Evaluation of Potential Mediators in Randomized Trials of Complex Intervention (Psychotherapies)

Session 2 Extending Principal Stratification for Characterising Trajectories of Change

Richard Emsley

Research funded by:

MRC Methodology Grants G0600555 G0900678, G0800606, G0802418 MHRN Methodology Research Group Methodology Research Group

Outline for session 2

• More examples of Principal Stratification • Introduction to growth curve models • Mediation analysis using growth mixture models • Latent class trajectory models

Principal Stratification – SoCRATES example

• Considering therapeutic alliance as our mediator/process variable.

• We can postulate the existence of two principal strata: –

High alliance

participants – those observed to have a high alliance in the therapy group together with those in the control group who would have had a high alliance had they been allocated to receive therapy.

–

Low alliance

participants – those observed to have a low alliance in the therapy group together with those in the control group who would have had a low alliance had they been allocated to receive therapy.

3 Frangakis C & Rubin D, Biometrics (2002); Jo B, Psych. Methods (2008).

Principal Strata – SoCRATES example

Therapeutic Alliance β α PANSS Randomisation γ

High Alliance

Randomisation ITT effect PANSS

Low Alliance

Randomisation ITT effect PANSS 4

SoCRATES - results

Estimated ITT effects on 18 month PANSS

Missing data ignorable (MAR)

Low alliance High alliance

+7.50 (8.18) -15.46 (4.60) Missing data latently ignorable (LI)

Low alliance High alliance

+6.49 (7.26) -16.97 (5.95) 5

Principal Strata – SoCRATES example

d X

High Alliance

U α Sessions Attended β Randomisation Psychotic Symptoms

d Y d X

Low Alliance

α Sessions Attended β U Randomisation Psychotic Symptoms

d Y

6

SoCRATES – effect of Sessions

Standard Structural Equation Model

Missing data assumption:

MAR

(uncorrelated errors α

β

– no hidden confounding)

Low alliance

+14.96 (0.96)

+0.59 (0.38) High alliance

+16.91 (0.45)

-0.75 (0.23) IV Structural Equation Model

(with correlated errors – hidden confounding) α

β Low alliance

+14.90 (0.97)

+0.37 (0.47) High alliance

+16.95 (0.46)

-0.80 (0.29) α - effect of randomisation on sessions β - effect of sessions on 18-month PANSS

7

Mechanisms Evaluation

Compliance with allocated treatment

Does the participant turn up for any therapy?

How many sessions does she attend?

Fidelity of therapy

How close is the therapy to that described in the treatment manual? Is it a cognitive-behavioural intervention, for example, or merely emotional support?

Quality of the therapeutic relationship

What is the strength of the therapeutic alliance? Is it associated with the effect of treatment?

8

Mechanisms Evaluation

What is the concomitant medication?

Does psychotherapy improve compliance with medication which, in turn, leads to better outcome? What is the direct effect of psychotherapy? Is there any?

What is the concomitant substance abuse?

Does psychotherapy reduce cannabis use, which in turn leads to improvements in psychotic symptoms?

What are the participant’s beliefs?

Does psychotherapy change attributions (beliefs), which, in turn, lead to better outcome? How much of the treatment effect is explained by changes in attributions?

9

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.

10

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.

11

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.

12

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.

13

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. 14

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!

15

Notation

Z i

– treatment group: the outcome of randomisation (Z for treatment, 0 for controls).

i

=1

X i

′ = X1 i

, X2

i

… Xp

i

– baseline covariates.

Y i

– observed outcome.

M i

– intermediate outcome that is a putative mediator of the effects of treatment on outcome (either a quantitative measure or binary).

R i

– response: missing value indicator (R missing, 1 if observed).

i

=0 if Y

i

is We also define the following potential (counterfactual) outcomes:

M i

(0) – mediator if randomised to the control condition.

M i

(1) – mediator if randomised to treatment.

16

Principal Strata with a binary mediators

Class C (stratum) 1 2 3 4

M i (1)

0 1 1 0

M i (0)

0 1 0 1

(M i (1), M i (0))

(0,0) (1,1) (1,0) (0,1) Proportion P 1 P 2 P 3 p 4 Treatment Effect ITT 1 ITT 2 ITT 3 ITT 4 • • • There are now four distinct possibilities (classes) for the joint combination of M

i

(1) and M

i

(0).

