Clustered Encouragement Designs with Individual Noncompliance: Bayesian Inference with Randomization, and Application to Advance Directive Forms.

Transcription

1 To appear in Biostatistics (with Discussion). Clustere Encouragement Designs with Iniviual Noncompliance: Bayesian Inference with Ranomization, an Application to Avance Directive Forms. CONSTANTINE E. FRANGAKIS Department of Biostatistics, The Johns Hopkins University Baltimore, MD 21205, U.S.A. DONALD B. RUBIN Department of Statistics, Science Center 709, Harvar University Cambrige, MA 02138, U.S.A. an XIAO-HUA ZHOU Division of Biostatistics, Iniana University School of Meicine Inianapolis, IN 46202, U.S.A. September 17, 2001 For corresponence: 615 N. Wolfe St., Baltimore, MD, 21205, tel: (410) ; fax: (410) ;

2 SUMMARY In many stuies comparing a new target treatment with a control target treatment, the receive treatment oes not always agree with assigne treatment that is, the compliance is imperfect. An obvious example arises when ethical or practical constraints prevent even the ranomize assignment of receipt of the new target treatment but allow the ranomize assignment of the encouragement to receive this treatment. In fact, many ranomize experiments where compliance is not enforce by the experimenter (e.g., with non-bline assignment) may be more accurately thought of as ranomize encouragement esigns. Moreover, often the assignment of encouragement is at the level of clusters (e.g., octors) where the compliance with the assignment varies across the units (e.g., patients) within clusters. We refer to such stuies as clustere encouragement esigns (CED) an they arise relatively frequently (e.g., Sommer an Zeger, 1991; McDonal, Hiu, an Tierney, 1992; Dexter et al., 1998). Here, we propose Bayesian methoology for causal inference for the effect of the new target treatment vs. the control target treatment in the ranomize CED with all-or-none compliance at the unit level, which generalizes the approach of Hirano, Imbens, Rubin, an Zhou (2000) in important an surprisingly subtle ways, to account for the clustering, which is necessary for statistical valiity. We illustrate our methos using ata from a recent stuy exploring the role of physician consulting in increasing patients completion of Avance Directive forms. Keywors: Avance Directive; Causal Inference; Clustering; Noncompliance; Phenomenological Bayesian Moel; Rubin Causal Moel. 1

3 1. INTRODUCTION AND PURPOSE Motivating stuies an ata features. When evaluating treatment options, irect assignment an enforcement of treatment receipt may not be ethical or feasible. In such cases, it is more realistic to view the esign as involving the ranomization of encouragement, as oppose to receipt, of the two target treatments, new an stanar, where in some esigns the encouragement is explicit an no enforcement is even attempte. Commonly, moreover, this encouragement is applie to clusters (e.g., physicians or villages) of subjects (e.g., patients). An example of such a clustere-encouragement-esign (CED) was reporte by Sommer an Zeger (1991) where investigators ranomize villages in Inonesia to offer or not vitamin A supplements to all their infants, but not all infants in the villages assigne to get vitamin A actually receive it. Another example of the CED was a stuy to evaluate a vaccine for influenza, where any ranomize withholing of the vaccine was consiere unethical (McDonal, Hiu, an Tierney, 1992; Hirano, Imbens, Rubin, an Zhou, 2000); for this reason, investigators ranomize physicians to receive or not receive encouragement to vaccinate their patients, but many patients of the encourage octors i not receive a flu shot, an some patients of the not-encourage octors i receive the shot. A more recent example of a CED was conucte on Avance Directive (AD) forms (Dexter et al., 1998), which are intene to be complete by patients to allow them to make early ecisions about meical treatments at the late stages of life (instructional irectives), an esignate a representative ecision maker (proxy irectives) (Wenger et al., 1993). Dexter et al. (1998) ranomize physicians to receive or not receive encouragement to iscuss AD with patients; the outcome was patient completion of AD, an the original stuy aresse the effect of encouragement on AD completion (Dexter et al., 1998). For our purpose, however, an equally important substantive research goal is to assess the effect of physicians iscussion of AD as the new target treatment for potentially increasing patients completion rates of the forms relative to the control target treatment of no such iscussion (e.g., Miles et al., 1996). 2

