G , G , G MHRN

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University of Manchester Research funded by: MRC Methodology Grants G0600555

1 Estimation of treatment-effects from randomized controlled trials in the presence non-compliance with randomized treatment allocation Graham Dunn University of Manchester Research funded by: MRC Methodology Grants G G , G , G MHRN Methodology Research Group Methodology Research Group

2 Plan for this talk Efficacy estimation The role of random allocation in removing effects of confounders Efficacy in the presence of departures from randomisation (non-compliance) The Complier-Average Causal Effect (CACE) CACE estimation via Principal Stratification Sensitivity analyses Missing data mechanisms (MAR and LI) Adding covariates to the models Dropping exclusion restrictions (ER)

3 Efficacy Estimation We have carried out a randomized controlled trial (RCT) for the treatment t t of depression: versus Treatment As Usual (TAU) Control Cognitive Behaviour Therapy (CBT) plus TAU Everyone in the CBT arm receives the allocated treatment (and none in the Control arm). Everyone in the trial provides a measure of outcome depression as measured by the Beck Depression Inventory (the BDI score). Efficacy (the treatment effect) is estimated by comparing the average BDI in the CBT arm with the average BDI in the controls. It is the effect of receiving treatment.

4 The Role of Random Allocation Receipt of treatment Clinical outcome (CBT) (BDI score, say) A B Path blocked C No confounding: Correlation between A and B implies a treatment effect

5 Noncompliance - The return of confounding Not everyone receives the allocated treatment. Treatment receipt is now subject to selection effects. Observational study nested within a trial. Randomised Treatment Allocation Received (rgroup) (g (CBT) Outcome (BDI) Omitted variables (Confounders) 5

6 Efficacy Estimation: starting with work of Bloom (1984) The Complier-Average Causal Effect (CACE) estimate is the comparison of the average outcome of the compliers in the CBT arm with the average outcome of the comparable group of would-be compliers in the Control arm. This is a randomisation-respecting estimate. It is the ITT effect in the sub-group of participants i who would always comply with their treatment allocation. It is not subject to confounding But how is it calculated? First, we make some explicit assumptions. 6

7 How do we define a treatment effect? It is a comparison between what is and what might have been. It is counterfactual. t We wish to estimate the difference between a patient s observed outcome and the outcome that would have been observed if, contrary to fact, the patient s treatment or care had been different (Rubin, 1974). Without the possibility of comparison the treatment effect is not defined. 7

8 Notation 1. Let s assume, for simplicity, that we can offer two alternatives: Treatment t( (psychotherapy or hospital admission, i for example). Control (treatment as usual or community care, for example). We indicate Treatment by T, and the Control condition by C. 8

9 Notation 2 Let s assume that we can measure outcome of treatment by the variable, Y. For each patient - In principle it is possible to measure Y after receipt of treatment: In principal it is possible to measure Y after being in the control condition: Y C Y T The effect of treatment is defined by the difference: = Y T Y C 9

10 Average Treatment Effects All = E(Y T -Y C ) Treated = E[(Y T -Y C ) D = 1] C = CACE = E[(Y T -Y C ) U = 1] N.B. These three average treatment effects are not necessarily equal there may be treatment effect heterogeneity determined by patient preferences. Key: D = 1 if received treatment, 0 if not. U = 1 if Complier, 0 if not. 10

11 Efficacy (CACE) Estimation Assumptions 1. There are two latent classes of participants (Principal Strata): Compliers and Non-compliers. Compliers get therapy if and only if allocated to the treatment. Non-compliers never get the therapy, Regardless of allocation. 2. As a consequence of randomisation, on average, the proportion of Compliers is the same in the two arms of the trial. 3. In the absence of treatment (i.e. for the Non-compliers) there is no effect of randomisation (i.e. treatment arm) on outcome. This assumption is often called an exclusion restriction. 11

12 The ODIN Trial (Dowrick et al., 2000) Eire: Finland: Norway: Spain: UK: Patricia Casey Ville Lehtinen Odd Stephan Dalgard Jose Luis Ajuso-Mateos Jose Luis Vazquez-Barquero Chris Dowrick (Liverpool) Graham Dunn (Manchester) Mohammad Maracy (Manchester) Helen Page (Liverpool) Clare Wilkinson (U. of Wales) Greg Wilkinson (Liverpool) 12