It is sometimes assumed that only one of classes 3 and 4 is present (known as the monotonicity assumption). Individuals’ class membership is not known: for example, an individual with Z

i

=1 and M

i

=1 is only known to belong to class 2 or class 3. 17

Principal Strata with a binary mediators

Class C (stratum) 1 2 3 4

M i (1)

0 1 1 0

M i (0)

0 1 0 1

(M i (1), M i (0))

(0,0) (1,1) (1,0) (0,1) Proportion P 1 P 2 P 3 p 4 Treatment Effect ITT 1 ITT 2 ITT 3 ITT 4 • • If the mediator is only in the treatment condition (e.g. therapeutic alliance when no intervention offered in the control group), then M

i

(0)=0 for everyone.

Classes 2 & 4 do not exist, so we ’ re only considering the existence of (0,0) and (1,0) classes.

–

Similar to only thinking about compliers & non compliers when no active control

18

Estimation of stratum-specific treatment (ITT) effects Let’s say there are two principal strata, with proportions p 1 and p 2 (with p 1 + p 2 =1).

Let ITT

all

be the overall ITT effect (which can be estimated directly in the conventional way) Similarly let ITT

1

effects.

and ITT

2

be the stratum-specific ITT Then

ITT all = p 1 ITT 1 + p 2 ITT 2

19

The identification problem

If

ITT all = p 1 ITT 1 + p 2 ITT 2

and we are not prepared to make any further assumptions, then we cannnot get unique estimates of

ITT 1

and ITT

2

. If we increase ITT

1

then ITT

2

will decrease to compensate (giving the same value for ITT

all

).

What can we do? 20

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. 21

With the exclusion restriction we have an identifiable (estimable) stratum-specific treatment effect Now

ITT all = p 1 .0 + p 2 ITT 2 ITT all = p 2 ITT 2

And therefore

ITT 2 = ITT all /p 2

This is the instrumental variable estimator as seen earlier. CACE = Overall ITT effect/Proportion of Compliers 22

Two exclusion restrictions for the Hospital admission/Community care trial

ITT all = p 1 ITT1 + p 2 ITT 2 + p 3 ITT 3 (p 1 +p 2 +p 3 =1) ITT all = p 1 ITT 1 + p 2 .0 + p 3 .0

And therefore

ITT 1 = ITT all /p 1

This is again the instrumental variable estimator (p 1 is fairly straightforward to estimate). CACE = Overall ITT effect/Proportion of Compliers 23

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. 24

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.

25

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.

26

Outline for session 2

• More examples of Principal Stratification • Introduction to growth curve models • Mediation analysis using growth mixture models • Latent class trajectory models 27

Situation considered so far…

d X

Mediator U Randomisation Outcomes

d Y

28

Extending to repeated measures on the outcome variable

Time 1 Time 2 Time 3

Mediator Treatment Outcomes Outcomes Outcomes 29

30

Simple linear regression

y i



i

|

x i

  1 2

x i

~

N

(0,   2 ) 

i

where ε i is the residual or error term for the ith participant, assumed to be independent of all the residuals for other participants.

β 1 + β 2

x

i is the

fixed

part of the model ε i is the

random

part of the model We also assume that the variance of Var(

y i

|

x i )

= Var(

ε i

|

x i

)= σ 2 31

Variance Components

y ij y ij

 

1

ij

  

j ij

We split the residual from the first model into two components: ζ j is specific to participant j and constant across occasions ε ij is specific to each participant j at each occasion i ζ j is a random effect or random intercept: mean=0, variance= ψ ε ij is the random deviation of y ij variance= θ.

from participant j’s mean: mean=0, 32

Mixed Models

• Mixed models are characterized as containing both fixed effects and random effects.

• The fixed effects are analogous to standard regression coefficients and are estimated directly. • The random effects are not directly estimated but are summarized according to their estimated variances and covariances.

