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1 Universiy of California, Berkeley U.C. Berkeley Division of Biosaisics Working Paper Series Year 2017 Paper 357 Evaluaion of Progress Towards he UNAIDS HIV Care Cascade: A Descripion of Saisical Mehods Used in an Inerim Analysis of he Inervenion Communiies in he SEARCH Sudy Laura Balzer Joshua Schwab Mark J. van der Laan Maya L. Peersen Deparmen of Biosaisics, Harvard T.H. Chan School of Public Heah, lbbalzer@hsph.harvard.edu Division of Biosaisics, Universiy of California, Berkeley, joshuaschwab@yahoo.com Division of Biosaisics, Universiy of California, Berkeley, laan@berkeley.edu Division of Biosaisics, Universiy of California, Berkeley, mayaliv@berkeley.edu This working paper is hosed by The Berkeley Elecronic Press (bepress) and may no be commercially reproduced wihou he permission of he copyrigh holder. hp://biosas.bepress.com/ucbbiosa/paper357 Copyrigh c 2017 by he auhors.

2 Evaluaion of Progress Towards he UNAIDS HIV Care Cascade: A Descripion of Saisical Mehods Used in an Inerim Analysis of he Inervenion Communiies in he SEARCH Sudy Laura Balzer, Joshua Schwab, Mark J. van der Laan, and Maya L. Peersen Absrac WHO guidelines call for universal anireroviral reamen, and UNAIDS has se a global arge o virally suppress mos HIV-posiive individuals. Accurae esimaes of populaion-level coverage a each sep of he HIV care cascade (esing, reamen, and viral suppression) are needed o assess he effeciveness of es and rea sraegies implemened o achieve his goal. The daa available o inform such esimaes, however, are suscepible o informaive missingness: he number of HIV-posiive individuals in a populaion is unknown; individuals esed for HIV may no be represenaive of hose whom a esing inervenion fails o reach, and HIV-posiive individuals wih a viral load measured may no be represenaive of hose for whom no viral load is obained. We provide an in-deph descripion of he saisical mehods (arge parameers, assumpions, saisical esimands, and algorihms) used in an inerim analysis of he inervenion arm of he SEARCH Sudy (NCT ) o analyze progress owards he UNAIDS arge a sudy baseline and afer one and wo years. We describe he mehods used o accoun for informaive measuremen in all analyses as well as for informaive censoring in longiudinal analyses. We use argeed maximum likelihood esimaion (TMLE) wih Super Learning o generae semi-parameric efficien and double robus esimaes of he care cascade among a open cohor of prevalen HIV-posiive aduls and among a closed cohor of baseline HIV-posiive aduls. TMLE is also used o evaluae predicors of poor oucomes.

3 Conens 1 Overview 2 2 Observed daa 3 3 Analysis of he open cohor of prevalen HIV-posiive aduls Overview Primary analysis HIV prevalence Proporion HIV-posiive, and previously diagnosed Proporion HIV-posiive, previously diagnosed, and ever on ART Proporion HIV-posiive, previously diagnosed, ever on ART, and virally suppressed The cascade and populaion-level viral suppression Unadjused secondary analysis Sraified analysis Addiional sensiiviy analyses Worked example: esimaing populaion-level viral suppression Targe parameer Unadjused (secondary) analysis approach Examining and relaxing assumpions Primary analysis approach Analysis of he closed cohor of baseline HIV-posiive aduls Suppression success and failure among baseline known HIV-posiive aduls Unadjused secondary analysis Sraified analysis Closed cohor analyses reaing deah and oumigraion as righ-censoring evens Targe populaion and oucomes of ineres Targe parameers Cascade probabiliies over ime Predicors of cascade failures Idenificaion Esimaors Acknowledgemens: 22 1 Hosed by The Berkeley Elecronic Press

4 1 Overview In 2014, UNAIDS issued an ambiious new arge for he HIV care cascade worldwide: by 2020, a leas 90% of HIV-posiive individuals should be diagnosed, a leas 90% of hose diagnosed should be receiving anireroviral herapy (ART), and a leas 90% of hose on ART should have suppressed viral replicaion, for an overall arge of 73% of all HIV-posiive individuals virally suppressed (UNAIDS, 2014). The Susainable Eas Africa Research in Communiy Healh (SEARCH) Sudy is a cluser randomized rial o evaluae he healh, economic, and educaional impacs of a Universal Tes and Trea HIV inervenion compared o he counry sandard-of-care in 32 pair-mached communiies in rural Kenya and Uganda (NCT ). A sudy baseline, a household census was used o enumerae all communiy residens. Immediaely following he baseline census and annually hereafer, populaion-wide HIV esing was conduced in he inervenion communiies hrough a hybrid esing model consising of a muli-disease communiy healh campaign (CHC) followed by racking and home-based esing for all enumeraed residens who did no aend he CHC (Chamie e al., 2016). Plasma HIV RNA levels (viral loads) were also measured on all HIV-posiive aduls during hese hybrid esing campaigns. We esimaed progress owards he UNAIDS care cascade arge among adul ( 15 years of age) residens of he SEARCH inervenion communiies a sudy baseline and afer one and wo years of he inervenion. We also evaluaed he exen o which specific subgroups were a risk of poor cascade oucomes. In his documen, we provide an in-deph descripion of he saisical mehods (arge parameers, assumpions, saisical esimands, and algorihms) used o generae hese esimaes. Full R code is available a hps://gihub.com/laurabalzer/esimaing in-search. We conduced hree ses of complimenary analyses. Firs, as described in Secion 3, we evaluaed an open cohor of prevalen HIV-posiive adul communiy residens a he ime of he hree annual rounds of populaion-wide HIV anibody and plasma HIV RNA level esing. In his open cohor, we esimaed: 1) he proporion of HIV-posiive individuals who had been previously diagnosed wih HIV; 2) he proporion of previously diagnosed HIV-posiive individuals who had previously been or were currenly reaed wih ART; 3) he proporion of previously or currenly reaed HIV-posiive individuals who were virally suppressed (viral load < 500 copies/ml); and 4) he proporion of all HIV-posiive individuals who were virally suppressed. In hese analyses, he number of HIV-posiive individuals was esimaed accouning for incomplee and possibly informaive HIV esing coverage, and he number of HIV-posiive individuals wih viral suppression was esimaed accouning for incomplee and possibly informaive viral load esing coverage. We repeaed hese analyses wihin a priori-specified subgroups of sex, age, and counry. Secion 3.6 provides a worked example demonsraing he approach used o adjus for missing measures. Second, as described in Secion 4, we evaluaed a closed cohor of adul residens wih an HIV diagnosis a or before sudy baseline. In his cohor, we esimaed he proporion who a baseline and afer one and wo years 1) had died; 2) had ou-migraed from he communiy; 3) were newly diagnosed; 4) had never iniiaed ART; 5) had iniiaed bu were unsuppressed; and 6) had iniiaed ART and were virally suppressed. In hese analyses, we adjused for incomplee and possibly informaive viral load esing coverage among HIV-posiive residens. We repeaed hese analyses wihin a priori-specified subgroups of sex, age, and counry. Third, as described in Secion 5, we again analyzed he closed cohor of adul residens wih an HIV diagnosis a or before sudy baseline. We esimaed he proporion of his cohor who were virally unsuppressed afer one and wo years, while reaing deah and ou-migraion as righ-censoring evens. We adjused for poenially informaive censoring in addiion o poenially informaive missing viral load measures. We repeaed hese analyses wihin a priori-specified subgroups defined by baseline prior HIV diagnosis, ART use, and viral suppression. In he same cohor of baseline HIV-posiive aduls, analogous analyses were performed o esimae he proporion of he cohor who had never iniiaed ART. In an expanded closed cohor of all adul communiy residens wihou an HIV diagnosis prior o baseline es- 2 hp://biosas.bepress.com/ucbbiosa/paper357