13 Example: The ODIN trial (Dowrick et al, 2000) Trial of 2 psychological interventions to reduce depression Randomised individuals: 236 to the psychological interventions (P) 128 to treatment as usual (C) Outcome: Beck Depression Inventory (BDI) at 6 months recorded on 317 randomised individuals ITT results Mean (SD) Difference in BDI6 P (n=177) C (n=140) (std error) Unadjusted (9.85) (10.42) (1.14) Adjusted for (1.02) baseline BDI 13

14 ODIN Results (complete cases) # participants Compliers Non-compliers All mean BDI Therapy (P) 118 (66.7%) Control (C)?? 140??

15 CACE estimation # participants Compliers Non-compliers All mean BDI Therapy (P) 118 (66.7%) Complier-Average randomisation balance Causal Effect(CACE) (59*140/177) Control (C) 93.3 (66.7%) exclusion restriction CACE = = (cf ITT = = -1.87) 15

16 CACE estimation ITT All = P C ITT Compliers + (1-P C )ITT Non-compliers = P C ITT Compliers CACE estimate = ITT estimate for outcome Proportion of Compliers = -1.87/0.667 =

17 Instrumental Variables (IV) Treatment Received as a Treatment Effect Mediator The exclusion restriction is now equivalent to saying that there is no direct effect of randomisation on outcome. Randomised Allocation (IV) Treatment Received Outcome (BDI) Omitted variables (Confounders) 17

18 Instrumental Variable (IV) Regression - example using Stata ivregress 2sls bdi6 (treat = rgroup) Instrumental variables (2SLS) regression Number of obs = 317 Wald chi2(1) = 2.66 Prob > chi2 = R-squared =. Root MSE = bdi6 Coef. Std. Err. z P> z [95% Conf. Interval] complya _cons Instrumented: complya Instruments: rgroup CACE estimate (s.e. 1.72) Similarly, to look at the effect of sessions attended, we would use ivregress 2sls bdi6 (sessions = rgroup) 18

19 More subtle variants of noncompliance Variable number of sessions attended (here looking for some sort of dose -response relationship). How does the treatment effect vary with number of sessions attended? The strength of the engagement between patient and therapist (the therapeutic alliance) varies from one patient to another. What is the effect of the therapeutic alliance on the treatment effect? Does it interact with number of sessions attended? 19

20 ODIN: Missing Outcome Data (loss to follow-up) Loss to follow-up strongly related to non-compliance with allocated treatment (Compliers 92%; Non-compliers 55%; Controls 73%). Possible to extend estimation procedures to allow for a credible missing data mechanism: Missing data mechanism ignorable (Missing at Random or MAR) Missing i data jointly determined d by allocation and the latent would-be compliance status (Latently Ignorable or LI with a compound exclusion restriction). 20

21 CACE analysis under MAR (Outcome data Missing i At Random) Mean BDI Compliers Never-takers All participants Therapy (E) complier-average causal effect (CACE) randomisation balance (108*191/236) Control (S) exclusion restriction CACE (MAR) = = cf CACE (CC) = =

22 More complex models To improve precision and allow sensitivity analyses Simultaneously predict principal strata and outcome using relevant baseline covariates (e.g. finite mixture models in Mplus Muthén). Assuming missing outcomes MAR (ignorable when using ML) or LI (i.e. include model of whether outcome observed as a function of principal stratum membership). Try relaxing exclusion restriction(s). 22

23 Returning to ODIN: Compliance rates vary with Centre Treatment Group C NC Control 1Eire 1. 6(40%) Spain 12 (63%) Finland 17 (74%) Finland 20 (71%) Norway 22 (52%) N 6. Norway 17 (47%) UK 19 (40%) UK 15 (58%)

