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.
Frangakis C & Rubin D, Biometrics (2002); Jo B, Psych. Methods (2008).
3
Principal Strata – SoCRATES example
Therapeutic
Alliance
β
α
PANSS
Randomisation
High Alliance
γ
Randomisation
PANSS
ITT effect
Low Alliance
Randomisation
PANSS
ITT effect
4
SoCRATES - results
Estimated ITT effects on 18 month PANSS
Missing data ignorable (MAR)
Low alliance
+7.50 (8.18)
High alliance
-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
dX
High Alliance
U
α
Sessions
Attended
β
Psychotic
Symptoms
Randomisation
dY
dX
Low Alliance
U
α
Randomisation
Sessions
Attended
β
Psychotic
Symptoms
dY
6
SoCRATES – effect of Sessions
Standard Structural Equation Model
Missing data assumption: MAR
(uncorrelated errors – no hidden confounding)
Low alliance
High 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)
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:
PS2:
PS3:
PS4:
take medication irrespective of allocation.
never take medication irrespective of allocation.
take medication only if allocated to CBT.
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
Zi – treatment group: the outcome of randomisation (Zi=1
for treatment, 0 for controls).
Xi′ = X1i, X2i … Xpi
– baseline covariates.
Yi – observed outcome.
Mi – intermediate outcome that is a putative mediator of
the effects of treatment on outcome (either a
quantitative measure or binary).
Ri – response: missing value indicator (Ri=0 if Yi is
missing, 1 if observed).
We also define the following potential (counterfactual)
outcomes:
Mi(0) – mediator if randomised to the control condition.
16
Principal Strata with a binary mediators
Class C
(stratum)
Mi(1)
Mi(0)
(Mi(1), Mi(0))
Proportion
Treatment
Effect
1
0
0
(0,0)
P1
ITT1
2
1
1
(1,1)
P2
ITT2
3
1
0
(1,0)
P3
ITT3
4
0
1
(0,1)
p4
ITT4
• There are now four distinct possibilities (classes) for the
joint combination of Mi(1) and Mi(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 Zi=1 and Mi=1 is only
known to belong to class 2 or class 3.
17
Principal Strata with a binary mediators
Class C
(stratum)
Mi(1)
Mi(0)
(Mi(1), Mi(0))
Proportion
Treatment
Effect
1
0
0
(0,0)
P1
ITT1
2
1
1
(1,1)
P2
ITT2
3
1
0
(1,0)
P3
ITT3
4
0
1
(0,1)
p4
ITT4
• If the mediator is only in the treatment condition (e.g.
therapeutic alliance when no intervention offered in the
control group), then Mi(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 & noncompliers when no active control
18
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 ITT2 be the stratum-specific ITT
effects.
Then
ITTall = p1ITT1 + p2ITT2
19
The identification problem
If
ITTall = p1ITT1 + p2ITT2
and we are not prepared to make any further
assumptions, then we cannnot get unique estimates of
ITT1 and ITT2. If we increase ITT1 then ITT2 will
decrease to compensate (giving the same value for ITTall).
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 Noncompliers (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
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
22
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
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…
dX
U
Mediator
Randomisation
Outcomes
dY
28
Extending to repeated measures on the
outcome variable
Time 1
Time 2
Time 3
Outcomes
Outcomes
Outcomes
Mediator
Treatment
29
30
Simple linear regression
yi  1   2 xi   i
 i | xi ~ N (0,  )
2
where εi is the residual or error term for the ith participant, assumed to be
independent of all the residuals for other participants.
β1+β2xi is the fixed part of the model
εi is the random part of the model
We also assume that the variance of Var(yi|xi)= Var(εi|xi)=σ2
31
Variance Components
yij  1   ij
yij     j   ij
We split the residual from the first model into two components:
ζj is specific to participant j and constant across occasions
εij is specific to each participant j at each occasion i
ζj is a random effect or random intercept: mean=0, variance=ψ
εij is the random deviation of yij from participant j’s mean: mean=0,
variance=θ.
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.
• Occasion i, subject j
Outcome 1
Outcome 2
1
1
1
I
Outcome 3
1
2
S
yij  1   2 xij   1 j   2 j xij   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 [subjectmatter?] 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
Outcomes
Outcomes
Outcomes
Mediator
Treatment
45
Principal Strata – with growth curves
High Alliance
Outcome 1
Outcome 2
I
Outcome 3
S
Low Alliance
Outcome 1
Outcome 2
I
Outcome 3
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
47
growth modeling.
Growth Mixture Model
Y1
Y2
I
X
Y3
Y4
Y5
S
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
2
138
0.31343
0.68657
Average Latent Class Probabilities for Most Likely Latent
Class Membership (Row) by Latent Class (Column)
1
2
1
0.859
0.114
2
0.141
0.886
51
SoCRATES Analysis in MPlus
Fixed Part of the Model
MODEL RESULTS
Estimate
Bootstrap
S.E.