5 ing, analogous analyses were performed o esimae he proporion of residens who were esed for HIV a leas once during follow-up. For each of hese oucomes, we furher evaluaed univariable and mulivariable associaions beween demographic facors and poor cascade oucomes (failing o suppress viral replicaion, never iniiaing ART, and never esing for HIV, respecively). 2 Observed daa We consider ime poins = {0, 1, 2}, corresponding o he communiy-specific daes of he annual populaion-wide HIV and viral load esing rounds a sudy baseline, follow-up year 1, and follow-up year 2, respecively. Le B denoe baseline variables measured on all individuals during he household census enumeraion conduced immediaely prior o he firs round of esing. These include age, sex, marial saus, educaion, occupaion, mobiliy, household wealh index, communiy of residence, and counry. Le D be an indicaor ha an individual has died by ime, and M be an indicaor ha an individual has migraed ou of he communiy by ime. We define C as an indicaor of censoring by deah or ou-migraion by ime. A baseline, D 0 = M 0 = C 0 = 0 deerminisically. Le Y and V LV denoe he underlying values of HIV serosaus and viral load a ime, irrespecive of wheher hese values are measured. Furher, le Supp be an indicaor ha an individual s viral load a ime is <500 copies/ml: Supp = I(V LV < 500). Throughou, we use viral loads measured a CHC/racking raher han hose measured a clinic o minimize he poenial for informaive missingess, as in-clinic measures inherenly depend on an individual s reenion in HIV care. Le pdx be an observed indicaor ha an HIV-posiive individual has been diagnosed prior o ime. Le eart be an observed indicaor ha an HIV-posiive individual has ever iniiaed ART prior o ime. Boh variables are sep funcions, jumping o one as soon as here is evidence of prior diagnosis or ART use, respecively, and remaining equal o one hereafer. Classificaion of an individual as previously diagnosed or previously reaed requires eiher healh record documenaion or a suppressed viral load in an individual documened o be HIV-posiive. If an individual does no have evidence of prior diagnosis, hen he or she is assumed no be previously diagnosed as HIVposiive. Likewise, if an individual does no have evidence of ART use, hen he or she is assumed no o have iniiaed ART. Le CHC and T r denoe indicaors ha an individual was seen a a CHC or a subsequen racking a ime, respecively. Le T shiv denoe an indicaor ha HIV saus is known a ime. HIV saus is considered known if an individual was esed for HIV by SEARCH a ime or already had a known HIVposiive saus from previous healh records or documened ess. Le denoe an indicaor ha a individual was conaced a he CHC/racking and had a known HIV saus a ime : = I[(CHC = 1 or T r = 1) & T shiv = 1]. Le Y = T shiv Y denoe observed HIV serosaus a ime. Le T sv L denoe an indicaor ha a viral load was measured a CHC/racking a ime. Le Supp = T sv L Supp denoe observed viral suppression (as measured a CHC/racking) a ime. 3 Hosed by The Berkeley Elecronic Press

6 Le ever.supp be an indicaor of having a leas one measured viral load < 500 copies/ml a or afer sudy baseline. This variable uses addiional viral load measures made in he clinic, beyond he annual measures made a CHC/racking. This variable reas he absence of a measured suppressed viral load as evidence of failure o suppress. The observed daa on a given individual a ime poin are O =(D, M, pdx, eart, CHC, T r, T shiv,, T shiv Y, T sv L, T sv L Supp, ever.supp ) =(C, pdx, eart, CHC, T r, T shiv,, Y, T sv L, Supp, ever.supp ). All variables in O are indicaor variables. The observed daa on a given individual consis of O = (B, O 0, O 1, O 2 ) where for ease of noaion, we assume ha variables afer censoring by deah or ou-migraion are equal o heir las observed values. We use overbars o denoe a variable s hisory (e.g. Ō (O 0,..., O ) for > 0). Throughou we make use of he following relaionships beween variables. One can only have a previous diagnosis if HIV-posiive: P(pDx = 1 Y = 0) = 0. One can only iniiae ART afer receiving a diagnosis: P(eART = 1 pdx = 0) = 0. We also assume no HIV-posiive individuals conrol viral replicaion below 500 copies/ml in he absence of reamen: P(Supp = 1 eart = 0, Y = 1) = 0. 3 Analysis of he open cohor of prevalen HIV-posiive aduls 3.1 Overview For hree ime poins = {0, 1, 2}, we aim o esimae cascade coverage and populaion-level viral suppression in an open cohor consising of all prevalen HIV-posiive adul residens of he communiy a ha ime poin. In oher words, our -specific arge populaion is all individuals who are alive, no ou-migraed, 15 years of age a ime, and HIV-posiive a ime. All parameers below are hus condiional on D = M = 0, and age 15 (in addiion o Y = 1). We suppress he former condiioning in our noaion o simplify presenaion. In he primary analysis, we resric our arge populaion o baseline enumeraed sable ( 6 monhs of pas year in he communiy during he census) residens. In sensiiviy analyses, we include non-sable residens and in-migrans. When esimaing serial cascade coverage in his open cohor, we leverage he full longiudinal daa srucure o adjus for poenially informaive missing measures of HIV saus and viral load. For example, individuals esing for HIV a baseline may also be more likely o be successfully conaced a CHC or racking in subsequen years han individuals who were no esed a baseline, poenially inflaing cascade coverage esimaes unless we adjus for pas aendance. As deailed in he following secions, our general approach is o idenify and esimae he following four populaion-level proporions: 1. HIV prevalence: P(Y = 1) 2. Proporion who are HIV-posiive and previously diagnosed: P(pDx = 1, Y = 1) 3. Proporion who are HIV-posiive, previously diagnosed, and ever on ART: P(eART = 1, pdx = 1, Y = 1) 4 hp://biosas.bepress.com/ucbbiosa/paper357

7 4. Proporion who are HIV-posiive, previously diagnosed, ever on ART, and virally suppressed: P(Supp = 1, eart = 1, pdx = 1, Y = 1) As explicily discussed below, probabiliies #2 and #3 are esimaed as empirical proporions, while probabiliies #1 and #4 are esimaed wih argeed maximum likelihood esimaion (TMLE), a double robus and semi-parameric efficien approach ha allows adjusmen for missing measuremens and censoring (van der Laan and Rubin, 2006; van der Laan and Rose, 2011). To minimize model misspecificaion bias and opimize esimaor performance, nuisance parameers are esimaed using Super Learner, an ensemble machine learning mehod (van der Laan e al., 2007). When implemening Super Learner, we use 5-fold cross-validaion and specify an algorihm library consising of general addiive models, sepwise regression, and logisic regression, all wih and wihou pre-screening based on univariae oucome correlaions. Esimaors are implemened using he lmle v (Schwab e al., 2016) and SuperLearner v packages (Polley and van der Laan, 2014) in R (R Core Team, 2015). Code o implemen hese esimaors is available a hps://gihub.com/laurabalzer/esimaing in-search. Given esimaes of probabiliies #1-4, our populaion-level esimaes of he cascade and viral suppression are obained by aking he following raios. Proporion of all HIV-posiive individuals who have a prior diagnosis a ime : P(pDx = 1 Y = 1) = P(pDx = 1, Y = 1) P(Y. = 1) Proporion of all HIV-posiive individuals wih a prior diagnosis who have ever sared ART a ime : P(eART = 1 pdx = 1, Y = 1) = P(eART = 1, pdx = 1, Y = 1) P(pDx = 1, Y. = 1) Proporion of all HIV-posiive individuals wih prior ART iniiaion who are virally suppressed a ime : P(Supp = 1 eart = 1, pdx = 1, Y = 1) = P(Supp = 1, eart = 1, pdx = 1, Y = 1) P(eART = 1, pdx = 1, Y. = 1) Proporion of all HIV-posiive individuals who are virally suppressed a ime : P(Supp = 1 Y = 1) = P(Supp = 1, eart = 1, pdx = 1, Y = 1) P(Y. = 1) In unadjused (secondary) analyses, we also implemen simple esimaors of each cascade sep, equivalen o he empirical proporions among individuals who have measured values and are conaced a he CHC/racking (Secion 3.3). We also refer he reader o Secion 3.6 for a worked example of our approach and illusraion of why oher mehods (e.g. unadjused esimaors) migh fall shor. Saisical inference, including Wald-Type 95% confidence inervals, are based on influence curve sandard error esimaors ha rea households as he uni of independence (van der Laan and Rubin, 2006; van der Laan and Rose, 2011). 5 Hosed by The Berkeley Elecronic Press

