An introduction to principal stratification

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An introduction to principal stratification

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Title: An introduction to principal stratification

1
Methods of explanatory analysis for psychological
treatment trials workshop

Session 4
An introduction to principal stratification
Graham Dunn

Funded by MRC Methodology Grant G0600555 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 strataPotential
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 strataPotential
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
strataNon-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.
Well leave this one for another day!

10
Estimation of stratum-specific treatment (ITT)
effects

Lets say there are two principal strata, with
proportions p1
and p2 (with p1 p21).
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

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
ITT1 and 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. Lets assume
that
allocation also has no effect on outcome in the
Non-
compliers (an exclusion restriction).
Example
If you dont take the tablets it doesnt 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
(p1p2p31)
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
Whats 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) model to 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 alliance High alliance
a 14.96 (0.96) 16.91 (0.45)
ß 0.59 (0.38) -0.75 (0.23)
IV Structural Equation Model
(with correlated errors hidden confounding)
Low alliance High alliance
a 14.90 (0.97) 16.95 (0.46)
ß 0.37 (0.47) -0.80 (0.29)

a - effect of randomisation on sessions ß -
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 alliance High alliance
a 14.94 (0.95) 16.92 (0.46)
ß 0.55 (0.42) -0.78 (0.28)
IV Structural Equation Model
(with correlated errors hidden confounding)
Low alliance High alliance
a 14.85 (0.98) 16.98 (0.47)
ß 0.34 (0.50) -0.88 (0.37)

a - effect of randomisation on sessions ß -
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 its 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 TYPEMIXTURE
STARTS 100 10
MODEL OVERALL
resp ON logdup pantot yearsed c1 c2
rgroup
pant18 ON logdup pantot yearsed c1 c2
rgroup
C1 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

32
Mplus code (contd.)

C1 ! Low Alliance
alliance1_at_15
! A declared threshold to force participants with
recorded alliance0 ! into this class.
resp1
resp ON rgroup0
pant18
pant18 ON rgroup0
! These statements release the equality
constraints on the relevant
! model intercept terms for the effects of the
randomized
! intervention.
C2 ! High alliance
alliance1_at_-15
! A declared threshold to force participants with
recorded alliance1 ! into this class.
resp1
resp ON rgroup0