4 Generally, CED stuies share two specific ata-structure aspects. First, there is frequent noncompliance of iniviual subjects - not clusters for the new target treatments within ranomize encouragement arms. Secon, the istribution of noncompliance an outcomes frequently varies within an between clusters, which are the units of ranomization, rather than the iniviual subjects. We consier CED stuies where the compliance for target treatments is by efinition (or for practical purposes) all or none (for extensions, see Sec. 5). This type of noncompliance means that there exist, essentially, two subgroups of patients who are not fully ientifiable from the ata: those who woul not change their actual behavior concerning the target treatment no matter what their physician s assignment the noncompliers, an those who woul comply uner both assignment the compliers (e.g., Imbens an Rubin, 1994; Baker an Lineman, 1994; Angrist, Imbens, an Rubin, 1996; Baker, 1998; Frangakis an Rubin, 1999). These efinitions are local to this particular stuy an o not suggest compliance or not in other stuies. An intention-to-treat (ITT) analysis is especially appropriate when the ranomize intervention is the scientific intervention - the target treatment of interest. However, the CED uses ranomize encouragement only as a surrogate to inuce the new target treatment, an ITT analysis is not as appropriate for two reasons. First, the noncompliers arguably o not carry information about the comparison between the target treatments (e.g., biological efficacy or sie effects) because, by efinition of this group, the ranomization cannot change receipt of target treatment; for relevant iscussion between explanatory an pragmatic comparisons, see Sheiner an Rubin (1996) an Armitage (1998). Secon, the noncompliers may experience effects of encouragement. For example, in the stuy on flu shots (McDonal, Hiu, an Tierney, 1992), it is possible that, for noncomplying physician-patient pairs, the encouragement has triggere physicians to suggest to their patients a number of other precautions against flu in aition to vaccination, an which might not have been taken in the absence of the encouragement; These are pure encouragement effects that confoun the effect of vaccination if the noncompliers are inclue in the ITT analysis. 3

5 The secon aspect common in CED stuies, the clustere structure of units, also has methoological implications. Because the assigne encouragement is at the cluster level, assignment is ignorable (Rubin, 1978) only conitionally on the clusters. An, because noncompliance an outcomes can vary both within an between clusters, the interactions between clustering, noncompliance an outcomes nee to be aresse Aressing clustering with noncompliance at the iniviual level. The problem of noncompliance has receive increasing attention recently. In particular, it is now generally recognize that the approach of focusing on the compliers, who are not generally fully ientifiable from observe ata (e.g., Sommer an Zeger, 1991; Baker an Lineman, 1994; Angrist et al., 1996), is critically ifferent from approaches that use the treatment actually receive as if it were ranomize, such as as-treate or per-protocol approaches, whose bias has been well ocumente (e.g., Rubin, 1991; Mark an Robins, 1993; Robins an Greenlan, 1994; Sheiner an Rubin, 1996). Moreover, implicit assumptions in the stanar instrumental variables analyses, such as the a priori exclusion of effects of assignment, have now formal expressions (Angrist et al., 1996), thereby allowing researchers to avoi such exclusion assumptions when they are not plausible. In relate work without the exclusion restriction, Robins (1989) erive estimate bouns for treatment effects, Imbens an Rubin (1997a) evelope an appropriate Bayesian approach for istinct patient-physician pairs, an, for the latter case, recently, Hirano et al. (2000) have moele covariate information. Research on clustere ata, on the other han, has a long history in interconnecte literatures incluing: survey methoology, ating back at least to Neyman (1934), an Hansen an Hurwitz (1943); ranom effects, ating to Hartley an Rao (1967), Harville (1976) an Lair an Ware (1982); estimating equations methos (e.g., Liang an Zeger, 1986); hierarchical Bayesian an Empirical Bayesian methos ating to James an Stein (1961), Efron an Morris (1973), Rubin (1981), an others. In more recent work on clustere ranomization with noncompliance, Frangakis, Rubin, an Zhou (1998) relaxe the exclusion restrictions but offere 4

6 " limite information on the role of covariates an on the influence of prior istributions, whereas Korhonen et al. (2000) focuse on analyses uner the exclusion restrictions. Here, we investigate the broaer combine problem of clustere encouragement followe by iniviual noncompliance, an thereby propose general methoology for causal inference in stuies where these two ata structures are present together, an where structural exclusion restrictions are relaxe. In the next section we introuce notation an formalize our goal. In Sec. 3 we iscuss moels an methoology: within an abstract phenomenological Bayesian moel (Rubin, 1978), we introuce an appropriate framework for causal inference with clustere ata suffering from noncompliance. We iscuss the critical role of clustering an covariates, an propose a flexible submoel. In Sec. 4 we illustrate our methos by analyzing ata from the stuy on AD forms. The final section offers concluing remarks. The appenix gives etails on our moels. 2. CLUSTERED ENCOURAGEMENT DESIGN 2 1. Setting. Consier a hospital serving a group of patients,, the th patient with physician, where, so that each physician may serve more than one patient. In orer to compare two target treatments, a new one versus a control one, assume that the hospital consiers two possible actions for each physician: (i) encouraging the physician to aminister the new target treatment, an (ii) no encouragement. In either case, however, patients within a physician may not comply with their physician s assignment. To allow for this, we aopt the formulation of Angrist et al. (1996) for all-or-none compliance, although we iscuss extensions in Sec.5. We assume that patient will either receive the new target treatment, inicate by, where, or the control one,! for encouragement for the new target treatment an, when is assigne action otherwise. Similarly let # be patient s outcome of interest, e.g., occurrence/absence of the isease, when physician $ is assigne action. Covariate information about patient an physician % is collectively 5

