Session 4 An introduction to principal stratification Graham Dunn

1. MethodologyResearch Group Methods of explanatory analysis for psychological treatment trials workshop Session 4 An introduction to principal stratification Graham Dunn Funded by: MRC Methodology GrantG0600555 MHRN Methodology Research Group

2. Randomisation-respecting inference: no confounding • Aim to estimate and compare effects of post-randomisation variables via the comparison of randomised sub-groups of patients (within-class or stratum-specific ITT effects). These effects are not subject to confounding. • For example, we would like to compare the outcome of treatment in those participants who develop a given level of alliance/quality of therapy with the outcome in the control patients who would have developed the same level of alliance/quality of therapy if they had, contrary to fact, been allocated to receive therapy. • Rationale for estimation through the use of Principal Stratification – a generalization of CACE (Complier-Average Causal Effect) estimation.

3. Principal Strata Defined by response to random allocation, as determined by the intermediate outcome - examples to follow Strata are wholly or partially hidden (latent). Often analysed using a latent class (or finite mixture) model.

4. Simple example – Two principal strata:Potential Compliers vs Non-compliers Random allocation to treatment or no treatment (control). Those allocated to no treatment cannot get access to therapy. Principal stratum 1: Compliers Treated if allocated to the treatment arm, not treated otherwise. Principal stratum 2: Non-compliers Never receive treatment, regardless of allocation. Possible to identify these two classes in those allocated to treatment; but they remain hidden in the control group.

5. Simple example – Two principal strata:Potential Low alliance vs Potential High alliance Random allocation to treatment or no treatment. Those allocated to no treatment cannot get access to therapy. Principal stratum 1: Low alliance Treated with low alliance if allocated to the treatment arm, not treated otherwise. Principal stratum 2: High alliance Treated with high alliance if allocated to the treatment arm, not treated otherwise. Possible to identify these two classes in those allocated to treatment; but they remain hidden in the control group.

6. Simple example – Three principal strata:Non-compliers vs Low alliance vs High alliance Random allocation to treatment or no treatment. Those allocated to no treatment cannot get access to therapy. Principal stratum 1: Non-compliers Never receive treatment, regardless of allocation. Pricipal stratum 2: Low alliance (Partial compliance) Treated with low alliance if allocated to the treatment arm, not treated otherwise. Principal stratum 3: High alliance (Full compliance) Treated with high alliance if allocated to the treatment arm, not treated otherwise. Possible to identify these three classes in those allocated to treatment; but they remain hidden in the control group.

7. Three principal strata: Compliers vs Always admitted vs Never admitted Random allocation to Hospital admission or Community care. Some of those allocated to Hospital admission never get admitted because of bed shortages. Some allocated to Community care have a crisis and have to be admitted. Principal stratum 1: Compliers Hospital admission if allocated to hospital, Community care, otherwise. Principal stratum 2: Always admitted Hospital admission, regardless of allocation. Principal stratum 3: Never admitted Community care, regardless of allocation. If allocated to Hospital admission and admitted then either Complier or Always admitted. If allocated to Hospital and receive Community care, then Never admitted. If allocated to Community care and receive Community care then either Complier or Never admitted. If allocated to Community care and admitted then always admitted.

8. Four principal strata based on a potential mediator. Random allocation to CBT or no CBT (control). Those allocated to no CBT cannot get access to therapy. Intermediate outcome – taking antidepressant medication. PS1: take medication irrespective of allocation. PS2: never take medication irrespective of allocation. PS3: take medication only if allocated to CBT. PS4: take medication only if allocated to control. ITT effects in PS1 and PS2 tell us about direct effects of CBT. ITT effects in PS3 and PS3 tell us about the joint effects of CBT and medication.

9. Principal strata based on remission Participants recruited to the trial during a psychotic episode. Random allocation to CBT or no CBT (control). Those allocated to no CBT cannot get access to therapy. Intermediate outcome –remission of psychotic symptoms. PS1: remission, irrespective of allocation. PS2: no remission, irrespective of allocation PS3: remission only if allocated to CBT. PS4: remission only if allocated to control (PS4 ruled out a priori? – the monotonicity assumption) What if our final outcome is relapse? Only makes sense to look at relapse rates in PS1. No-one to relapse in PS2. No controls for those in PS3. We’ll leave this one for another day!

