazard functon and statstcally sgnfcant n models usng wder CD4 count bandwdths. In models wth smaller bandwdths, the coeffcents remaned essentally unchanged 2014 Lppncott Wllams & Wlkns 733

6 Bor et al Epdemology Volume 25, Number 5, September 2014 TABLE. Intent-to-Treat Estmates: The Causal Effect of ART Elgblty on Mortalty a Frst CD4 Count (a) Exponental (b) Cox Range (cells/μl) Predctor HR (95% CI) HR (95% CI) Sample No D 0.59 ( ) 0.62 ( ) 4, ( ) ( ) D * ( ) ( ) D 0.65 ( ) 0.67 ( ) 3, ( ) ( ) D * ( ) ( ) D 0.66 ( ) 0.67 ( ) 2, ( ) ( ) D * ( ) ( ) D 0.68 ( ) 0.72 ( ) 1, ( ) ( ) D * ( ) ( ) D 0.54 ( ) 0.55 ( ) ( ) ( ) D * ( ) ( ) a Z s frst CD4 count. D = 1[ Z < c] s an ndcator for whether the patent s elgble for ART accordng to CD4 count. Table dsplays the results of 10 regressons 5 ranges of CD4 counts and 2 statstcal models for the survval tmes. Mortalty rate (per person year) Frst CD4 count (cells/µl) FIGURE 4. Frst CD4 count and mortalty hazard rate. Predcted hazards from the Table, model 2a are dsplayed as sold lnes. Dashed lne shows extrapolated predcton f all patents were treatment elgble at frst CD4 count. Dots are hazards predcted for CD4 count bns of wdth 10 cells. although, as expected, the estmates were less precse. Vsual nspecton of Fgure 4 shows no evdence of msspecfcaton. The hazard ratos on the nteracton terms were close to 1.0, suggestng that treatment effects were not heterogeneous close to the threshold (Table). The ITT FRD estmate s arguably the parameter of nterest from a polcy perspectve: t s the causal effect of ART elgblty for all patents seekng care wth CD4 counts close to the threshold. However, clncans may also be nterested n CACE FRD, the causal effect of rapd ART ntaton on survval among patents who ntated based on ther CD4 count. To obtan CACE FRD, we scaled the dfference n mortalty hazards at the threshold by the dfference n the probablty of ART ntaton wthn 3 months at the threshold. In both cases, we used models wth separate lnear terms on ether sde of the threshold, estmated on the range of cells. Ths yelded a causal dfference n hazards of CACE FRD = / = fewer deaths per personyear for patents who ntated because they were CD4-count elgble, compared wth those who were precluded from ntatng because they were nelgble. Mortalty hazards for treatment and control complers were calculated to be and 0.040, respectvely, resultng n a compler causal hazard rato of (See eappendx for detals on these calculatons and robustness checks, Rapd ART ntaton thus causally reduced mortalty by 72% among patents who ntated ART because CD4 < 200 cells/μl. DISCUSSION Regresson dscontnuty desgns present an opportunty for causal nference n epdemology when randomzaton s beyond the control of the researcher. As a quas-expermental study desgn, regresson dscontnuty offers sgnfcant benefts over nonexpermental approaches based on regresson adjustment or matchng. Contnuty n the assgnment varable at the Lppncott Wllams & Wlkns