8 3.2 Primary analysis HIV prevalence Our goal is o idenify and hen esimae he proporion of he arge populaion who are HIV-posiive a ime : P(Y = 1). To idenify his parameer, we assume ha wihin sraa defined by baseline covariaes, pas esing, and cascade hisory, he prevalence of HIV among hose seen a CHC/racking is represenaive of he prevalence of HIV among hose no seen: Y B, Ō 1. As our missingess indicaor, we use raher han T shiv, because T shiv is a direc funcion of underlying HIV saus Y. A previously diagnosed HIV-posiive individual s saus can be ascerained hrough records wihou aending he CHC/racking, whereas an HIV-negaive individual s saus canno be. By definiion, he above assumpion holds for individuals known o be HIV-posiive a he previous ime poin (i.e. wih Y 1 = 1): P(Y = 1 Y 1 = 1) = 1. Furhermore, among hose no previously known o be HIV-posiive (Y 1 = 0), here is only variabiliy in baseline demographics, pas CHC/racking aendance, and pas HIV esing hisory. Therefore, for he subgroup of individuals wihou a prior HIV diagnosis by he close of he previous year s esing (Y 1 = 0), we assume ha wihin sraa defined by baseline demographics, pas CHC/racking conac, and HIV esing hisory (e.g. number of prior negaive ess), HIV prevalence among hose esed a CHC/racking is represenaive of HIV prevalence among hose no esed: Y Y 1 = 0, CHC 1, T r 1, T shiv 1, B. We furher assume posiiviy: here are no sraa (defined by baseline demographics, CHC/racking hisory, and esing hisory) in which zero previously undiagnosed individuals are esed a he CHC/racking a ime : P( = 1 Y 1 = 0, CHC 1, T r 1, T shiv 1, B) > 0. Le us denoe hese sraa as L ( CHC 1, T r 1, T shiv 1, B). Under he above assumpions and using he above deerminisic knowledge, we have he following idenifiabiliy resul: P(Y = 1) = P(Y = P(Y 1 = 1) = 1 Y 1 = 1)P(Y 1 = 1) + P(Y = 1 Y 1 = 0)P(Y 1 = 0) + P(Y 1 = 0) l [ P(Y = 1 = 1, L = l, Y 1 = 0)P(L = l Y 1 = 0) ]. Our esimand can be inerpreed as he proporion known o be HIV-posiive a he prior ime poin plus he adjused proporion no previously known o be HIV-posiive who are known o be HIV-posiive a. The laer accouns for incomplee measuremen of HIV saus among his group. We sraify he populaion no previously known o be HIV-posiive on baseline demographics, pas CHC/racking aendance, and HIV esing hisory; assume ha wihin each sraum, HIV prevalence among individuals esed a he CHC/racking is he represenaive of prevalence among hose wihou an HIV es resul, and hen combine hese sraum-specific esimaes ino a single sandardized esimae. For esimaion, we use TMLE wih as he single inervenion node, wih (Y 1, L ) as he adjusmen se, and knowledge ha P(Y = 1 = 1, L = l, Y 1 = 1) = 1 for individuals known o be HIV-posiive a he close of he prior round of esing. The esimaed number of prevalen HIV-posiive individuals is calculaed as he esimaed prevalence imes he arge populaion size a ime. 6 hp://biosas.bepress.com/ucbbiosa/paper357

9 3.2.2 Proporion HIV-posiive, and previously diagnosed Our goal is o idenify and hen esimae he proporion of he arge populaion who are previously diagnosed wih HIV a ime. P(pDx = 1, Y = 1) = P(pDx ), where we use ha prior diagnosis of HIV implies ha an individual is HIV-posiive a ime. Esimaion is based on he empirical proporion of he arge populaion wih evidence of a prior diagnosis a ime : a posiive HIV es prior o ime, a record of HIV care prior o ime, or a viral load < 500 copies/ml a or before ime among individuals who are confirmed HIV-posiive. The number of previously diagnosed HIV-posiive individuals is calculaed as he proporion wih evidence of a prior diagnosis imes he arge populaion size a ime. Failing o idenify records of prior diagnosis among HIV-posiive individuals will resul in underesimaion of prior diagnosis. This moivaes our use of a suppressed viral load in a confirmed HIV-posiive individual as evidence of being on ART a ha ime poin and hus having been previously diagnosed. However, his approach assumes ha no HIV-posiive individuals conrol viral replicaion below 500 copies/ml in he absence of reamen, and resuls in poenially differenial reducion in misclassificaion (missed ART records are only correced among hose individuals wih suppressed viral loads). In secondary analyses, we employ an alernaive assumpion ha prior diagnosis among HIV-posiive individuals seen a he CHC/racking is represenaive of prior diagnosis among HIV-posiive individuals no seen. Finally, in recogniion of he increased poenial for missing records of prior diagnosis a sudy baseline, we conduc an addiional sensiiviy analysis in which self-repor is used as evidence of prior diagnosis Proporion HIV-posiive, previously diagnosed, and ever on ART Our goal is o idenify and hen esimae he proporion of he arge populaion who are HIV-posiive, previously diagnosed, and have iniiaed ART by ime. P(eART = 1, pdx = 1, Y = 1) = P(eART = 1) where we have used ha ART iniiaion implies ha an individual has been previously diagnosed and is HIV-posiive. Esimaion is based on he empirical proporion of he arge populaion who have evidence ART iniiaion by ime : a healh care record wih an ART sar dae prior o ime, or a viral load < 500 copies/ml a or before ime among individuals who are confirmed HIV-posiive. The number of HIV-posiive individuals ever on ART is calculaed as he proporion wih evidence of prior ART iniiaion imes he arge populaion size as ime. Failing o idenify records of ART use among HIV-posiive individuals will resul in underesimaion of he proporion ever on ART. This moivaes our use of a suppressed viral load in a confirmed HIVposiive individual as evidence of being on ART a ha ime poin. Again, his approach assumes ha no HIV-posiive individuals conrol viral replicaion below 500 copies/ml in he absence of reamen, and resuls in poenially differenial reducion in misclassificaion (missed ART records are only correced among hose individuals wih suppressed viral loads). In secondary analyses, we employ an alernaive assumpion ha ART use among HIV-posiive individuals seen a he CHC/racking is represenaive of use among HIV-posiive individuals no seen Proporion HIV-posiive, previously diagnosed, ever on ART, and virally suppressed Our goal is o idenify and hen esimae he proporion of he arge populaion who are HIV-posiive, previously diagnosed, ever on ART, and virally suppressed a ime : P(Supp = 1, eart = 1, pdx = 1, Y = 1). Now we have missing measures on boh HIV saus and viral load. To simplify noaion, le Z I(Supp = 1, eart = 1, pdx = 1, Y = 1) 7 Hosed by The Berkeley Elecronic Press

