Transcript Slide 1

Methods of explanatory analysis
for psychological treatment
trials workshop
Session 3
Analysis of mediation and moderation using
instrumental variables
Richard Emsley
Funded by:
MRC Methodology Grant G0600555
MHRN Methodology Research Group
Methodology Research Group
Plan for session 3
• Quick review of instrumental variables from Ian’s
talk.
• Why do we use instrumental variables?
• Where do we find instrumental variables?
• Examples:
– PROSPECT mediator example
– SoCRATES S+A*S model.
• Designing trials with instruments in mind.
2
Quick review of IVs from Ian’s talk…
• Ian has demonstrated how we can use instrumental
variable methods to infer a causal effect of treatment
in the presence of departures from randomised
intervention.
• This utilises randomisation as the instrumental
variable. As we will see, randomisation meets the
assumptions required for an IV…
• But we will also need to consider the situation where we
cannot use randomisation as an instrument…
3
Instrumental Variables (IVs)
• In a standard regression model, if an explanatory variable
is correlated with the error term (known as endogeneity) its
coefficient cannot be unbiasedly estimated.
• An instrumental variable (IV) is a variable that does not
appear in the model, is uncorrelated with the error term
and is correlated with the endogenous explanatory variable;
randomisation, where available, often satisfies this criteria.
• A two stage least squares (2SLS) procedure can then be
applied to estimate the coefficient. At its simplest, the first
stage involves using a simple linear regression of the
endogenous variable on the instrument and saving the
predicted values. In the second stage the outcome is then
regressed on the predicted values, with the latter
4
regression coefficient being the required estimate of the
Some notation
• Ri – treatment group: the outcome of randomisation
(Ri=1 for treatment, 0 for controls).
• Xi′ = X1i, X2i … Xpi
– baseline covariates.
• Yi – observed outcome.
• Di – actual treatment received. This is an intermediate
outcome that is a putative mediator of the effects of
treatment on outcome (either a quantitative measure or
binary).
5
Instrumental variables (IV) (from session 1)
• Popular in econometrics
• Simplest idea is:
– Outcome:
Yi = a + b Di + ei
– Treatment: Di = g + d Ri + fi
– Allow error ei to be correlated with Di but assume it’s
independent of Ri
» randomisation Ri only affects outcome through its
effect on compliance Di
• Estimation by “two-stage least squares”:
– E[Yi | Ri] = a + b E[Di | Ri]
– so first regress Di on Ri to get E[Di | Ri]
– then regress Yi on E[Di | Ri]
– NB standard errors not quite correct by this method:
general IV uses different standard errors
6
Simple Mediation Idea (from session 2)
dX
Mediator
α
Treatment
γ
β
Outcomes
dY
The total effect is the sum of the direct effect (γ) and the indirect effect (α*β)
7
Confounded Mediation Diagram
dX
U – the unmeasured confounders
U
Mediator
α
Treatment
γ
β
Outcomes
If treatment is randomised then assumption of no confounding
of treatment and other variables (outcomes) is justified.
dY
8
Confounded Mediation Diagram
dX
U
U
Mediator
α
Treatment
γ
β
Outcomes
dY
U
If treatment is not randomised then there is likely to be even more
unmeasured confounding.
9
Confounded Mediation Diagram
dX
U
Mediator
α
Randomisation
γ
β
Outcomes
dY
Thankfully we’re talking about randomised trials!
10
Linking the two previous sessions:
Compliance as a mediator
dX
Treatment
Received
Randomisation
Outcomes
dY
11
Linking the two previous sessions:
Randomisation as an IV
dX
Treatment
Received
Randomisation
Outcomes
dY
By assuming the absence of a direct path from randomisation
to outcome, we assume the entire effect of randomisation acts
through receipt of treatment.
→ randomisation is an instrumental variable.
12
Plan for session 3
• Quick review of instrumental variables from Ian’s talk.
• Why do we use instrumental variables?