7 " enote by a & -imensional vector that inclues the vector of ones to allow for an intercept. Note that if compliance behavior were always the same within physicians, it woul be more relevant to formulate the problem with the physicians as units efining variables, in aition to their being units of esign. Assume, for simplicity, that physicians assignments are ecie by complete ranomization, where we let ( (* + ) if patient s physician,, is ranomize to encouragement, an otherwise, an note that, since ranomization is in clusters, (, (.- whenever The Bayesian analysis is unchange if the ranomization epene on fully observe covariates that escribe physicians or their patients. Finally, note that only the values " " an ( 7 are observe; the values uner the alternative assignment, " /: (= are missing. /-. 6 (* an ;: ( < 2 2. Compliance principal strata. Each patient belongs to one of the following four groups: >? )@ by 6 6 AB target treatment uner both assignments of physician 6 ; >C ED for a complier, efine, that is, a patient who woul comply with respect to the for a never-taker, that is, a patient who, in this stuy, woul never take the new target treatment no matter the physician s assignment, so that A ; >C GF always take the new target treatment, so that woul act opposite to the assignment, so that for an always-taker, one who, in this stuy, woul B ; an >H JI for a efier, one who (e.g., Imbens an Rubin, 1994; Pearl, 1994; Baker an Lineman, 1994; Angrist et al., 1996; Baker, 1998; Barnar et al., 2001). Because membership to those four strata, unlike membership to observe compliance strata, is unaffecte by encouragement, we call them compliance principal strata (for manifestations of principal strata in more general settings, see Frangakis an Rubin, 2000). Although the encouragement to use the new target treatment may or may not inuce its actual receipt for some patients, we assume that, in this context, the encouragement woul not reverse a ecision to take the new target treatment. This assumption was terme monotonicity 6

8 X c " L by Imbens an Angrist (1994), an allows only for the first three principal strata, that is, efiers are not allowe Goal. By efinition, the principal strata of noncompliers the never-takers an always-takers, are those for whom ifferent assignment in this experiment woul not change their behavior with respect to the target treatments, an therefore those groups are not relevant for comparing the target treatments (e.g., Sommer an Zeger, 1991; Sheiner an Rubin, 1996). Moreover, if the ranomize treatment is encouragement for vaccination, then other effects may be present if the encourage physicians suggest to patients alternative preventive measures, such as recommenations to reuce the patients exposure, prescriptions of other meicines, or earlier taking of the vaccine. Such pure encouragement effects are arguably more ominant for noncompliers than for compliers, because compliers experience both the effect of encouragement an the effect of the target treatment. Also, iscerning pure encouragement effects among compliers requires assumptions not verifiable even with knowlege of all memberships to the compliance principal strata in the stuy. Nevertheless, because any effects of assignment (/ on outcome " for noncompliers must be ue to sources other than the target treatment, such effects for these patients can be remove simply by focusing analyses on the principal stratum of compliers. For these reasons, we istinguish between the effect of encouragement on outcomes among noncompliers an among compliers. Using notation consistent with Imbens an Rubin (1997a), we let K ML TS NB 5 >C LPO, the subset of subjects with compliance status L be the number of patients in that group. The stanar ITT estiman U equals the mixture U SY[Z]\^ _B^ `[a bs ITTc Se ITTc Sgf Se YRh Se, where : " +@RFiD= Q@RD= " V or F, an let are the average effects of assignment within compliance principal strata. In the remainer of our : " 7WMX (2 1) 7

9 l l w " w iscussion, we focus on estimating the principal strata-specific ITT effects (2 1), an, in particular, the ITT effect among the compliers, ITTc \j, the complier average causal effect (CACE). 3. MODELING IN THE CED 3 1. Role of clustering in the phenomenological Bayesian approach. All potentially observable ata can be expresse by the matrix k l m n poqpr, whose th row is the vector " (=. Here, enotes the pair of potential receipts 6 6 " for patient, an is the pair of potential outcomes ". Although the quantities l m n po in k may be assume fixe over hypothetical replications of the experiment (as in a permutation-base analysis, Rubin, 1990), we will more generally allow them to be consiere ranom, with a istribution pr l. The joint probability istribution of k an the clustere ranomization, pr rts that is inuce by the istribution pr l m n Ao m n po m n po, will still be enote by pr. We assume that the matrix l m n po contains all the information on observable ata an on the esign, so that the rows of k are exchangeable (Rubin, 1978). Following results on exchangeability (e Finetti, 1974), we may write essentially without loss of generality, write prl m n po for some istributions pr QuJv an pr " N pr " 7 s w O pr I#wx (3 1) s w w, where can be thought of as representing the characteristics of a larger reference population from which the stuy units, physicians an patients, are rawn. We stress that, although the clustere ranomization on o is likely to give less precise estimates of effects than a complete ranomization at the patient level, the clustere ranomization oes not affect the joint exchangeability in (3 1). Rather, the clustere ranomization is relate to the assignment mechanism: because the assignment is at the level of physicians, o, we have that pr rts m n Ao pr rtsyo, so that assignment is ignorable (Rubin, 1976, 1978) with 8