10 denoe a binary indicaor he oucome of ineres, and le Z I(Supp = 1, eart = 1, pdx = 1, Y = 1) be is observed analog. We formulae he idenifiabiliy of our arge parameer P(Z = 1) as a longiudinal dynamic regime (e.g. Hernán e al. (2006); van der Laan and Peersen (2007); Robins e al. (2008)). Oher approaches are also possible, and several alernaive formulaions resul in he same esimand. Here, we consider a wo componen hypoheical inervenion on he missingness mechanism: d0: se = 1. In oher words, ensure ha all individuals aend he CHC or racking a ime and have known HIV saus. d1(y ): if (Y = 1), se T sv L = 1; else se T sv L = 0. In oher words, if an individual is known o be HIV-posiive a ime, ensure ha his or her viral load is measured. Under his hypoheical join inervenion, we would have complee measuremen of underlying HIV saus and complee measuremen of viral loads among all HIV-posiive individuals. To idenify he probabiliy of suppression under such a hypoheical inervenion, we assume he following (sequenial) randomizaion assumpions (Robins, 1986): Z L Z T sv L Y, = 1, L (1) where L (B, Ō 1, pdx, eart ). Togeher wih he corresponding posiiviy assumpions, our arge parameer is idenified as P(Z = 1) = l,y P(Z = 1 T sv L = d1(y ), Y = y, = 1, L = l )P(Y = y = 1, L = l )P(L = l ) = P(Supp = 1 T sv L = 1, Y = 1, = 1, eart = 1, pdx = 1, ō 1, b) b,ō 1 P(Y = 1 = 1, eart = 1, pdx = 1, ō 1, b) P(eART = 1, pdx = 1, ō 1, b). For he second equaliy, we use ha he join oucome Z is deerminisically 0 if Y = 0 OR pdx = 0 OR eart = 0. Also in he second equaliy, we use our definiion of dynamic reamen rule d1(y = 1) = 1. The final equaliy is now in erms of he observed daa disribuion and can be esimaed wih longiudinal TMLE wih a dynamic reamen regime. Insead, we simplify our approach by noing he following. Boh randomizaion assumpions (Eq. 1) hold deerminisically among individuals never on ART (eart = 0). Likewise, ever ART use (eart = 1) implies prior diagnosis (pdx = 1) and an HIV-posiive saus (Y = 1). Finally, having a viral load measured a he CHC/racking T sv L = 1 implies ha he individual was known o be HIV-posiive and aended he CHC/racking = 1. Therefore, our arge parameer is idenified 1 as P(Z = 1) = P(Supp = 1 T sv L = 1, eart = 1, ō 1, b) P(eART = 1, ō 1, b) b,ō 1 = P(eART = 1) P(Supp = 1 T sv L = 1, eart = 1, ō 1, b) b,ō 1 P(ō 1, b eart = 1). 1 Our idenifiabiliy assumpions reduce o Supp T sv L eart = 1, Ō 1, B. For HIV-posiive individuals who have iniiaed ART, we assume ha wihin sraa defined by baseline demographics, pas HIV esing, and cascade hisory, suppression among hose wih a viral load measured a CHC/racking is represenaive of suppression among hose wih a missing viral load. We assume he corresponding posiiviy assumpion: P(T sv L = 1 eart = 1, Ō 1, B) > 0. We require some posiive probabiliy of having viral load measured a he CHC/racking, given he HIV-posiive individual has iniiaed ART, regardless of baseline demographics or he observed pas. 8 hp://biosas.bepress.com/ucbbiosa/paper357

11 Thus, our esimand is he proporion of individuals known o have sared ART muliplied by he adjused probabiliy of being suppressed given prior ART iniiaion. The laer is esimaed by sraifying he populaion wih prior ART iniiaion on baseline demographics and cascade hisory, assuming ha wihin each sraum he proporion suppressed among hose wih a viral load es available is represenaive of he proporion suppressed wihou a viral load es, and combining hese sraum-specific proporions ino a single sandardized esimae. Esimaion is based on a TMLE wih a inervenion node as T sv L and adjusmen se as (eart, Ō 1, B). During esimaion, we use knowledge ha he oucome is 0 if eart = 0. The esimaed number of virally suppressed HIV-posiive individuals is calculaed as he esimaed proporion suppressed imes he arge populaion size a ime The cascade and populaion-level viral suppression The previous subsecions described our procedure for obaining esimaes of 1. The proporion of he arge populaion who is HIV-posiive: P(Y = 1) 2. The proporion of he arge populaion who is HIV-posiive and previously diagnosed: P(pDx = 1, Y = 1) 3. The proporion of he arge populaion who is HIV-posiive, previously diagnosed, and ever on ART: P(eART = 1, pdx = 1, Y = 1) 4. The proporion of he arge populaion who is HIV-posiive, previously diagnosed, ever on ART, and virally suppressed: P(Supp = 1, eart = 1, pdx = 1, Y = 1). These esimaes can be ranslaed ino esimaes of he UNAIDS cascade arge by aking he following raios. Inference is obained by he Dela Mehod. Proporion of all HIV-posiive individuals who have a prior diagnosis a ime : P(pDx = 1 Y = 1) = P(pDx = 1, Y = 1) P(Y. = 1) Proporion of all HIV-posiive individuals wih a prior diagnosis who have ever sared ART a ime : P(eART = 1 pdx = 1, Y = 1) = P(eART = 1, pdx = 1, Y = 1) P(pDx = 1, Y. = 1) Proporion of all HIV-posiive individuals wih prior ART iniiaion who are virally suppressed a ime : P(Supp = 1 eart = 1, pdx = 1, Y = 1) = P(Supp = 1, eart = 1, pdx = 1, Y = 1) P(eART = 1, pdx = 1, Y. = 1) Proporion of all HIV-posiive individuals who are virally suppressed a ime : P(Supp = 1 Y 3.3 Unadjused secondary analysis = 1) = P(Supp = 1, eart = 1, pdx = 1, Y = 1) P(Y. = 1) In an unadjused analysis, we esimae each cascade sep and overall populaion-level suppression using simple empirical proporions among individuals seen a he CHC/racking wih known HIV saus ( = 1) and wih viral load measured (T sv L = 1) for he suppression oucome. We make he following assumpions and noe ha several alernaive formulaions resul in he same esimands. 9 Hosed by The Berkeley Elecronic Press

12 1. We assume HIV prevalence among hose wih known HIV saus seen a he CHC/racking is represenaive of HIV prevalence in he arge populaion: Y. 2. In hese secondary analyses, we assume ha prior diagnosis is compleely measured only among HIVposiive individuals seen a he CHC/racking. We use pdx o denoe underlying prior diagnoses and pdx = pdx as is observed analog. We assume he proporion of individuals wih a prior diagnosis seen a he CHC/racking is represenaive of he proporion of individuals wih a prior diagnosis in he arge populaion: pdx. 3. In hese secondary analyses, we assume ha prior ART use is compleely measured only among HIV-posiive individuals seen a he CHC/racking. We use eart o denoe underlying prior ART use and eart = eart as is observed analog. We assume he proporion of individuals seen a he CHC/racking who have ever used ART is represenaive of he proporion of individuals who have ever used ART in he arge populaion: eart. 4. Finally, we assume ha for HIV-posiive individuals who have iniiaed ART, suppression among hose wih viral loads measured a he CHC/racking is represenaive of suppression among hose wih missing viral load measures: Supp T sv L eart = 1. Under hese assumpions, we esimae cascade coverage and populaion-level suppression in he secondary analysis as follows. 1. Proporion of all HIV-posiive individuals who have a prior diagnosis a ime : Under hese assumpions and using ha prior diagnosis implies an HIV-posiive saus, we have ha P(pDx = 1 Y = 1) = P(pDx = 1 = 1) P(Y = 1 = 1) The righ hand side is esimaed as he empirical proporion of individuals seen a CHC/racking who have a prior diagnosis, divided by he empirical proporion of individuals seen a CHC/racking who are known o be HIV posiive. This is equivalen o he number of previously diagnosed HIV-posiive individuals aending CHC/racking, divided by he number of HIV-posiive individuals aending CHC/racking. 2. Proporion of all HIV-posiive individuals wih a prior diagnosis who have ever sared ART a ime : Under hese assumpions and using ha prior ART iniiaion implies a prior diagnosis and an HIV-posiive saus, we have P(eART = 1 pdx = 1, Y = 1) = P(eART = 1 = 1) P(pDx = 1 = 1). The righ hand side is esimaed as he empirical proporion of individuals seen a CHC/racking who have ever iniiaed ART, divided by he empirical proporion of individuals seen a CHC/racking who have a prior diagnosis. This is equivalen o he number of HIV-posiive individuals aending CHC/racking who have ever iniiaed ART, divided by he number of HIV-posiive individuals aending CHC/racking who have a prior diagnosis. 10 hp://biosas.bepress.com/ucbbiosa/paper357