33

Stata commands

xtmixed Multilevel mixed-effects linear regression xtmelogit Multilevel mixed-effects logistic regression Can also use: xtreg Fixed-, between-, and random-effects, and population-averaged linear models gllamm – Generalised Linear Latent and Mixed Models 34

Growth Curves for repeated outcome measures

• Diagram assumes balanced occasions.

• This is the random part of a random coefficient model, which is the same as a traditional linear growth curve model.

Outcome 1 1 I • Occasion i, subject j 1 Outcome 2 1 1 S Outcome 3 2

y ij

   1  2

x ij

  1

j

  2

j x ij

 

ij

fixed part random part 35

Are missing data a problem?

• Are missing data a problem? Why can’t we simply analyse the observed data using an appropriate analysis model? • There are a number of reasons – Invalidity of the estimation method – – – Loss of precision Violation of the intention-to-treat principle Lack of generalisability • The main issue is selection bias.

36

Missingness mechanisms

• Subjects in the sample provide two types of information – The pattern of missing values – The observations actually made • The missing value mechanism is the probability that a set of values are missing given the values taken by the (later) observed and missing observations. • For a number of analysis approaches based on the observed data it will then be possible to specify the MV mechanism or assumptions under which they provide valid inferences.

37

Rubin’s classification

• Rubin’s classification of MV generating mechanisms has proved useful for this purpose – Missing Complete at Random (MCAR) – Missing at Random (MAR) – Not Missing at Random (NMAR) 38

How to decide between them?

• • It is not possible to determine empirically which of the three classes we are in – We can never rule out NMAR empirically Although the data can tell us some things – Can choose between MCAR and MAR • The final decision has to be made on empirical [subject matter?] as well as theoretical grounds 39

Some Good News!

• Mixed models or random effects models estimated using maximum likelihood estimation is also VALID under the missing at random assumption.

• So we can model ALL the data – including the intermediate outcomes.

• Ensures participants with some missing values are NOT dropped from the subsequent analysis (as would be performed in SPSS – akin to complete case analysis).

40

Example: SoCRATES Trial again

• We examine the primary outcome, The Positive and Negative Syndromes Schedule (PANSS), an interview-based scale for rating 30 psychotic and non-psychotic symptoms administered blind to condition allocation. • The PANSS was administered – at baseline, – once a week over the first 6 weeks (we just use 6 weeks here) – and then at 3 months, 9 months and 18 months. • In analyses we log transformed the timescale, and considered linear and quadratic terms.

41

Observed trajectories for 30 patients

42

Model fitted quadratic curves for 30 same patients 43

Outline for session 2

• More examples of Principal Stratification • Introduction to growth curve models • Mediation analysis using growth mixture models • Latent class trajectory models 44

Extending to repeated measures on the outcome variable

Time 1 Time 2 Time 3

Mediator Treatment Outcomes Outcomes Outcomes 45

Principal Strata – with growth curves

High Alliance

Outcome 1 Outcome 2 Outcome 3

Low Alliance

I S Outcome 1 Outcome 2 Outcome 3 I S 46

Mixture Modelling

• Mixture modeling refers to modeling with categorical latent variables that represent subpopulations where population membership is not known but is inferred from the data – such as principal strata.

• The simplest longitudinal mixture model is latent class growth analysis (LCGA). In LCGA, the mixture corresponds to different latent trajectory classes. No variation across individuals is allowed within classes.

• Another longitudinal mixture model is the growth mixture model (GMM). In GMM, within class variation of individuals is allowed for the latent trajectory classes. The within-class variation is represented by random effects, that is, continuous latent variables, as in regular growth modeling. 47

Growth Mixture Model

Y 1 Y 2 Y 3 Y 4 Y 5 I S X C Z • Since c is a categorical latent variable, the interpretation of this GMM is not the same as for a continuous latent variable.

• The arrows from c to the growth factors indicate that the intercepts in the regressions of the growth factors on Z vary across the classes of c.

48

SoCRATES Analysis in MPlus

• We simultaneously fit the following models using maximum likelihood: – Principal strata membership on covariates (log of duration of untreated psychosis, centre, years of education) – Quadratic growth curve model within each class, allowing the variances of the intercept – Effect of randomisation on the slope within each class • Bootstrap the procedure to obtain valid standard error estimates.