10 5 " m m L : " L but not without conitioning on o. Consequently, inference on the potential outcomes also nees to be conitional on o, which, in this case, is expecte to reuce precision because no physician has patients in both assignment arms. If the compliance principal strata >z " an the potential outcomes # were known for Se all an, then the principal strata-specific effects ITTc coul be compute from efinition (2 1). Although the values l9{} [~ N O m an { [~ N " O are known, the values lƒ N ~ O m an $ N " ~ O are unknown (efine at the en of Section 2.1). From (3 1), the missing values $ ~ l $ ~ have a joint posterior preictive istribution pr $ ~ s l $ ~ k {} [~ w prl $ ~ w s k { [~ (3 2) where k9{} [~ 5 estimans ITTc Se l {} [~ m {} [~ n poqpr, the observe ata. Then, Bayesian inference on the follows from their posterior preictive istributions inuce by (3 2) Role of covariates for relaxing exclusion restrictions. Because membership to the compliance principal strata is not fully ientifiable from observe ata, it is important to unerstan how information on the principal strata-specific causal effects is recoverable. " For simplicity, assume # is binary, e.g., 1 for occurrence of isease, 0 otherwise, an S^ ˆ pr Šs > w pr Šs > w, the average causal w in the reference population efine by (3 1) \^ ˆ let ITTc within compliance principal stratum L (unconitionally on physicians). The estiman ITT c is estimable consistently, as grows, uner the so-calle exclusion restrictions (e.g., Bloom, 1984; Sommer an Zeger, 1991; Angrist et al. 1996). However, for the reasons in Sec. 1.1, we o not wish to impose a priori these S^ ˆ assumptions for the CED. In the absence of exclusion restrictions an of covariates, the effects ITTc are not consistently estimable, but bouns are (e.g., Robins, 1989; Manski, 1990; Balke an Pearl, 1997) uner the stanar asymptotics with sample size growing. Never- 9

11 L theless, when covariates are available, Frangakis (1999, PhD thesis) shows that the preictive moel pr >H s % w of compliance principal strata from the covariates is estimable with no exclusion restrictions, an that for the effects ITT c covariates are use than if they are not use, so that ITTc S^ ˆ, asymptotic bouns are narrower if the S^ ˆ are consistently estimable uner an asymptotic argument that has the number of covariates growing in an appropriate sequence in aition to sample size. In practice, the above iscussion oes not change the way Bayesian inference is rawn, but helps critically in unerstaning the role of covariates in recovering information. Uner (3 1), Bayesian inference is base irectly on the posterior preictive istribution of the target estimans ITTc Se inuce by (3 2), where (i) in the absence of covariates, the sprea of the posterior istribution will reflect the relatively large uncertainty in preicting compliance principal strata, whereas (ii) when a covariate that preicts compliance principal strata is moele, the sprea of the posterior istribution of the estimans will be narrower. In the remaining part of this paper, we escribe a specific Bayesian moel an apply it to the stuy of AD introuce in Sec Asymptotic inference using estimate covariate-ajuste bouns an comparisons with small sample Bayesian inference will be iscusse in etail in a subsequent article. We moel the joint istribution pr " 3 3. Specific moel. $ w of (3 1) by a sequence of conitional istributions. First, because we focus on the finite stuy estimans ITT c \7 an, because an $ are known for all, we take the marginal istribution of an to be the observe n po istribution. Conitionally on, we assume the following structure for the other moel components. Compliance Principal Strata. Define the vector Œ to be an Ž subset of that inclues the vector of ones but exclues characteristics that are constant across patients clustere within physician, because those are 10

12 " L ` : L aliase with the intercept for that physician. We moel the compliance principal strata with two probit submoels, pr >H pr >H +DCs $ $ w z: QNŠ?: < Œ < pr >H +DCs $ w O NŠ/: (C,2) < (C,2) Œ < O (3 3). In these expres- an pr >C sions, +Fšs $ w (C,2) are &œž *: pr >C +@ s w pr >C +DCs $ w parameter vectors that moel the association between compliance principal strata an covariates. The corresponing parameters š š are Ž (C,2) vectors specific to physician, an these moel the association between compliance principal strata an physicians. The function 4 is the stanar normal cumulative istribution. To facilitate computations later, we augment the above moel with two latent variables for each person, ž, an Ÿ, efine via the relations >C >C D +@ if > if > ƒ Œ š žx A ` \ (C,2) an > < (C,2) Œ < š Ÿ A an where ži. AB an Ÿ š AB inepenently. Potential Outcomes. For our application in Sec. 4 we have binary potential outcomes so we posit the probit moel, for L pr G@RD=F # an Šs > respectively; is a& Ž «AB w. Here, ce tn parameter vector; an cª ce cª Œ L š O are link vector functions of imensions &. are Ž $. This formulation assumes, for parsimony, that the secon term c (3 4) R, parameters specific to physician in (3 4) is not a function of assignment arm. As note by a referee, such issue of parsimony also arises when moelling many covariates. 11