24 ODIN: 6-month follow-up rates vary with Centre No. Observations (%) Centre C NC Control 1 6 (100%) 2 (22%) 12 (52%) 2 12 (100%) 3 (43%) 7 (64%) 3 17 (100%) 2(33%) 17 (71%) 4 18 (90%) 6 (75%) 20 (91%) 5 20 (91%) 11 (55%) 17 (68%) 6 17 (100%) 15 (79%) 18 (72%) 7 15 (79%) 16 (57%) 31 (84%) 8 13 (87%) 4 (36%) 18 (75%) 24

25 Missing Data Model: Assumption 1. Latent Ignorability (LI) Compliance class independent of randomisation Assumption 2. Given both Treatment Arm (randomization) and Compliance class (Complier, Non-Complier), outcome is independent of whether it is actually observed or missing The Latent in Latent Ignorability comes from the fact that we cannot observe Compliance class completely. 25

26 The Compound Exclusion Restriction Assumption 3. For Non-Compliers, the drop-out rate is the same in the two arms of the trial. That is, the offer of treatment, in itself, does not influence loss to follow-up. For Non-Compliers Compliers, the outcome is the same in the two arms of the trial. That is, the offer of treatment, in itself, does not influence outcome. 26

27 Joint Models for Outcome and Non- Response (if LI): ML Estimation in Mplus Random allocation: Response observed: Latent Compliance class: Observed outcome: Z (1 if treatment, 0 control) R (1 if observed, 0 otherwise) U (1 for Complier, 0 otherwise) BDI6 BDI6 * = α + βu + Δ C Z.U logit (R=1)* * = ε + δu + Δ CZ.U logit(u-1) = λ * Absence of a main effect of Z corresponds to an exclusion restriction * Model dropped from the analysis when missing data MAR. 27

28 ODIN Results Ignoring Covariates Estimate (s.e.) Δ C (MAR) (2.20*) (253) (2.53) Δ C (LI) (1.80*) C (1.88) Black method of moments; Blue maximum likelihood * from a simple bootstrap 28

29 Joint Models for Outcome and Non- Response (if LI): now add covariates Random allocation: Response observed: Latent Compliance class: Observed outcome: Baseline Covariates (Centre): Z (1 if treatment, 0 control) R (1 if observed, 0 otherwise) U (1 for Complier, 0 otherwise) BDI6 X j BDI6 = α + β 0 U + Δ C Z.U + Σ j β j X j logit (R=1) = ε + δ 0 U + Δ C Z.U + Σ j δ j X j logit(u-1) = λ 0 + Σ j λ j X j 29

30 ODIN: Maximum likelihood (ML-EM) estimates of c With covariates (BDI0 and Centre) Est. Δ C (s.e.) MAR (1.86) without t ER on BDI (185) (1.85) LI (1.56) without ER on BDI (1.66) without ER on RESP (2.52) without ER on both (2.44) 30

31 Concluding thoughts Estimating average treatment effects from trials with substantial non- compliance and loss to follow-up is fraught with difficulties. CACE estimation should complement ITT, not replace it. Estimation of the causal effect of treatment received requires assumptions only some of which h can be tested. We have assumed homogeneity of Δ C across treatment centres. This could be tested in principle but the test would have very low power. The average treatment effect in the compliers is not necessarily the same as that in the non-compliers. It is still difficult to get a handle on Δ All. If we can persuade more participants i t to comply with their treatment t t allocation then Δ C might change. This might have interesting implications for the investigation of treatment effect heterogeneity in a meta-analysis of data from multiple trials. 31

32 Efficacy Evaluation and Non-compliance Further Reading Dunn, G., Maracy, M., Dowrick, C et al. (2003). Estimating psychological l treatment effects from an RCT with both noncompliance and loss to follow-up. British Journal of Psychiatry 183, 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 finstrumental variable methods. Statistical i Methods in Medical Research 14, Maracy, M. & Dunn, G. (2010). Estimating dose-response effects in psychological treatment trials: the role of instrumental variables. Statistical Methods in Medical Research. In press: first published online on November 26, 2008 as doi: / /

Demystifying causal inference in randomised trials

Demystifying causal inference in randomised trials Ian White 1, Sabine Landau 2, Graham Dunn 3,4 and Richard Emsley 3,4 1 MRC Biostatistics Unit Cambridge 2 Department of Biostatistics, Institute of Psychiatry,