13 3. Proporion of all HIV-posiive individuals wih prior ART iniiaion who are virally suppressed a ime : Under hese assumpions and using ha prior ART iniiaion implies a prior diagnosis and an HIV-posiive saus, we have P(Supp = 1 eart = 1, pdx = 1, Y = 1) = P(Supp = 1 T sv L = 1, eart = 1) = P(Supp = 1 T sv L = 1, eart = 1) (Recall having a viral load measured a he CHC/racking T sv L = 1 implies he individual was known o be HIV-posiive and aended he CHC/racking = 1.) The righ hand side is esimaed as he empirical proporion of individuals wih a measured viral load who are currenly suppressed, divided by he empirical proporion of individuals wih a measured viral load who have ever iniiaed ART. This is equivalen o he number of HIV-posiive individuals wih a measured viral load who are currenly suppressed, divided by number of HIV-posiive individuals wih a measured viral load who ever iniiaed ART. 4. Proporion of all HIV-posiive individuals who are virally suppressed a ime : As deailed in he worked example (Secion 3.6), we can idenify he proporion of all HIV-posiive who are suppressed in his secondary analysis as P(Supp = 1 Y = 1) = P(Supp = 1 T sv L = 1). The righ hand side is esimaed wih he empirical proporion of all HIV-posiive individuals wih a viral load measured a he CHC/racking who are currenly suppressed. This is equivalen o he number of HIV-posiive individuals wih observed viral suppression, divided by number of HIVposiive individuals wih a measured viral load. See Secion 3.6 for furher deails. 3.4 Sraified analysis We esimae he same parameers as in Secion 3.2, bu now wihin he following sraa defined by baseline variables included in B. 1. Sex (Male vs. Female) 2. Age (Younger: years vs. Older: > 24 years) a ime 3. Counry of residence (Uganda vs. Kenya) 3.5 Addiional sensiiviy analyses We implemen he following sensiiviy analyses: 1. Expanding he arge populaion o include non-sable residens and in-migrans 2. Including self-repor as evidence of prior diagnosis a baseline 3.6 Worked example: esimaing populaion-level viral suppression To illusrae he approach aken in he primary analysis and deailed in Secion 3.2, we provide he following worked example Targe parameer Our goal is o esimae he populaion-level viral suppression among HIV-posiive aduls: P(Supp = 1 Y = 1) = P(Supp = 1, Y = 1) P(Y. = 1) 11 Hosed by The Berkeley Elecronic Press

14 The numeraor is he join probabiliy of suppressed and being HIV-posiive (irrespecive of wheher HIV saus or viral load is measured), and he denominaor is he underlying prevalence of HIV. We firs focus on esimaing he denominaor and hen he numeraor. Finally, we combine hese esimaes o obain an esimae of populaion-level viral suppression among HIV-posiive aduls Unadjused (secondary) analysis approach Denominaor - HIV Prevalence: Our goal is o esimae populaion-level HIV prevalence: P(Y = 1). One approach would be o assume HIV prevalence is he same among hose wih known saus and hose wih unknown saus. Under his assumpion, he esimaed HIV prevalence a baseline ( = 0) is P(Y 0 = 1) = P(Y 0 = 1 T shiv 0 = 1) = # known o be HIV-posiive a baseline populaion wih known saus a baseline = = 10.3% This approach is poenially problemaic for he following reasons. Firs, he saus of an HIV-uninfeced individual can only be ascerained hrough esing a he CHC/racking, while he saus of a previously diagnosed HIV-infeced individual can be ascerained hrough healh records wihou aending he CHC/racking. This assumpion is furher problemaic a laer ime poins. For > 0, we are more likely o know a previously HIV-infeced individual s saus due o boh healh records and because he/she only needed o be esed for HIV once. In oher words, he missingness variable T shiv is a funcion of underlying HIV saus Y. Therefore, in all analyses we rely on HIV saus as ascerained a he CHC/racking. This incorporaes aendees who es for HIV (boh wih posiive and negaive resuls) and aendees who are previously diagnosed as HIV-infeced (and hus no reesed). In he secondary analyses (Secion 3.3), we assume ha he prevalence of HIV among hose esing a he CHC/racking or aending he CHC/racking wih a documened prior HIV-posiive es resul ( = 1) is represenaive of prevalence among hose no ( = 0). (We relax his assumpion below.) Wih his assumpion, we can idenify HIV prevalence as P(Y = 1) = P(Y = 1 = 1) = P(Y = 1, = 1) P( = 1) By his approach (Table 1), he esimaed baseline prevalence of HIV in SEARCH is (# HIV-posiive & seen wih known saus)/77774 (# seen wih known saus)/77774 = = 10.0%. 0 = 0 0 = 1 oal Y 0 = Y 0 = oal Table 1: Observed HIV saus by CHC/racking aendance wih known saus a baseline ( = 0) Numeraor - Suppression & HIV-posiive: Now our goal is o esimae populaion-level probabiliy of being suppressed and HIV-posiive: P(Supp = 1, Y = 1). This parameer is subjec o missing measures on boh HIV saus and viral load. To simplify noaion, le us denoe a binary indicaor he oucome of ineres as Z I(Supp = 1, Y = 1). Likewise, le us denoe is observed analog as Z I(Supp = 1, Y = 1). Analogous o Secion 3.2.4, we formulae he idenifiabiliy of he arge parameer P(Z = 1) as a longiudinal dynamic regime problem. Again, we consider a wo componen hypoheical inervenion on he missingness mechanism: 12 hp://biosas.bepress.com/ucbbiosa/paper357

15 d0: se = 1. In oher words, ensure ha all individuals aend he CHC/racking a ime and have known HIV saus. d1(y ): if (Y = 1), se T sv L = 1; else se T sv L = 0. In oher words, if an individual is known o be HIV-posiive a ime, ensure ha his or her viral load is measured. Under his hypoheical join inervenion, we would have complee measuremen of underlying HIV saus and complee measuremen of viral load among all HIV-posiive individuals. To idenify he probabiliy of suppression under such a hypoheical inervenion, suppose we are willing assume 1. HIV saus and suppression among hose seen a he CHC/racking and wih HIV saus measured is represenaive of HIV saus and suppression among hose no seen; 2. Among HIV-posiive individuals seen a he CHC/racking, suppression among hose wih viral load measured is represenaive of suppression among hose missing a viral load measure. These assumpions are equivalen o he following (sequenial) randomizaion assumpions (Robins, 1986): Z Z T sv L Y, = 1 Togeher wih he corresponding posiiviy assumpions, we have P(Z = 1) = y P(Z = 1 T sv L = d1(y ), Y = y, = 1)P(Y = y = 1) = P(Z = 1 T sv L = 1, Y = 1, = 1)P(Y = 1 = 1) = P(Supp = 1 T sv L = 1)P(Y = 1 = 1) In he second equaliy, we have used ha he oucome Z is deerminisically zero if Y = 0. In he final equaliy, we used ha he only HIV-posiive individuals have heir viral loads measured he CHC or racking. In oher words, T sv L = 1 implies (Y = 1, = 1). Therefore, our esimand for he probabiliy of being suppressed and HIV-posiive P(Supp = 1, Y = 1) is he proporion of HIV-posiive individuals wih measured suppression imes he prevalence of HIV among hose wih a known saus aending he CHC/racking. As shown in Table 2, he firs erm can be esimaed a baseline as ˆP(Supp 0 = 1 T sv L 0 = 1) = ˆP(Supp 0 = 1, T sv L 0 = 1) ˆP(T sv L 0 = 1) (# HIV-posiive w/ measured VL < 500 copies/ml) = (# HIV-posiive w/ measured VL) = = 51.2%. The second erm is our esimand for baseline prevalence. (See above.) T sv L 0 = 0 T sv L 0 = 1 oal Supp 0 = Supp 0 = oal Table 2: Table of viral load esing and observed suppression among known HIV-posiive (Y 0 = 1) a baseline ( = 0). 13 Hosed by The Berkeley Elecronic Press