7 Epdemology Volume 25, Number 5, September 2014 Regresson Dscontnuty Desgns n Epdemology Threshold rules nfluence a wde range of epdemologcal exposures. In these settngs, regresson dscontnuty desgns offer epdemologsts a smple but rgorous approach to causal nference from observatonal data. Potental applcatons nclude: Clncal dagnoses:effect of beng dagnosed wth hgh blood pressure on sodum ntake Clncal treatments: effect of cholesterol lowerng medcatons on cardovascular dsease Access to harmful substances: effect of beng of drnkng age on rsk of sucde Elgblty for programs: effect of ncome-elgblty for Medcad on health spendng Government regulatons: effect of workng for a busness wth less than 50 employees (exempt from health nsurance mandate) on nsurance status and emergency room use Electons: effect of unon representaton (>50% of votes n electon) on workplace njury Envronmental hazards: effect of water polluton on brth defects downstream from source FIGURE 5. Potental applcatons of regresson-dscontnuty desgns n epdemology. threshold breaks all lnks between treatment assgnment and both observed and unobserved confounders. Nether ex-post-covarate adjustment nor assumptons about the absence of resdual confoundng s requred for causal nference, smlar to an RCT. Regresson dscontnuty desgns also have mportant benefts over other quas-expermental approaches. In most studes usng nstrumental varables, the assumpton that treatment assgnment s as-good-as-random s an artcle of fath; n dscontnuty desgns, ths assumpton follows drectly from random nose n measurements of the assgnment varable and can be assessed through tests of contnuty n varables observed at baselne. 2 There are also mportant cases where regresson dscontnuty desgns may be preferred to RCTs, such as when t s unethcal to deny a randomzed nterventon to a control group or when an experment s too expensve or logstcally dffcult to mplement. Addtonally, regresson dscontnuty desgns can evaluate the real-world effectveness of nterventons as mplemented, provdng causal effect estmates that are often more relevant for polcy decsons than those derved under the hghly controlled condtons of an RCT. Although regresson dscontnuty desgns requre larger sample szes than RCTs to acheve a gven level of power, 52 they can often be mplemented usng routne clncal or admnstratve data, whch are comparatvely cheap to collect. Regresson dscontnuty desgns are also more lkely to be generalzable to the populaton seekng care than RCTs wth opt-n partcpant recrutment and a range of partcpant ncluson and excluson crtera. Fnally, regresson dscontnuty desgns dentfy a type of causal effect that s of partcular nterest for polcy and clncal practce: the effect for patents near the threshold (whch s also the effect of margnally rasng or lowerng the threshold). In contrast, RCTs estmate average causal effects across a wder range of data and thus do not provde the specfc nformaton needed for optmzng treatment thresholds (eappendx, Regresson dscontnuty desgns can be mplemented whenever an exposure s assgned at least n part by a threshold rule. In spte of many potental applcatons (Fgure 5), regresson dscontnuty has yet to make substantal nroads n epdemology. 17 Ths may be due to (ms)perceptons that the range of applcatons s lmted or that the assumptons requred for causal nference are mplausble. Some of the early lterature on regresson dscontnuty (and smlar desgns under other names) proposed that (1) treatment assgnment must be based solely on the threshold rule 7,29 ; (2) treatment assgnment must be under control of the researcher 53 ; (3) the functonal form of the relatonshp between the outcome and assgnment varable must be known 21,29,54 ; (4) treatment effects must be constant 21 ; and (5) measurement error n the assgnment varable s a source of bas. 54,55 In fact, as descrbed n ths paper, assgnment need not be determnstc nor under control of the researcher; causal nference can be conducted at the threshold usng local lnear regresson, wthout functonal form assumptons; and treatment effects may be heterogeneous, wth the provso that effects are local to observatons near the threshold. Rather than beng a threat to valdty, random nose n the assgnment varable ensures contnuty n potental outcomes the key assumpton requred for causal nference and attenuates effect heterogenety, ncreasng the generalzablty of the estmates. In our llustraton of regresson dscontnuty, we found large survval benefts to early versus delayed ART ntaton at the CD4 count threshold of 200 cells/μl. Our results are smlar n magntude to those reported by Severe et al 41 the only RCT to report survval mpacts of delayng ART untl a patent s CD4 count s below 200 cells/μl. Several factors support our nterpretaton of these results as causal. By desgn, our analyss s robust to any unobserved factors that are correlated both wth tmng of treatment ntaton and ndependently correlated wth survval. Causal dentfcaton depends only on the assumpton that these factors are smooth at the threshold, and ths s guaranteed by random nose n measurements of CD4 counts. Our results are unlkely to be based due to systematc msclassfcaton, selecton nto the sample, or attrton. Mortalty data were collected through 2014 Lppncott Wllams & Wlkns 735

8 Bor et al Epdemology Volume 25, Number 5, September 2014 semannual demographc survellance; CD4 counts were reported drectly from the laboratory; and dates of ART ntaton were captured from clncal records. The study ncluded all patents who sought care, not just those who ntated ART. And we observed survval n the survellance system even for patents who were not retaned clncally. Although we beleve the nternal valdty of our results to be hgh, they may not be generalzable to persons who dd not seek care and to patents presentng wth CD4 counts far from 200 cells/μl. The beauty of the regresson dscontnuty desgn les n ts smplcty: causal effects can be estmated wth very few assumptons, and the source of causal dentfcaton s transparent and easy to communcate graphcally. These qualtes stand out compared wth other nonexpermental methods that rely on ex-post statstcal adjustment. Threshold rules are ubqutous n clncal practce, n determnng elgblty for programs, and exposure to rsk factors. 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