49

SoCRATES Analysis in MPlus

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASSES BASED ON THE ESTIMATED MODEL Latent Classes 1 69.89513 2 131.10487 0.34774

0.65226

FINAL CLASS COUNTS AND PROPORTIONS FOR THE LATENT CLASS PATTERNS BASED ON ESTIMATED POSTERIOR PROBABILITIES Latent Classes 1 69.89513 2 131.10487 CLASSIFICATION QUALITY Entropy 0.34774

0.65226

0.641

50

SoCRATES Analysis in MPlus

CLASSIFICATION OF INDIVIDUALS BASED ON THEIR MOST LIKELY LATENT CLASS MEMBERSHIP Class Counts and Proportions Latent Classes 1 63 0.31343

2 138 0.68657

Average Latent Class Probabilities for Most Likely Latent Class Membership (Row) by Latent Class (Column) 1 1 0.859 2 0.114 2 0.141

0.886

51

SoCRATES Analysis in MPlus

Fixed Part of the Model MODEL RESULTS Estimate INTER | PANTOT 1.000 PAN1 1.000 PAN3 1.000 PAN9 1.000 PANT18 1.000 Bootstrap S.E. 0.000 0.000 0.000 0.000 0.000 SLOPE | PANTOT 0.000 PAN1 1.946 PAN3 2.565 PAN9 3.611 PANT18 4.369 0.000 0.000 0.000 0.000 0.000 Est./S.E. 999.000 999.000 999.000 999.000 999.000 Two-Tailed P-Value 999.000

999.000

999.000

999.000

999.000

999.000 999.000 999.000 999.000 999.000 999.000

999.000

999.000

999.000

999.000

52

SoCRATES Analysis in MPlus

Estimate B ootstrap S.E. Est./S.E. Two-Tailed P-Value Quadratic | PANTOT 0.000 0.000 999.000 PAN1 3.787 0.000 999.000 PAN3 6.579 0.000 999.000 PAN9 13.039 0.000 999.000 PANT18 19.092 0.000 999.000 999.000

999.000

999.000

999.000

999.000

53

SoCRATES Analysis in MPlus

Latent Class 1 – Low Alliance Group SLOPE ON GROUP 1.808 1.644 1.100 0.271

Q WITH INTER 0.543 1.628 0.334 0.739

Means INTER 90.444 3.441 26.281 0.000

Q 2.269 0.460 4.928 0.000

Intercepts SLOPE -17.118 2.697 -6.346 0.000

Variances INTER 101.743 53.936 1.886 0.059

Q -0.149 0.419 -0.356 0.722

54

SoCRATES Analysis in MPlus

Latent Class 1 – Low Alliance Group

Residual Variances PANTOT 155.448 25.475 6.102 0.000

PAN1 225.422 33.614 6.706 0.000

PAN3 134.520 29.049 4.631 0.000

PAN9 96.053 33.022 2.909 0.004

PANT18 37.002 24.554 1.507 0.132

SLOPE 9.209 8.681 1.061 0.289

55

Estimated means for class 1 and observed trajectories 56

SoCRATES Analysis in MPlus

Latent Class 2 – High Alliance Group SLOPE ON GROUP -2.843 1.136 -2.502 0.012

Q WITH INTER 0.543 1.628 0.334 0.739

Means INTER 87.844 1.954 44.955 0.000

Q 1.784 0.309 5.769 0.000

Intercepts SLOPE -11.558 1.763 -6.556 0.000

Variances INTER 163.426 33.921 4.818 0.000

Q 0.625 0.298 2.095 0.036

57

SoCRATES Analysis in MPlus

Latent Class 2 – High Alliance Group

Residual Variances PANTOT 155.448 25.475 6.102 0.000

PAN1 225.422 33.614 6.706 0.000

PAN3 134.520 29.049 4.631 0.000

PAN9 96.053 33.022 2.909 0.004

PANT18 37.002 24.554 1.507 0.132

SLOPE -4.728 5.382 -0.878 0.380

58

Estimated means for class 2 and observed trajectories 59

Sample and estimated means by class

60

Outline for session 2

• More examples of Principal Stratification • Introduction to growth curve models • Mediation analysis using growth mixture models • Latent class trajectory models 61

Growth Curves for repeated outcome measures

• Diagram assumes balanced occasions.