13 " Also, we assume that, given the covariates, physician inicators, compliance principal strata, an parameters, the two potential outcomes " an " are inepenent. This moel, say M-in, can be extene to a moel, say M-ep, to allow conitional association between the two potential outcomes, but the parameters of such a moel that control that association, e.g., the partial correlation between " " an, o not appear in the observe ata likeli- S^ ˆ hoo of moel M-ep given the parameters, or in the estimans ITTc in the larger reference population, regarless of the assume or the true population istribution. For these reasons, the posterior istribution for these estimans base on moel M-in is ientical to the posterior istribution base on the moels M-ep; for etails, see Imbens an Rubin, 1997a, Sec. 3, paragraphs 6-7. For the finite population estimans, ITT c Se, there will generally be some sensitivity of the posterior istribution to the choice of moel M-ep, which, to focus on main points here, can be explore in applications. As with the moel for compliance, to facilitate computations, we augment the outcome moel as follows: # if " # c >C # c Œ >C ± #,² where, for each, ± # AB inepenently of Ÿ* our assumption of conitional inepenence between " an " at this stage implies inepenence between ± an ± ž³ for %. Furthermore, (computations nee only consier the term ± ± (* ; see appenix, item 4). We emphasize that this inepenence is not relate to inepenence (or epenence) between outcome an compliance principal strata. Parameters. w N The parameter inclues µ (C,2)µ µ (C,2) µ c 2 O, where c 2 is a variance-covariance matrix parameter efine below. We choose the prior istributions for w to be proper but iffuse, in orer to ensure proper posterior istributions an relatively fast convergence of the fitting algorithms, but, at the same time, to be relatively noninformative for 12

14 5 5 L our application (Sec. 4). Given the physician-specific parameters, we posit prior istributions ¹ Iº (C,2) A Iº A Iº (3 6) inepenently. Here, I is the ientity matrix, an º is an inflating factor, which is set by the analyst. The component of ¹ :t that correspons to the intercept is set to» ¼ º an the remaining components are set to 0 to represent a prior proportion of approximately 33% for each of the three compliance principal strata. The physician-specific parameters are assume ranom with (C,2) s c 2 inepenently across physicians ƒ½ A c 2. In our application of Sec. 4 we ha a relatively low proportion of always-takers, so we constraine c 2 (3 7) so that an (C,2) were a priori inepenent. For the other components of c 2, we assume that the inverses of c var (C,2) an c var a priori have Wishart istributions with scale 4 7 matrices M¾ c c À an M¾ cª cª À, respectively, an egrees of freeom, ¾ ce an ¾ cª respectively, equal to the imensions of (C,2) an (in orer to introuce relatively little prior information, e.g., Wakefiel et al., 1994; for c an c see appenix), although it woul also be interesting to stuy more informative priors istributions. Informative priors istributions can be fitte, for example, using the same moel but after introucing pseuo-subjects in the ataset (e.g., Hirano, et al., 2000), where we can express the prior information by attaching weights to the pseuo-subjects contribution in the likelihoo base on the configuration in their covariates, cluster inicators, compliance principal strata, assignment, an outcomes. Using moel (3 3)-(3 7), inference for the estimans of interest, ITT c Se Á@ F D, in the finite stuy population is base on (3 2). The appenix outlines an algorithm for simulating these istributions for our moels. 13

15 4. APPLICATION TO ADVANCE DIRECTIVES 4 1. Proceures. We now return to the stuy of AD forms introuce in Sec AD forms are esigne to increase patient s autonomy, an although they enjoy some support by ethicists an physicians (Hughes an Singer, 1992), few patients complete them in practice, an few physicians iscuss the role of these forms with their patients. It has been hypothesize that if physicians briefly iscusse the role of AD forms with their patients, this woul cause completion rates to substantially increase (Miles et al. 1996). The iscussion effect is very important because, if shown large, coul help convince physicians to spen the brief time neee to iscuss AD forms with even more of their eligible patients. So, our goal is to aress this hypothesis with a vali analysis of a CED. The problem of more flexible alternative esigns in this application is stuie by Frangakis an Baker (2001). The ata we use are a subset from the stuy on AD forms by Dexter et al. (1998), who analyze the ata by ITT analyses. In that stuy the researchers ranomly ivie eligible physicians of an urban hospital into four groups: one group routinely receive computer reminers to iscuss instructional irectives with their patients; another group receive reminers on proxy irectives; a thir group receive both reminers, an a fourth group receive no reminers. Here, the subset of ata is from the extreme arms: the control group, enote by (, an the group receiving reminers for both AD forms, enote by ( Â, where only patients eligible for AD iscussion an completion were inclue. To use these ata in our framework, let 0243 be the inicator for actual iscussion of AD, equal to 1 if patient s physician iscusses either of the two AD forms with patient (the new target treatment), 0 otherwise (the control target treatment), an let " 0243 be the inicator for completion of AD, equal to 1 if the patient completes either AD form, 0 otherwise. Patient age ata are also of interest because age has been previously shown to be associate with iscussions of AD forms (Duffiel an Pozamsky, 1996; Boy et al., 1996; Hakim et al., 1996). Moreover, because in a preliminary 14