16 Proporion - Suppression among HIV-posiives: Under hese assumpions, we can wrie he populaion-level probabiliy of viral suppression among HIV-posiive individuals as P(Supp = 1 Y = 1) = P(Supp = 1, Y = 1) P(Y = 1) = P(Supp = 1 T sv L = 1)P(Y = 1 = 1) P(Y = 1 = 1) = P(Supp = 1 T sv L = 1) Wih his approach, he esimaed baseline viral suppression among HIV-posiive is 51.2% Examining and relaxing assumpions As discussed above, his approach o esimae HIV prevalence (i.e. he number of HIV-posiive individuals in he populaion) and populaion-level viral suppression requires on wo poenially srong assumpions: 1. HIV prevalence among hose aending he CHC/racking wih known saus is represenaive of HIV prevalence among hose no aending; 2. For known HIV-posiive individuals, suppression among hose wih a viral load measured a he CHC/racking is represenaive of suppression among hose wihou a viral load measured. However, cerain groups may be over-represened and oher groups under-represened. If facors affecing CHC/racking aendance are also associaed wih underlying HIV saus, his will bias our esimaes of HIV prevalence. Likewise, if facors affecing viral load measuremen are also associaed wih viral suppression, his will bias our esimaes of populaion-level suppression. In his subsecion, we relax hese assumpions by assuming hey hold wihin a binary sraa. In he following Secion 3.6.4, we presen our fully adjused approach, corresponding o he primary analysis. Denominaor - HIV Prevalence: Suppose ha we assume wihin counry, HIV prevalence among hose esing a he baseline CHC/racking or aending he baseline CHC/racking wih a documened prior HIV-posiive es resul ( = 1) is represenaive of HIV prevalence among hose no ( = 0). Then we can conrol for missing HIV ess by esimaing HIV prevalence for each counry separaely and hen sandardizing o he disribuion of hese sraa in he populaion: P(Y = 1) = P(Y = 1 = 1, counry)p(counry) counry = P(Y = 1 = 1, Ugandan)P(Ugandan) + P(Y = 1 = 1, Kenyan)P(Kenyan). Wih his approach, he esimaed baseline prevalence of HIV in SEARCH is ( ) ( ) # HIV-posiive Ugandans & seen wih known saus # Ugandans # Ugandans & seen wih known saus # oal pop ( ) ( ) # HIV-posiive Kenyans & seen wih known saus # Kenyans + # Kenyans & seen wih known saus # oal pop ( ) ( ) ( ) ( ) = + = 10.1% In his simple scenario, our esimaor (G-compuaion wih a sauraed parameric regression model) is idenical o he inverse probabiliy weighing approach (as used by BCPP in Gaolahe e al. (2016)) and o he argeed maximum likelihood approach (used in his paper). 14 hp://biosas.bepress.com/ucbbiosa/paper357

17 Numeraor - Suppression & HIV-posiive: Along wih he above assumpion on HIV prevalence, suppose ha we assume for known HIV-posiive individuals and wihin counry, suppression among hose wih a measured viral load is represenaive of suppression among hose wihou a measured viral load. Then our esimand is he sraa-specific proporion of HIV-posiive individuals wih measured suppression imes he sraa-specific prevalence of HIV and hen sandardized o he disribuion of sraa in he populaion: P(Z = 1) = P(Z = 1 T sv L = d1(y ), Y = y, = 1, counry)p(y = y = 1, counry)p(counry) counry,y = P(Z = 1 T sv L = 1, Y = 1, = 1, counry)p(y = 1 = 1, counry)p(counry) counry = counry P(Supp = 1 T sv L = 1, counry)p(y = 1 = 1, counry)p(counry) Each erm can be esimaed wih he empirical proporion (i.e. coningency ables). Proporion - Suppression among HIV-posiives: To esimae he populaion-level viral suppression among all HIV-posiive aduls, we conrol for missing HIV ess and viral loads by esimaing suppression among HIV-posiive residens for each counry separaely and hen sandardizing o he disribuion of residency: P(Supp = 1 Y = 1) = counry P(Supp = 1 Y = 1, counry)p(counry) = P(Supp = 1 T sv L = 1, Ugandan)P(Y = 1 = 1, Ugandan) P(Ugandan) P(Y = 1 = 1, Ugandan) + P(Supp = 1 T sv L = 1, Kenyan)P(Y = 1 = 1, Kenyan) P(Kenyan) P(Y = 1 = 1, Kenyan) = P(Supp = 1 T sv L = 1, Ugandan)P(Ugandan) + P(Supp = 1 T sv L = 1, Kenyan)P(Kenyan) Wih his approach, he esimaed baseline viral suppression among HIV-posiive is 48.9%: ( ) ( ) # HIV-posiive Ugandans w/ measured VL < 500 copies/ml # Ugandan # HIV-posiive Ugandans w/ measured VL # oal pop ( ) ( ) # HIV-posiive Kenyans w/ measured VL < 500 copies/ml # Kenyan + # HIV-posiive Kenyans w/ measured VL # oal pop ( ) ( ) ( ) ( ) = + = 48.9% Again in his simple scenario, our esimaor (G-compuaion wih a sauraed parameric regression model) is idenical o he inverse weighing approach (as used by BCPP in Gaolahe e al. (2016)) and o he argeed maximum likelihood approach (used in his paper) Primary analysis approach While sraifying on counry weakened our assumpions on missing HIV ess and missing viral loads, hey remain quie srong. In pracice, here are many poenial adjusmen variables ha could impac he probabiliy of being esed, underlying HIV saus, and viral suppression among HIV-posiive individuals. Examples include age, sex, occupaion, educaion, socio-economic saus, mobiliy, and communiy. As he number of poenial adjusmen variables grow (or we consider coninuous variables), he above approach based on coningency ables breaks down due o sparse cells. This problem is furher inensified afer baseline when he hisory of esing, care, or suppression could impac curren esing, HIV saus, and suppression among HIV-posiive individuals. 15 Hosed by The Berkeley Elecronic Press

18 As an alernaive, one could use inverse weighing as illusraed in Gaolahe e al. (2016). Targeed maximum likelihood esimaion (TMLE), however, is more a efficien and robus approach (van der Laan and Rose, 2011). TMLE combines esimaes of expeced oucome for poenial sraa wih an esimae of he propensiy score (quanificaion of which ypes of people more or less represened). As a resul, TMLE is a double robus; we will have a consisen esimae if eiher he sraa-specific expeced oucome is consisenly esimaed or he propensiy score is consisenly esimaed. TMLE is also a subsiuion (plug-in) esimaor providing robusness under srong confounding or rare oucomes. Furhermore, raher han rely on parameric regression models (e.g. main erms logisic regression), TMLE furher improves robusness by using he machine learning algorihm Super Learner (van der Laan e al., 2007). Super Learner uses cross-validaion (i.e. sample spliing) o build he bes weighed combinaion of esimaes from a library of candidae algorihms. Denominaor - HIV Prevalence: We refer he reader o Secion for full deails on esimaing he populaion-level HIV prevalence in he primary analysis. Our fully adjused esimae of baseline prevalence is 10.0%. Numeraor - Suppression & HIV-posiive: We refer he reader o Secion for full deails on esimaing he populaion-level probabiliy of being suppressed and HIV-posiive in he primary analysis. Our fully adjused esimae of he proporion of he populaion who are HIV-posiive and suppressed is 4.5%. Proporion - Suppression among HIV-posiives: In our primary analysis, we esimae he populaionlevel viral suppression among HIV-posiive residens by dividing of he esimaed probabiliy of being suppressed by he esimaed prevalence of HIV (Secion 3.2.5). Boh he numeraor and denominaor are fully adjused for missing ess. Wih his approach, he esimaed baseline viral suppression among HIV-posiive is 44.7% - subsanially more conservaive han he secondary analysis making sronger assumpions abou HIV esing and viral load measuremen (51.2%). 4 Analysis of he closed cohor of baseline HIV-posiive aduls Among enumeraed sable residens who are 15 years of age and known o be HIV-posiive a he close of baseline esing (n = 7108), we conduc longiudinal analyses evaluaing he change in cascade saus over ime. Specifically, we esimae he probabiliy ha such a individual falls ino one of six exhausive and muually exclusive caegories a each ime = {0, 1, 2}: 1. Dead. 2. Alive and ou-migraed. 3. Alive, no ou-migraed, and newly diagnosed. 4. Alive, no ou-migraed, previously diagnosed, bu never on ART. 5. Alive, no ou-migraed, wih previous ART iniiaion, bu no currenly virally suppressed. 6. Alive, no ou-migraed, wih previous ART iniiaion, and currenly virally suppressed. Specifically, we esimae he following quaniies: 1. Died. P(D = 1 Y 0 = 1). By definiion of he arge populaion, he probabiliy of being dead a = 0 is Ou-migraed. P(M = 1, D = 0 Y 0 = 1). By definiion of he arge populaion, he probabiliy of being ou-migraed a = 0 is hp://biosas.bepress.com/ucbbiosa/paper357