Est./S.E.
Two-Tailed
P-Value
INTER
|
PANTOT
PAN1
PAN3
PAN9
PANT18
1.000
1.000
1.000
1.000
1.000
0.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
999.000
SLOPE
|
PANTOT
PAN1
PAN3
PAN9
PANT18
0.000
1.946
2.565
3.611
4.369
0.000
0.000
0.000
0.000
0.000
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
Bootstrap
Estimate
Quadratic |
PANTOT
PAN1
PAN3
PAN9
PANT18
0.000
3.787
6.579
13.039
19.092
S.E.
Est./S.E.
Two-Tailed
P-Value
0.000
0.000
0.000
0.000
0.000
999.000
999.000
999.000
999.000
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
Q
WITH
INTER
Means
INTER
Q
Intercepts
SLOPE
Variances
INTER
Q
1.808
1.644
1.100
0.271
0.543
1.628
0.334
0.739
90.444
2.269
3.441
0.460
26.281
4.928
0.000
0.000
-17.118
2.697
-6.346
0.000
101.743
-0.149
53.936
0.419
1.886
-0.356
0.059
0.722
54
SoCRATES Analysis in MPlus
Latent Class 1 – Low Alliance Group
Residual Variances
PANTOT
PAN1
PAN3
PAN9
PANT18
SLOPE
155.448
225.422
134.520
96.053
37.002
9.209
25.475
33.614
29.049
33.022
24.554
8.681
6.102
6.706
4.631
2.909
1.507
1.061
0.000
0.000
0.000
0.004
0.132
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
Q
WITH
INTER
Means
INTER
Q
Intercepts
SLOPE
Variances
INTER
Q
-2.843
1.136
-2.502
0.012
0.543
1.628
0.334
0.739
87.844
1.784
1.954
0.309
44.955
5.769
0.000
0.000
-11.558
1.763
-6.556
0.000
163.426
0.625
33.921
0.298
4.818
2.095
0.000
0.036
57
SoCRATES Analysis in MPlus
Latent Class 2 – High Alliance Group
Residual Variances
PANTOT
PAN1
PAN3
PAN9
PANT18
SLOPE
155.448
225.422
134.520
96.053
37.002
-4.728
25.475
33.614
29.049
33.022
24.554
5.382
6.102
6.706
4.631
2.909
1.507
-0.878
0.000
0.000
0.000
0.004
0.132
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.
• Occasion i, subject j
Outcome 1
Outcome 2
1
1
1
I
Outcome 3
1
2
S
yij  1   2 xij   1 j   2 j xij   ij
fixed part
random part
62
Latent Trajectory Models
yij  1   2 xij   1 j   ij
fixed part
random part
• Here the ς1j are represented by discrete trajectory
classes c with probability πc:
(ςit | c) = e1c + e2ct
where
• e1c is the trajectory origin or intercept for class c
• e2c is the trajectory slope for class c
• Prevalence of trajectory class is πc
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. Unconditional trajectory classes and conditional class
probabilities
– Allow the probability to depend on covariates
3. 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)
y3
y2
y1
1
1
1
1
y4
b2
b1
1
S
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 overestimating 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 k1 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)
: better but not good
BIC = -2logL+plog(n)
: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
70
Methods in Medical Research. 19(3), pp 271-290.
SoCRATES example: Increasing classes
and fit criteria
71
25
20
15
10
Positive symptoms
30
Trajectories for the 4-class model for the
SoCRATES study
0
1
2
logweek
Class 1 (42%)
Class 3 (37%)
3
4
Class 2 (6%)
Class 4 (15%)
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Class assignment certainty
Assigned class
Average
probability
Class 1
Average
probability
Class 2
Average
probability
Class 3
Average
probability
Class 4
Class 1
0.84
0.02
0.14
0.00
Class 2
0.00
0.87
0.09
0.04
Class 3
0.18
0.03
0.75
0.04
Class 4
0.00
0.03
0.11
0.86
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Prevalence of treatment by class
Class
Control
Treatment
Total
1
39 (30%)
90 (70%)
129
2
8 (62%)
5 (32%)
13
3
37 (31%)
83 (69%)
120
4
19 (40%)
28 (60%)
47
Total
103 (33%)
206 (67%)
309
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Latent Class Growth Model (LCGM) with
predictors of class membership
y3
y2
y1
1
1
1
1
y4
b2
b1
1
S
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)
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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.
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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
Symptoms 0
Substance
Use 6
Substance
Use 12
Substance
Use 18
I
S
I
S
Symptoms 12
Substance
Use 24
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.
84
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.
85