16 analysis, we foun none of the available covariates other than age to be as useful in preicting compliance status, here we use age as the only covariate. Table 1 gives some basic characteristics of our ata that can be summarize easily using theory for finite population cluster sampling (Cochran, 1963). Table 1 about here Table 1 shows that 3% of patients complete AD forms uner control ( when their physicians were encourage to iscuss them ( à versus 14%. However, approximately three quarters of the physician-patient pairs i not iscuss the forms when encourage, an so, following Sec. 2.2, they must be never-iscussants in the sense that they woul not iscuss the forms whether encourage or not in this stuy. Analogously, approximately 5% of the pairs are always-iscussants. The estimate remaining approximately one fifth of physician-patient who are neither always-iscussants nor never-iscussants are iscussion-compliers, in the sense that they woul iscuss AD if an only if encourage. Therefore, although the moest 11 % ITT effect of encouragement on completion rates may suggest to physicians that iscussing AD forms with their patients has no practical effect, the majority of physician-patient counte in that estimate are not relevant to the effect of AD iscussion on AD completion. That is, evience for the effect of AD iscussion on AD completion nees to be sought among iscussion compliers. Generally, it is not known whether physicians or patients initiate the iscussions. Nevertheless, focusing on the iscussion-complier pairs almost ensures that the iscussions uner encouragement are initiate by physicians, because we know they woul not have occurre without encouraging the physicians. Therefore, we focus on estimating the effect of encouragement on completion among the subset of iscussion-compliers, ITT c effect of iscussion on form completion. To estimate ITT c moels of the form (3 3)-(3 7). \7 \j, which, here, we call the, we use two proceures base on The first moel-base proceure uses the specifications (3 3)-(3 7) with no covariates. For this moel, an Œ are 1. The outcome link function c saturates the patterns of compliance principal strata by treatment arm: ce 15 >C # V Ä >C ÁD ;Å? Ä >C ÁD ;Åb :

17 L # Ä >C +@ function c šå* Œ Ä >C +@ šåz,: >C to be V Ä >C # D Ä >C +F šå= Ä >C +F ÆÅ/,: #7W. We take the vector Ä >C +@ Ä >C +F 7W. The hyperparameter º in (3 6) is set here to 5, although other values were also trie (see also Sec. 4.3). The secon proceure uses patients age in the moel components (3 3)-(3 7). Specifically, for this moel, both an Œ log-log scale. The link function ce are set to V age W, where age is the patient s age in stanarize now fits separate intercepts an slopes on age for each of the six combinations of compliance principal strata >? crosse by assignment arm. The link function cª fits separate intercepts an slopes on age for each compliance principal strata. The values of the hyperparameters are as in the moel without age. For further etails on the fitting methos, see the appenix Results. Table 2 about here In Table 2, we report estimates of the estimans: compliance principal strata specific percentages of AD completion uner control, uner encouragement, an the between-arm (encouragement - control) ifference ITTc Se Ç@RD=F using the two moel-base proceures. For each moel, we obtaine the posterior preictive istributions of these estimans as inuce by (3 2). The table summarizes these istributions by means (within treatment arms, the posterior preictive means are the posterior preictive probabilities of AD completion average over È w an from its posterior istribution), stanar eviations, an 95% intervals (2.5% an 97.5% quantiles). The moel that accounts for clustering but oes not moel age gives, mostly, unhelpfully broa answers, consistent with the role of clustering an covariates iscusse, respectively, in Secs. 3.1 an 3.2. An exception in this ataset is inference for the never-iscussants, which occurs because the observe completion rate for the non-iscussants uner control arm (Table 1) is zero. This is, generally, a mixture of completion rates of iscussion-compliers an neveriscussants assigne control, an therefore, here, both of these rates are zero. However, most 16

18 " L s L w of the other 95% intervals, incluing for the effect on iscussion-compliers, are too wie for practical use or interpretation with this moel. In contrast, the moel that uses age in both the compliance an the outcome components provies 95% posterior intervals that are quite usefully narrower than those of the moel without age. In particular, among compliers, the effect of assignment on completion rates has a posterior mean of 62% an is most likely at least 34% (2.5% posterior quantile). These estimates are also substantially higher than those reporte by the ITT analyses in Table 1, an reflect in a principle way the uncertainty from the ifferent sources of missing information. The other results are generally not surprising, except perhaps the negative estimates for the effect of encouragement among always-iscussants. Nevertheless, because the zero effect is well within the posterior interval in both moels, this result is consistent with ranom fluctuation. Moreover, because in this application the proportion of always-takers is low, imposing a priori the exclusion restriction woul not substantially change the results. In other applications where this assumption woul be plausible an with larger proportions of always-takers, it woul be beneficial to formulate this assumption explicitly in the moel. Table 3 about here It is relevant to check the extent to which the increase precision of the secon moel-base proceure can be attribute to information in the ata, incluing the ability of age to preict compliance principal strata, or information supplie by the prior istribution. In Table 3 we report the probabilities of completion rates uner control, encouragement an their ifference as inuce solely by the prior istributions for each moel. For example, to get a raw " from the prior istribution for compliers, we (i) raw a subject s $ from pr (by assumption here, the observe istribution), (ii) raw from the efining moel pr w w s $ p pr, D=F (iii) raw compliance principal stratum >/ from the moel pr >, $ of (3 3), an (iv) if the stratum is complier, we calculate pr ÁŠs!@ w >C $ from (3 4) an, with this probability, raw a Bernoulli outcome ". The istributions for the other entries are erive analogously. None of these istributions appears particularly informative for 17