10. Estimation of stratum-specific treatment (ITT) effects Let’s say there are two principal strata, with proportions p1 and p2 (with p1 + p2=1). Let ITTall be the overall ITT effect (which can be estimated directly in the conventional way) Similarly let ITT1 and ITT2be the stratum-specific ITT effects. Then ITTall = p1ITT1 + p2ITT2

11. The Identification problem If ITTall = p1ITT1 + p2ITT2 and we are not prepared to make any further assumptions, then we cannnot get unique estimates of ITT1and ITT2. If we increase ITT1 then ITT2 will decrease to compensate (giving the same value for ITTall). What can we do?

12. Exclusion restrictions What if stratum 1 corresponds to the Non-compliers? These are participants who never receive treatment whatever the treatment allocation. Let’s assume that allocation also has no effect on outcome in the Non- compliers (an exclusion restriction). Example: If you don’t take the tablets it doesn’t matter whether you have been assigned to the placebo or the supposedly active drug.

13. With the exclusion restriction we have an identifiable (estimable) stratum-specific treatment effect Now ITTall = p1.0 + p2ITT2 ITTall = p2ITT2 And therefore ITT2 = ITTall/p2 This is the instrumental variable estimator as seen earlier. CACE = Overall ITT effect/Proportion of Compliers

14. Two exclusion restrictions for the Hospital admission/Community care trial ITTall = p1ITT1 + p2ITT2 + p3ITT3 (p1+p2+p3=1) ITTall = p1ITT1 + p2.0+ p3.0 And therefore ITT1 = ITTall/p1 This is again the instrumental variable estimator (p1 is fairly straightforward to estimate). CACE = Overall ITT effect/Proportion of Compliers

15. Principal strata based on therapeutic alliance are a problem An a priori exclusion restriction for the Low alliance stratum extremely difficult to justify. In the three-stratum setting there is also a problem unless we can introduce two exclusion restrictions. What is the solution? Answer: Find baseline variables that help predict stratum membership (i.e. help us to discriminate Low and High principal strata). Although they are not necessary for identification, baseline variables that help predict stratum membership are also useful in the presence of exclusion restrictions – they increase the precision of the estimates.

16. The SoCRATES Trial • SoCRATES was a multi-centre RCT designed to evaluate the effects of cognitive behaviour therapy (CBT) and supportive counselling(SC) on the outcomes of an early episode of schizophrenia. • Participants were allocated to one of three conditions: Treatment as Usual (TAU), CBT + TAU, SC + TAU. • For our illustrative purposes, we ignore the distinction between CBT and SC, using a binary variable to distinguish treatment and control.

17. SoCRATES(contd.) • 3 treatment centres: Liverpool, Manchester and Nottinghamshire. Other baseline covariates include logarithm of untreated psychosis and years of education. • Outcome (a psychotic symptoms score) was obtained using the Positive and Negative Syndromes Schedule (PANSS). We consider the 18 month PANSS total score here. • From an ITT analyses of 18 month follow-up data, both psychological treatment groups had a superior outcome in terms of symptoms (as measured using the PANSS) compared to the control group. There were no differences in the effects of CBT and SC, but there was a strong centre effect, with outcomes for the psychological therapies at one of the centres (Liverpool) being significantly better than at the remaining two.

18. SoCRATES (contd.) • Post-randomization variables that have a potential explanatory role in exploring the therapeutic effects include the total number of sessions of therapy actually attended and the quality or strength of the therapeutic alliance. • Therapeutic alliance was measured at the 4th session of therapy, early in the time-course of the intervention, but not too early to assess the development of the relationship between therapist and patient. We use a patient rating of alliance based on the CALPAS (California Therapeutic Alliance Scale). • Total CALPAS scores (ranging from 0, indicating low alliance, to 7, indicating high alliance) were used in some of the analyses reported below, but we also use a binary alliance variable (1 if CALPAS score ≥5, otherwise 0). .

19. SoCRATES (contd.) • 182 (88.3%) out of 206 patients in the treated groups provided data on the number of sessions attended. 56 patients from the CBT group and 58 from the SC group completed CALPAS forms at session 4 (overall 55.34%). • The analysis in this talk is based on all control participants but only those from treated groups who provide both a CALPAS and a record of the number of sessions (missing sessions/alliance data another potential source of bias that will be ignored here).

20. SoCRATES - Summary Statistics Lewis et al, BJP (2002); Tarrier et al, BJP (2004); Dunn & Bentall, Stats in Medicine (2007).

21. SoCRATES “dose”-response model: complete mediation Offer of Treatment (random) Sessions Attended Psychotic Symptoms U What’s the role of the therapeutic alliance? Does Alliance modify the effect of randomisation on sessions attended? Does Alliance modify the effect of treatment received on outcome?