19 3. New Diagnosis: P(pDx = 0, M = 0, D = 0 Y 0 = 1). By definiion of he arge he populaion as individuals wih known baseline HIV-posiive saus, he probabiliy of being a new diagnosis for > 0 is Diagnosed, never on ART: 5. Suppression failure: 6. Suppression success: P(eART = 0, pdx = 1, M = 0, D = 0 Y 0 = 1). P(Supp = 0, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1). P(Supp = 1, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1). The only missingness in hese analyses arises from incomplee measuremen of viral loads. (Our arge populaion condiions on known baseline HIV-posiive saus, and deah and ou-migraion are no reaed as forms of missingness.) Therefore, he firs four quaniies can be esimaed as empirical proporions. As deailed below, esimaors of he probabiliy of suppression failure or success, however, mus accoun for missing viral load measures. As for he open cohor analysis, we use TMLE wih Super Learning (wih he same library and cross-validaion scheme) o adjus for he poenially informaive measuremen of viral loads. Saisical inference is based on he esimaed influence-curve, reaing he household as he uni of independence. The corresponding number of HIV-posiive individuals in each caegory is esimaed by muliplying he esimaed probabiliy imes he populaion size (n = 7108). 4.1 Suppression success and failure among baseline known HIV-posiive aduls Suppression success: We firs focus on idenificaion of he probabiliy of being alive, no ou-migraed, previously diagnosed, on ART, and currenly virally suppressed. Le We assume Z = I(Supp = 1, eart = 1, pdx = 1, M = 0, D = 0). Z T sv L eart, pdx, M, D, Ō 1, B. This assumpion and he resuling esimaion scheme can be simplified by noing he following. Firs, he oucome Z is deerminisically 0 if D = 1 or M = 1 or pdx = 0 or eart = 0. Furhermore, ever ART use (eart = 1) implies previous diagnosis (pdx = 1). Our randomizaion assumpion hus simplifies o Supp T sv L eart = 1, M = 0, D = 0, Ō 1, B. For individuals who are no dead, no ou-migraed and who have previously iniiaed ART by ime, we assume ha wihin sraa defined by baseline demographics, pas esing and cascade hisory (e.g. suppression hisory), curren suppression among individuals who have viral load measured is represenaive of curren suppression among hose who are missing viral load measures. We also assume he following posiiviy assumpion: P(T sv L = 1 eart = 1, M = 0, D = 0, Ō 1, Y 0 = 1, B) > 0. We require individuals who are baseline HIV-posiive, no dead, no ou-migraed and who have previously iniiaed ART by ime o have some posiive probabiliy of having viral load measured during 17 Hosed by The Berkeley Elecronic Press

20 CHC/racking a ime, regardless of covariae values. Wih hese assumpions, our arge parameer is idenified using he G-compuaion formula (Robins, 1986): P(Z = 1 Y 0 = 1) = l P(Z = 1 T sv L = 1, L = l, Y 0 = 1) P(L = l Y 0 = 1) = ō 1,b P(Supp = 1 T sv L = 1, eart = 1, M = 0, D = 0, ō 1, b, Y 0 = 1) P(eART = 1, M = 0, D = 0, ō 1, b Y 0 = 1) = P(Supp = 1 T sv L = 1, eart = 1, M = 0, D = 0, ō 1, b, Y 0 = 1) ō 1,b P(ō 1, b eart = 1, M = 0, D = 0, Y 0 = 1)P(eART = 1, M = 0, D = 0 Y 0 = 1). where L (eart, pdx, M, D, Ō 1, B). Esimaion is based on poin reamen TMLE wih single inervenion node as T sv L and adjusmen se as (eart, M, D, Ō 1, B). During esimaion, we use knowledge ha he join oucome Z is deerminisically zero if eart = 0 OR D = 1 OR M = 1. Suppression failure: Idenificaion and esimaion of he probabiliy of suppression failure among individuals known o be HIV-posiive a baseline can be implemened analogously. Insead, we esimae his probabiliy by using ha hese quaniies are exhausive and muually exclusive: P(Supp = 0, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1) = 1 P(D = 1 Y 0 = 1) P(M = 1, D = 0 Y 0 = 1) P(pDx = 0, M = 0, D = 0 Y 0 = 1) P(eART = 0, pdx = 1, M = 0, D = 0 Y 0 = 1) P(Supp = 1, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1) We ake he laer approach and use he Dela Mehod for inference. 4.2 Unadjused secondary analysis As a secondary analysis, we conrol for missing viral load measures by esimaing he following 7 probabiliies 1. Died: P(D = 1 Y 0 = 1) 2. Ou-migraed: P(M = 1, D = 0 Y 0 = 1) 3. New Diagnosis: P(pDx = 0, M = 0, D = 0 Y 0 = 1) 4. Diagnosed, never on ART: P(eART = 0, pdx = 1, M = 0, D = 0 Y 0 = 1) 5. On ART, bu missed viral load measure: P(T sv L = 0, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1) 6. Measured suppression failure: P(Supp = 0, T sv L = 1, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1) 7. Measured suppression success: P(Supp = 1, T sv L = 1, eart = 1, pdx = 1, M = 0, D = 0 Y 0 = 1) All seven probabiliies can be esimaed wih empirical proporions. For example, he probabiliy of dying is esimaed as he number of baseline HIV-posiive subjecs who died by ime divided by he populaion size (n = 7108). 18 hp://biosas.bepress.com/ucbbiosa/paper357

21 4.3 Sraified analysis We implemen he same esimaors, bu now wihin he following sraa defined by baseline variables included in B. 1. Sex (Male vs. Female) 2. Age (Younger: years vs. Older: > 24 years) a baseline 3. Counry of residence (Uganda vs. Kenya) 5 Closed cohor analyses reaing deah and oumigraion as righcensoring evens In he previous secion, deah and ou-migraion are reaed as oucomes. As an alernaive, we conduc complimenary longiudinal analyses wih righ-censoring a deah or ou-migraion. These analyses adjus for he poenially informaive naure of his censoring in addiion o informaive missingness of viral load measures where applicable. Firs, in a closed cohor of baseline enumeraed, sable adul residens wihou an HIV diagnosis prior o he baseline esing round, we esimae he proporion who are esed a leas once by he close of wo and hree rounds of esing ( = {1, 2}). Second, among aduls diagnosed as HIVposiive by he close of baseline esing, we esimae he proporion who have iniiaed ART by = {1, 2}. Third, among he cohor of aduls diagnosed as HIV-posiive by he close of baseline esing, we esimae he proporion virally suppressed a = {1, 2}, while accouning for missing viral load measures. We furher evaluae evoluion in pos-baseline viral suppression wihin subgroups defined by baseline cascade saus (prior diagnosis, ART use, and viral suppression). Finally, we consider univariae and mulivariae predicors of hese oucomes. Adjusmen employs TMLE wih Super Learning (as described previously). 5.1 Targe populaion and oucomes of ineres We consider he following oucomes. 1. ever.es : indicaor ha an individual has ever had an HIV es (i.e. any T shiv j = 1, j ). This is a couning process ha jumps a mos once (i.e. if ever.es j = 1, hen ever.es = 1 deerminisically for > j). If an individual has no record of having esed, he or she is assumed no o have esed. 2. eart : indicaor ha an individual has ever been on ART. This is a couning process ha jumps a mos once (i.e. if eart j = 1, hen eart = 1 deerminisically for > j). If an individual has no record of having iniiaed ART, he or she is assumed no o have iniiaed ART. 3. Supp : indicaor ha an HIV-posiive individual has achieved viral suppression a year CHC/racking (i.e. if V LV < 500 copies/ml hen Supp = 1). This variable is no a couning process and is measured only if T sv L = 1 (i.e. Supp = Supp T sv L ). We define he following oucome-specific arge populaions: We resric he arge populaion for all oucomes o baseline enumeraed sable, adul ( 15 years of age) residens. For oucome ever.es, we furher resric he arge populaion o exclude hose wih a prior diagnosis a baseline (pdx 0 = 1). For oucomes (eart, Supp ), we resric he arge populaion o known baseline HIV-posiive individuals (Y 0 = 1). 19 Hosed by The Berkeley Elecronic Press