19 the estimans summarize in Table 2, suggesting that the increase precision in the moel with age in Table 2 is not particularly influence by the chosen prior istributions. Moreover, the posterior istributions of the moel parameters showe evience that the probability of being a never-iscussant ecrease with age (mean, [2.5%, 97.5%] quantiles for _ÉÊ (C,2) : [0.025, 0.094]), an most of the shift was to being a complier ( : [ ,-0.122]). We also calculate the posterior preictive istribution of the probability that the next new patient seen has each compliance principal stratum as a function of that patient s age. Means an stanar eviations from this istribution conitionally on age are isplaye in Fig. 1. Each raw from this istribution is obtaine as the efining probability of compliance (C,2) principal stratum in (3 3), where an are rawn from their joint posterior istribution, an (C,2) an are rawn from their efining moel (3 7) after having rawn c 2 from its posterior istribution. _ÉPÊ Figure 1 about here Base on these results, the estimates obtaine from the secon moel-base proceure are expecte to be more appropriate for the compliance principal stratum-specific effects than either the estimate bouns or the first moel estimates. Taking the effect of encouragement on the iscussion-compliers to be the relevant estiman for the effect of AD iscussions on AD completion, these results give support to, an consierably strengthen, the hypothesis that physician-iscussion can substantially increase patient-completion of AD forms Other moels. More than twenty other moels of the form (3 3)-(3 7) were fit, were we varie the link functions ce an c, the functional forms for age, an the inflating factor º in (3 6). In aition, in a preliminary effort in this work (Frangakis et al., 1998), we ha also trie logistic mixe effects analogues to the probit moels reporte here. Those moels gave results mostly similar to the ones presente in Table 2 here (see, for example, Table 2 of Frangakis et al., 1998), 18

20 although with varying performance in the criteria of (i) egree in which the prior istributions influence the results in the sense of the measures in Table 3; an (ii) convergence iagnostics. For example, in contrast to the moels in Sec. 3.3, the simulation stage for the logistic mixe effects moel, after incorporation of Metropolis-Hastings ajustments to aress lack of conjugacy, i not pass the convergence iagnostics of Gelman an Rubin (1992) (see also appenix) within a satisfactory time. Among the moels we trie, the two reporte in the previous section were the most acceptable with respect to these two criteria. We i not consier moels with resiual epenence between the potential outcomes conitionally on stage (3 4), or with assignment-arm ifferences in the term cª for physician-specific ranom parameters. 5. REMARKS We escribe methoology for causal inference in stuies with ranomization in clusters but noncompliance at the iniviual level. The propose metho improves upon current proceures, which face limitations with respect to either valiity or precision in estimation. Our proceure is esigne to be Bayesian for the input prior. In general, of course, a posterior interval oes not automatically share the property of a confience interval just as the latter oes not automatically share the property of the former. It woul be interesting to stuy frequency calibration of Bayesian proceures in this problem, for example, by asymptotics that allow information from covariates to grow with samples size, as mentione in Sec. 3.2, or by mixing permutation istributions with the Bayesian moel (e.g., Rubin, 1998). Our methos assume all-or-none observe compliance. For situations where observe compliance is continuous or multilevel, an approach that woul iscretize compliance to two (or few) levels can still be practically useful, as is often common practice with continuous variables (e.g., age simplifie to young/ol) where appropriate. Alternatively, irect moeling of the multiple compliance principal strata can be one by using appropriate assumptions on the parameters. For an example, suppose that from a stuy s continuous compliance measure we create 19

21 " " " " Ë0243 to have three orere levels, labele 2, 1, an 0, roughly representing, respectively, full ose, half ose, an, no ose of the new treatment. Then, there are generally 9 principal strata 6 6. Suppose further that the behavioral or pharmacological context of that stuy makes the following three assumptions plausible. First, the exclusion restriction hols, i.e., " " if 6 6. Secon, encouragement to the new treatment increases receipt of ose of the new treatment both within an across subjects in the following sense: Multilevel Monotonicity. (a) (within subjects): 6, an (b) (across subjects): If,Ì -# then -Š. Multilevel monotonicity allows for a total of 5 principal strata, three with treatment receipt unchangeable by encouragement ( =(0,0), (1,1), or (2,2)), an two with treatment receipt increasing by 1 ose with encouragement ( =(0,1) or (1,2)). A thir assumption, then, can be that for the last two groups, the effect of encouragement on outcomes is the same, e.g., in the scale of relative risk: pr pr ÍŠs ÍŠs AB AB pr pr Šs Šs Î Î Then it can be proven that, uner the above conitions, the causal effect, as well as all other unknown components of the istributions, are consistently estimable. As in Sec. 3.2, some of the assumptions can be relaxe with the help of covariates or by substitution with alternative assumptions. It is then relevant to explore the plausibility of such assumptions in the stuy context, an compare results to alternative methos. Details an applications of this approach to multilevel compliance will be iscusse in the future. More generally, it is precisely the emphasis of this approach on the existence of principal strata of compliance that allows the researcher to input into analyses explicit scientific assumptions. 20