• This is the random part of a random coefficient model, which is the same as a traditional linear growth curve model.

Outcome 1 1 I • Occasion i, subject j 1 Outcome 2 1 1 S Outcome 3 2

y ij

   1  2

x ij

  1

j

  2

j x ij

 

ij

fixed part random part 62

Latent Trajectory Models

y ij

   1  2

x ij

  1

j

 

ij

• fixed part random part Here the ς 1j are represented by discrete trajectory classes c with probability π c : (ς it | c) = e 1c + e 2c t • • • where e 1c e 2c is the trajectory origin or intercept for class c is the trajectory slope for class c Prevalence of trajectory class is π c Pickles and Croudace (2010). SMinMR.

63

Latent Trajectory Models

• Hypothetical example with 2 distinct classes

Class 1 Class 2

Outcome Time Time 64

Latent Trajectory Models

• Three type of models: 1.

Unconditional trajectory classes and unconditional class probabilities 2.

3.

Unconditional trajectory classes and conditional class probabilities –

Allow the probability to depend on covariates

Conditional trajectory classes and unconditional class probabilities – –

Covariate effects included in fixed part of the model Classes now represent groups having accounted for covariate differences

– Similarity with Cluster Analysis.

65

Latent Class Growth Model (LCGM)

y1 y2 y3 y4

1 1 1 1 1 b1 S b2 I Latent classes C 66

Posterior Class Membership

• For example, we can fit a growth model in which we postulate just two underlying patterns of change.

• The model identifies those two patterns of change and their relative size so as to best fit the data.

• It is then possible to calculate how likely each participant is, given their data, to belong to one or other class.

• We can then group participants according to their most likely class and display the data from those subjects and examine how membership of those groups are associated with other variables.

67

Deciding the number of classes-1

• Tests for k-1 versus k classes: – Standard LR test rejects far too often over estimating the number of classes as LR statistic is not chi-square distributed.

– – Lo-Mendell-Rubin LR test derives an approximate distribution as a mixture of LR comparisons. Performs much better.

Parametric Bootstrap LR test (standard LR test compared to LR-test distribution of k-1 versus k comparison of bootstrap samples simulated under k 1 model). Performs still better than LMR test for both Type-1 error rate and power. Performance declines in sample sizes <200.

• See Nylund, Asparouhov and Muthen, SEM, 2007 for details.

68

Deciding the number of classes-2

• Information Criteria (p=parameters, n=sample size) AIC = -2logL+2p : poor CAIC = -2logL+p(log(n)+1) BIC = -2logL+plog(n) : better but not good :performs well, less so in small samples Sample-size adjusted BIC = as above with n*=(n+2)/24 : generally performed worse than BIC Overall BLRT slightly better than BIC but computationally intensive and assumes k-1 model correctly specified.

69

Example: SoCRATES Trial again

• We examine the primary outcome, The Positive and Negative Syndromes Schedule (PANSS), an interview-based scale for rating 30 psychotic and non-psychotic symptoms administered blind to condition allocation. • The PANSS was administered – at baseline, – – once a week over the first 6 weeks (we just use 6 weeks here) and then at 3 months, 9 months and 18 months. • In analyses used log transformed the timescale, and considered linear and quadratic terms (GLLAMM).

• See Pickles, A. & Croudace, T. (2010) Latent Mixture Models for Multivariate and Longitudinal Outcomes. Statistical Methods in Medical Research. 19(3), pp 271-290. 70

SoCRATES example: Increasing classes and fit criteria

71

Trajectories for the 4-class model for the SoCRATES study

0 1 2 logweek Class 1 (42%) Class 3 (37%) 3 Class 2 (6%) Class 4 (15%) 4 72

Class assignment certainty

Assigned class Average probability Class 1 Average probability Class 2 Average probability Class 3 Average probability Class 4 Class 1 Class 2 Class 3 Class 4 0.84

0.00

0.18

0.00

0.02

0.87

0.03

0.03

0.14

0.09

0.75

0.11

0.00

0.04

0.04

0.86

73

Prevalence of treatment by class

Class 1 2 3 4 Total Control 39 (30%) 8 (62%) 37 (31%) 19 (40%) 103 (33%) Treatment 90 (70%) 5 (32%) 83 (69%) 28 (60%) 206 (67%) Total 129 13 120 47 309 74