22. Principal Stratification - by Therapeutic Alliance • For simplicity we assume that everyone allocated to psychotherapy actually receives it – everyone is a complier.* • We have one sub-group of participants who receive no therapy if allocated to the control condition but receive therapy with a low alliance if allocated to the treatment group. • We have a second sub-group who receive no therapy if allocated to the control condition but receive therapy with a high alliance if allocated to the treatment group. • Principal stratum membership is independent of treatment allocation • We can stratify by stratum membership and evaluate the effects of treatment allocation within them. • But we could easily add a third stratum i.e. Non-compliers

23. Model Identification – Principal Strata • We need baseline covariates that are good predictors of stratum membership. • With two principal strata (high vs low alliance), we would construct a logistic regression (latent class) modelto predict stratum membership using baseline covariates, X(particularly treatment centre, for example). • This approach (predicting principal strata from baseline covariates) is analogous to using the baseline covariate-randomization interactions as instrumental variables in 2SLS. • We simultaneously model the ITT effects on outcome within the two principal strata. • Estimation proceeds by specifying a full probability model, here, for example, using ML.

24. Model Identification – Principal Strata • It is possible to fit the latent class model for stratum membership and simultaneously a further regression model for the ITT effects of treatment within each of the principal strata, usually allowing for the same baseline covariates – for example, when using the finite mixture model option in Mplus (Muthén & Muthén). • If we have missing outcome data (with missing outcome indicator, Ri) we can also simultaneously fit a third model predicting missing outcomes, based on the assumptions of latent ignorability. • In our SoCRATES examples, we use treatment centre, logDUP, Years Education and baseline PANSS to predict stratum membership. We use the same covariates plus the effect of randomisation to model outcome within principal strata – assuming that there are no covariate by randomisation interactions in this part of the model (sensitivity of results checked by relaxing this constraint for selected variables). Bootstrapping used to get standard errors.

25. Extensions: explanatory models nested within principal strata • The basic idea of principal stratification is the estimation of ITT effects within principal strata. • Typically we are interested in a univariate response, but we could investigate the advantages of simultaneously estimating effects for two or more different outcomes (i.e. multivariate responses). • It is possible to look at binary outcomes and, of course, one of these binary outcomes might be a missing value indicator – as in models assuming latent ignorability (Frangakis and Rubin, 1999). • In the context treatment compliance, Jo and Muthen have investigated the use of latent growth curve/trajectory models for longitudinal outcome data. • We will illustrate the idea by looking at the effect of sessions attended on the effects of therapy.

26. SoCRATES - results Estimated ITT effects on 18 month PANSS Low alliance High alliance Missing data ignorable (MAR) +7.50 (8.18) -15.46 (4.60) Missing data latently ignorable (LI) +6.49 (7.26) -16.97 (5.95)

27. SoCRATES – effect of SessionsMissing data assumption:MAR Standard Structural Equation Model (uncorrelated errors – no hidden confounding) Low allianceHigh alliance α 14.96 (0.96) 16.91 (0.45) β+0.59 (0.38) -0.75 (0.23) IV Structural Equation Model (with correlated errors – hidden confounding) Low allianceHigh alliance α 14.90 (0.97) 16.95 (0.46) β+0.37 (0.47) -0.80 (0.29) α - effect of randomisation on sessions β - effect of sessions on 18-month PANSS

28. SoCRATES – effect of SessionsMissing data assumption:LI Standard Structural Equation Model (uncorrelated errors – no hidden confounding) Low allianceHigh alliance α 14.94 (0.95) 16.92 (0.46) β+0.55 (0.42) -0.78 (0.28) IV Structural Equation Model (with correlated errors – hidden confounding) Low allianceHigh alliance α 14.85 (0.98) 16.98 (0.47) β+0.34 (0.50) -0.88 (0.37) α - effect of randomisation on sessions β - effect of sessions on 18-month PANSS

29. Principal Stratification in Practice Problems: • Imprecise estimates – trials not large enough. • Missing data for intermediate variables (sometimes lots!) – source of imprecision and bias. • Difficult to find baseline variables that are good predictors of stratum membership. • Difficult-to-verify assumptions.

30. Principal Stratification in Practice • Imprecise estimates – trials not large enough. Combine data from several trials? Meta-regression. Need common outcomes. • Missing data for intermediate variables (sometimes lots!) – source of imprecision and bias. If you think it’s important then collect the data. • Difficult to find baseline variables that are good predictors of stratum membership. Novel designs: Incorporate multiple randomisations to specifically target the intermediate variables. • Difficult-to-verify assumptions. Sensitivity analyses.