22 N N ` " ACKNOWLEDGEMENT The authors thank the Eitors, an Associate Eitor, an two reviewers for penetrating comments. A MODEL FITTING Computation of the posterior istribution (3 2) of the missing compliance principal strata, say, Ï $ ~, missing potential outcomes m $ ~, an parameters w were base on simulations from a Gibbs sampler (Geman an Geman, 1984), which raws, in this orer: the missing compliance principal strata Ï $ ~ ; the missing potential outcomes m $ ~ ; the latent variables > the current set of never-takers, compliers, an always-takers; the latent variables " an > \ for for the outcome moel; the parameters (C,2) for the compliance moel; the outcome moel parameters ; the cluster-specific parameters matrices c c (C,2) (=, for ; an the variance. For all steps, rawing is one cyclically an each step conitions on all other unknowns, with the following exceptions: the first step must exclue > ` an > \ from the conitioning to allow the raws of Ï ~ to vary over their sample space; also, at this step, the conitional istribution on " 0]243 conitional istribution on " is relatively easy to simulate from, an, so, replaces the for algorithmic efficiency; an, the potential outcomes m $ ~, rawn at step 2 to calculate the estimans (2 1), are not inclue in any other conitioning, for algorithmic efficiency. The istributions involve in the Gibbs sampler are as follows. 1. Any missing compliance principal stratum is rawn at this step from pr > w (*. This istribution is obtaine from the joint istribution pr " >t For example, a subject with ( Ë0243 Í conitional Bernoulli istribution of >/ is proportional to 0243 s " 0243 g0] s w (= can be either a complier or a never-taker, an the. ( " 0243 w O}Ð cñ Ò \7 K D= ( " 0]243 w O}Ð cñ MÒ `[ 21

23 > " L ` L N " " " \ ¹ " L L where we efine K ML ¹ ¹ Ó ¹ ¹ Ô ¹ w to be pr > ¹ s +Ó ¹ + ¹ w pr +Ô ¹ s > ¹ Ó ¹ + ¹ w Therefore, the conitional probability of the subject being a complier at this step is (* " 0243 w (= $ " 0]243 w K D= (* " 0243 w O À The rawing of >, for subjects with (; 2. The missing potential outcome, " 8 3 Bernoulli istribution with probability 0243 " Õ: is one in a similar way. ( <, of each person is rawn from the pr [Öe Šs >C $ " /: Ö w pr [Öe ÍŠs >C w that is, the efining moel (3 4), where Ö /: is set to (. ` 3. The rawing of > is from pr > s $ >C w. This istribution is the same as the efining ` moel pr > s w but truncate either to the left or to the right of zero epening on >. The rawing of the truncate normal is one using its inverse istribution function, which is reaily calculable. For subjects that, in the previous cycle of the algorithm, ha been impute as always-takers or compliers, the rawing of > is one in a similar way. " 4. The rawing of is from pr # s " 0243 w is the same as the efining moel pr s w $, where is set to (. This istribution except that it is truncate to the right or " left of zero epening on The rawing is as with the compliance latent normals > \. 5. The rawing of the coefficients is from pr syn all > ` ` an š OR. This istribution is a Bayesian linear regression base on the efining likelihoo an prior with offsets \ Œ š (C,2) known at this step. The rawing of the coefficients is from pr (C,2)syN all > (C,2) $ 5 >C or OR, an the rawing of the coefficient is from pr syn " all (* >C š (C,2) OR, both of which are Bayesian linear regressions with offsets, respectively, Œ š 22

24 N O L ` O an cª Œ >C š. 6. Cluster-specific parameters are inepenently rawn for each cluster from pr c Ø syn " Ù all which is a Bayesian regression with offsets c >C (* >C 5 (* N. Then, the prior mean an variance matrix for O are ajuste to TÍ the conitional prior mean an variance matrix given the rawn values of use as prior istributions in the Bayesian regression pr syn with offsets, to raw (C,2). The parameters all > 5 an, are rawn analogously, except that the prior mean an variance matrix are not ajuste for the rawing of assume prior inepenence. because of the N 7. The rawing of c O À is from the Wishart istribution with ¾ ce egrees of freeom an scale matrix V U p 4 Ö ¾ ce c W<À (Gelman et al., 1995). The rawing of cª O À is from the Wishart istribution with ¾ cª egrees of freeom an scale matrix V U (C,2) (C,2) ¾ c c W À. For each moel, three chains were run. Inepenently among chains an across subjects, any unknown compliance principal stratum was initialize to a Bernoulli raw, conitionally on assignment arm, an with probabilities of compliance principal strata obtaine from the sampling point estimates erive from Table 1. Using the initialize principal strata for each chain, parameter estimates were subsequently initialize base on generalize linear moels estimates: for an, base on all subjects; for (C,2), base on the initialize set of compliers an always-takers; an for the physician-specific parameters, base on physicians with corresponing full rank esign matrices. were use to set the values of c an cª to the values of c an (e.g., Wakefiel et al., 1994). The matrices ce ce an c c The initialize physician-specific parameters, to represent, respectively, preliminary estimates of an c were also initialize, respectively. Each chain was run for iterations. At iterations, an base on the three chains for each moel, the potential scale reuction statistic (Gelman an Rubin, 1992) was compute for ITT c Se +@RD=F giving Ú [AB Ú ÎB Ú Î respectively, suggesting no evience against convergence. Inference for each moel of Table 2 23

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