Latent Class Growth Model (LCGM) with predictors of class membership

y1 y2 y3 y4

1 1 1 1 1 b1 S b2 I Latent classes C treatment 75

Extend model to include predictors of trajectory class membership

Predicting class from randomised group – gllamm output.

log odds parameters (SEs) class 1 rgrp: .37211754 (.44317846) _cons: .89785223 (.40652414) class 2 rgrp: -1.8011231 (.87633211) _cons: .03280307 (.51026448) class 3 rgrp: .45207293 (.49493849) _cons: .70044474 (.43137194) 76

Latent trajectory models for mediators

• We can also apply this methodology to repeated measures on the mediator.

• MIDAS (Motivational Intervention for Drug and Alcohol use in Schizophrenia) is an RCT of 327 patients with dual diagnosis.

• For example, in the MIDAS trial we are interested in the use of cannabis – the intervention aimed to reduce the amount of substance use over time, with the aim of this reducing the psychotic symptoms.

• Here we look at 160 cannabis users at baseline, and examine their cannabis use over time; measured as average daily weight per using day.

77

Observed trajectories for 20 random patients 78

Sample means over time for latent classes

79

Model based means over time for latent classes 80

Connecting mediators with outcomes

Substance Use 0 Substance Use 6 Substance Use 12 Substance Use 18 Substance Use 24 I I S S Symptoms 0 Symptoms 12 Symptoms 24 More work is needed to develop these models to allow for unmeasured confounders between the mediators and outcomes – we’re working on this now!

81

What do others do for causal mediation analysis?

• A group based at HSPH, led by Tyler VanderWeele and Stijn Vansteelandt, have developed methods based on regression models. These allow for interactions between the exposure/treatment and the mediators, but are still based on the no unmeasured confounders assumption.

• A group based in Penn State, led by Tom Ten Have and Marshall Joffe, have promoted the use of rank preserving structural models. We have demonstrated that these are merely an unnecessary complex use of the instrumental variables method we introduced earlier – currently being written up.

82

Summary

• There is a large literature of mixture models for analysing longitudinal data.

• Viewing the mixtures as being principal strata allows us to use these mixture models for mediation analysis without the unmeasured confounding assumption between mediator and outcomes.

• We can use repeated measures on the mediator to form latent classes based on their trajectories over time – the trick is to link this with outcome trajectories and classes.

83

Further Reading – principal stratification and Mplus Emsley RA, Dunn G & White IR (2010). Mediation and moderation of treatment effects in randomised trials of complex interventions. Statistical Methods in Medical Research 19 (3), pp237-270.

Emsley RA & Dunn G. (2010). Evaluation of potential mediators in randomized trials of complex interventions. In preparation for “Statistical Methods in Causal Inference”, Berzuini, C., Dawid, P. & Bernardinelli, L. (Eds).

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.

Mplus User’s Guide illustrates CACE estimation.

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Further Reading – growth mixtures

Pickles, A. & Croudace, T. (2010) Latent Mixture Models for Multivariate and Longitudinal Outcomes. Statistical Methods in Medical Research 19(3), pp271-290.

Emsley, R.A, Pickles, A. & Dunn, G. (2010) Mediation analysis with growth mixture modelling. In preparation for Statistics in Medicine.

Muthén B, Brown CH, Masyn K et al. General growth mixture modeling for randomized preventive interventions. Biostatistics 2002; 3: 459–75.

Muthén, B. & Brown, H. (2009). Estimating drug effects in the presence of placebo response: Causal inference using growth mixture modeling. Statistics in Medicine, 28, 3363-3385.

Stulz N, Lutz W, Leach C, Lucock M, Barkham M. Shapes of early change in psychotherapy under routine outpatient conditions. Journal of Consulting and Clinical Psychology 2007; 75: 864–74.

Tarrier N, Lewis S, Haddock G et al. Cognitive-behavioural therapy in early schizophrenia: 18 month follow up of a randomised, controlled trial. British Journal of Psychiatry 2004; 184: 231–9.

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