31. Appendix – for reference onlyMplus Code – ITT effects within PS: Missing data LI. TITLE: Principal stratification - SoCRATES DATA: FILE IS Socrates_alliance.raw; VARIABLE: NAMES logdup pantot pant18 yearsed c1 c2 rgroup alliance resp; CLASSES C(2); CATEGORICAL alliance resp; USEVARIABLES logdup pantot pant18 yearsed c1 c2 rgroup alliance resp; MISSING pant18(999) alliance(999); ANALYSIS: TYPE=MIXTURE; STARTS = 100 10; MODEL: %OVERALL% resp ON logdup pantot yearsed c1 c2 rgroup; pant18 ON logdup pantot yearsed c1 c2 rgroup; C#1 ON logdup pantot yearsed c1 c2; ! There are three models here. The first is a logistic regression to ! predict the indicator of a non-missing outcome (resp). The second ! is a multiple regression for the outcome itself. The third is is ! a logistic regression for latent class membership (high versus low ! alliance). All parameters for the missing data and outcome models ! are constrained to be equal for the two classes unless otherwise ! indicated below. !Continued on next slide ………..

32. Mplus code (contd.) %C#1% ! Low Alliance [alliance$1@15]; ! A declared threshold to force participants with recorded alliance=0 ! into this class. [resp$1]; resp ON rgroup*0; [pant18]; pant18 ON rgroup*0; ! These statements release the equality constraints on the relevant ! model intercept terms for the effects of the randomized ! intervention. %C#2% ! High alliance [alliance$1@-15]; ! A declared threshold to force participants with recorded alliance=1 ! into this class. [resp$1]; resp ON rgroup*0; [pant18]; pant18 ON rgroup*0;

33. Mplus for a dose-response model within PS:Missing data LI TITLE: Principal stratification - SoCRATES DATA: FILE IS Socrates_alliance.raw; VARIABLE: NAMES logdup pantot pant18 sessions yearsed c1 c2 rgroup alliance resp; CLASSES C(2); CATEGORICAL alliance resp; USEVARIABLES logdup pantot pant18 sessions yearsed c1 c2 rgroup alliance resp; MISSING pant18(999) alliance(999); ANALYSIS: TYPE=MIXTURE MISSING; starts = 100 10; estimator=ml; bootstrap=250; MODEL: %OVERALL% resp ON logdup pantot yearsed c1 c2 rgroup; sessions ON logdup pantot yearsed c1 c2 rgroup; pant18 ON sessions logdup pantot yearsed c1 c2; pant18 WITH sessions; C#1 ON logdup pantot yearsed c1 c2; ! Continued on next slide…..

34. Mplus code (contd.) %C#1% ! Low Alliance [alliance$1@15]; [resp$1]; resp ON RGROUP*0; [sessions]; sessions ON rgroup*0; [pant18]; pant18 ON sessions*0; %C#2% ! High alliance [alliance$1@-15]; [resp$1]; resp ON RGROUP*0; [sessions]; sessions ON rgroup*0; [pant18]; pant18 ON sessions*0;

35. Reference Emsley RA, Dunn G & White IR (2009). Mediation and moderation of treatment effects in randomised trials of complex interventions. Statistical Methods in Medical Research. In press – published online.

36. Further Reading (use of Mplus) Jo B (2008). Causal inference in randomized experiments with mediational processes. Psychological Methods 13, 314-336. Jo B & Muthén BO (2001). Modeling of intervention effects with noncompliance:a latent variable approach for randomized trials. In: Marcoulides GA, Schumacker RE, eds. New Developments and Techniques in Structural Equation Modeling. Mahwah, New Jersey: Lawrence Erlbaum Associates pp. 57-87. Jo B & Muthén BO (2002). Longitudinal Studies With Intervention and Noncompliance: Estimation of Causal Effects in Growth Mixture Modeling. In: Duan N, Reise S, eds. Multilevel Modeling: Methodological Advances, Issues, and Applications. Lawrence Erlbaum Associates pp. 112-39. Dunn, G., Maracy, M. & Tomenson, B. (2005). Estimating treatment effects from randomized clinical trials with non-compliance and loss to follow-up: the role of instrumental variable methods. Statistical Methods in Medical Research 14, 369-395. MplusUser’s Guide illustrates CACE estimation.