22 5.2 Targe parameers Cascade probabiliies over ime We consider he following hypoheical inervenions of ineres. For all oucomes, we aim o evaluae he counerfacual proporion ha would have been observed in he absence of deah/ou-migraion. In oher words, we inervene o se D = 0 and M = 0. For suppression oucome Supp, we furher inervene o se T sv L = 1 o ensure ha viral load is measured a he CHC/racking a ime. We generically denoe hese inervenions of ineres as d. For each oucome, we aim o esimae is ime poin-specific mean under an inervenion o preven censoring and for suppression, o ensure viral load measuremen. Le Z refer generically o each oucome above, and le Z (d) refer o he counerfacual value of his oucome under regime d. We consider he following arge parameers: For all oucomes, we esimae he inervenion-specific mean oucome E[Z (d)] for = {1, 2}. Addiionally for suppression oucome Supp, we esimae he pos-baseline ( = {1, 2}) probabiliy of suppression wihin he following subgroups defined by baseline cascade saus. Baseline new diagnoses: Among baseline new diagnoses (i.e. hose wih no record of prior care a baseline), we esimae he overall pos-baseline suppression probabiliy E[Supp (d) pdx 0 = 0]. Baseline prior diagnosis wihou ART: Among individuals wih a record of prior diagnosis bu no record of prior ART iniiaion a baseline, we esimae he overall pos-baseline suppression probabiliy E[Supp (d) pdx 0 = 1, eart 0 = 0]. Prior ART iniiaion: Among individuals wih a record of ART iniiaion prior o baseline, we esimae he overall pos-baseline suppression probabiliy E[Supp (d) eart 0 = 1]. We furher esimae pos-baseline suppression probabiliy wihin subgroups defined by viral suppression a baseline (measured suppression success, measured suppression failure, or missing baseline viral load) Predicors of cascade failures For all oucomes a = 2, we esimae variable imporance measures on an absolue scale (saisical analog of he causal risk difference), reaing each variable in B in urn as he inervenion variable, and he remainder as he adjusmen se. We consider hypoheical inervenions as above o preven censoring for all oucomes; for Supp we consider he addiional hypoheical inervenion o ensure ha viral load is measured (T sv L = 1). 5.3 Idenificaion In he primary analyses, we assume ha wihin sraa defined by pas esing, cascade hisory, and baseline covariaes, hose who remain alive and residen in he communiy are represenaive of hose who are censored wih respec o each of he oucomes considered. Z (d) C j B, Ōj 1, C j 1 = 0, for j and = {1, 2}. 20 hp://biosas.bepress.com/ucbbiosa/paper357

23 We furher assume posiiviy; here are no sraa (defined by esing hisory, cascade hisory and baseline demographics) in which all individuals are censored: P(C j = 0 B, Ōj 1, C j 1 = 0) > 0, for j. For he suppression oucome Supp, we also assume ha wihin sraa defined by pas esing, cascade hisory (including curren ART use), and baseline demographics, suppression among individuals wih a viral load measured a CHC/racking is represenaive of suppression among hose missing a viral load measuremen: Z (d) T sv L B, Ōj 1, eart, C = 0, for = {1, 2}. Finally, we assume ha among individuals who remain alive and residen in he communiy, here are no subsraa (defined by baseline covariaes, pas esing and cascade hisory) in which all individuals are missing a viral load es: P(T sv L = 1 B, Ō 1, eart, C = 0) > Esimaors For arge parameers corresponding o he inervenion-specific mean (overall and wihin he prespecified subgroups), we implemen he following esimaors. Adjused esimaors, which conrol for informaive censoring (and informaive missing viral load for he suppression oucome) using longiudinal TMLE. Adjusmen ses are implied by independence assumpions above. Nuisance parameers are esimaed using Super Learner, wih 5-fold cross-validaion and library as specified in he open cohor analysis. Unadjused esimaors correspond o simple empirical proporions among uncensored individuals (i.e. condiioning on C = 0) as well as measured viral loads (T sv L = 1) for he suppression oucome. We esimae univariae and mulivariae predicors of each oucome an absolue risk scale (corresponding o unadjused and adjused risk differences, respecively). We consider each of he following variables included in he baseline characerisics B in urn as he predicor of ineres: Age: 4-level caegorical variable Sex Mobiliy: > 1 mo/pas year away from communiy or no Marial Saus: ever vs. never married Educaion: less han primary, primary, secondary or higher Occupaion: formal secor, high risk informal secor, low risk informal secor, oher, no job/disabled Household wealh quinile based on a principal componens analysis of a baseline household consumpion survey (Chamie e al., 2016). Using longiudinal TMLE, mulivariae predicors adjus for he remaining variables in B, informaive censoring, and informaive missing viral load measures for he suppression oucome. Again, adjusmen ses for censoring and viral load esing are implied by independence assumpions above. Nuisance parameers are esimaed using Super Learner, wih 5 fold cross-validaion and library as specified in he open cohor analysis. Univariae predicors are evaluaed condiional on being uncensored (C = 0), as well as condiional on measured viral load (T sv L = 1) for suppression oucome, and wihou adjusing for oher variables in B. Saisical inference is based on he esimaed influence-curve, reaing he household as he uni of independence. 21 Hosed by The Berkeley Elecronic Press

24 6 Acknowledgemens: Research repored in his presenaion was suppored by Division of AIDS, NIAID of he Naional Insiues of Healh under award numbers (U01AI099959, R37AI051164, R01-AI074345) and in par by he Presiden s Emergency Plan for AIDS Relief, Bill and Melinda Gaes Foundaion, and Gilead Sciences. The conen is solely he responsibiliy of he auhors and does no necessarily represen he official views of he NIH, PEPFAR, Bill and Melinda Gaes Foundaion, or Gilead. The SEARCH projec graefully acknowledges he Minisries of Healh of Uganda and Kenya, our research eam, collaboraors and advisory boards, and especially all communiies and paricipans involved. References G. Chamie, T.D. Clark, J. Kabami, K. Kadede, E. Ssemmondo, R. Seinfeld, G. Lavoy, D. Kwarisiima, N. Sang, V. Jain, H. Thirumurhy, T. Liegler, L. Balzer, e al. A hybrid mobile HIV esing approach for populaion-wide HIV esing in rural Eas Africa. Lance HIV, January, T. Gaolahe, K.E. Wirh, M.P. Holme, J. Makhema, S. Moyo, e al. Boswana s progress oward achieving he 2020 UNAIDS anireroviral herapy and virological suppression goals: a populaion-based survey. Lance HIV, Online, doi: /S (16) M.A. Hernán, E. Lanoy, D. Cosagliola, and J.M. Robins. Comparison of dynamic reamen regimes via inverse probabiliy weighing. Basic & Clinical Pharmacology & Toxicology, 98(3): , E. Polley and M. van der Laan. SuperLearner: Super Learner Predicion, URL {hp://cran. R-projec.org/package=SuperLearner}. R package version R Core Team. R: A Language and Environmen for Saisical Compuing. R Foundaion for Saisical Compuing, Vienna, Ausria, URL hp:// J.M. Robins. A new approach o causal inference in moraliy sudies wih susained exposure periods applicaion o conrol of he healhy worker survivor effec. Mahemaical Modelling, 7: , doi: / (86) J.M. Robins, L. Orellana, and A. Ronizky. Esimaion and exrapolaion of opimal reamen and esing sraegies. Saisics in Medicine, 27(23): , Joshua Schwab, Samuel Lendle, Maya Peersen, and Mark van der Laan. lmle: Longiudinal Targeed Maximum Likelihood Esimaion, URL hp://cran.r-projec.org/package=lmle. R package version UNAIDS an ambiious reamen arge o help end he AIDS epidemic, URL hp: // M. van der Laan and S. Rose. Targeed Learning: Causal Inference for Observaional and Experimenal Daa. Springer, New York Dordrech Heidelberg London, M.J. van der Laan and M.L. Peersen. Causal effec models for realisic individualized reamen and inenion o rea rules. The Inernaional Journal of Biosaisics, 3(1):Aricle 3, M.J. van der Laan and D.B. Rubin. Targeed maximum likelihood learning. The Inernaional Journal of Biosaisics, 2(1):Aricle 11, doi: / M.J. van der Laan, E.C. Polley, and A.E. Hubbard. Super learner. Saisical Applicaions in Geneics and Molecular Biology, 6(1):25, doi: / hp://biosas.bepress.com/ucbbiosa/paper357