• Where do we find instrumental variables?
• Examples:
– PROSPECT mediator example
– SoCRATES S+A*S model.
• Designing trials with instruments in mind.
13
Why do we use instrumental variables?
• All available statistical methods we usually use (for any
standard analysis), including:
–
–
–
–
Stratification
Regression
Matching
Standardization
• require the one unverifiable condition we identified
previously:
NO UNMEASURED CONFOUNDING
14
Why do we use instrumental variables?
• Unlike all other methods, IV methods can be used to
consistently estimate causal effects in the presence
of unmeasured confounding AND measurement
error.
• SO WE CAN SOLVE THE PROBLEM OF…
dX
U
Mediator
α
Randomisation
γ
β
Outcomes
dY
15
Definition of an instrumental variable
A variable is an instrumental variable Z if:
i. Z has a causal effect on the mediator D;
This can be tested in the data.
ii. Z affects the outcome Y only through D
i.e. there is no direct effect of Z on Y;
This is an assumption (sometimes a strong assumption).
iii. Z does not share common causes with the outcome Y
i.e. there is no confounding for the effect of Z on Y.
This is another assumption which randomisation satisfies
but other IVs may not.
16
Assumptions for instrumental variables
• IV methods require FOUR assumptions
• The first 3 assumptions are from the definition:
– The association between instrument and mediator.
– no direct effect of the instrument on outcome.
– no unmeasured confounding for the instrument and
outcome.
• There are a wide variety of fourth assumptions and
different assumptions result in the estimation of
different causal effects:
– E.g. no interactions, monotonicity (no defiers).
17
Testing assumptions…
• There are a number of tests we can use for some of
these assumptions.
• Stata has three postestimation commands following
ivregress:
– estat overid
– estat endogenous
– estat firststage
• This final option is perhaps the most useful. It gives an
indication of whether the set of instruments strongly
predict the mediator – see PROSPECT example later on.
18
Advantages of IVs
• Can allow for unmeasured confounding;
• Can allow for measurement error;
• Randomisation meets the definition so is an ideal
instrument
– When available.
» Obviously not in observational studies.
19
Disadvantages of IVs
1. It is impossible to verify that Z is an instrument and
using a non instrument introduces additional bias.
2. A weak instrument Z increases the bias over that of
ordinary regression.
3. Instruments by themselves are actually insufficient to
estimate causal effects and we require additional
unverifiable assumptions such as the “no defiers”
assumption.
4. Standard IV methods do not cope well with time-varying
exposures/mediators…yet.
20
See Hernán and Robins (2006), Epidemiology for further details
Assumption trade-off
• IV methods replace one unverifiable assumption of no
unmeasured confounding between the mediator and
the outcome by other unverifiable assumptions
– no unmeasured confounding for the instruments, or
– no direct effect of the instruments.
• We need to decide which assumptions are more likely to
hold in our mediation analysis.
• An IV analysis will also increase the precision of our
estimates because of allowing for the unmeasured
confounding.
21
Also…
• What about if we want to estimate the direct effect of
randomisation in the presence of a potential mediator?
dX
U
Mediator
α
Randomisation
γ
β
Outcomes
Clearly we can’t use randomisation as an
instrument here…we need another instrument.
dY
22
Plan for session 3
• Quick review of instrumental variables from Ian’s talk.
• Why do we use instrumental variables?
• Where do we find instrumental variables?
• Examples:
– PROSPECT mediator example
– SoCRATES S+A*S model.
• Designing trials with instruments in mind.
23
Multiple instruments
• When we are trying to estimate the direct effect of
randomisation we need alternative instruments.
• Likewise, if we have more than one endogenous
variable (multiple mediators), then we need multiple
instruments.
• For IV model identification, we always need to have as
many instruments as we have endogenous variables.
– i.e. if considering two mediators in the model
(therapeutic alliance and number of sessions of
therapy attended), then we need at least two
instrumental variables.
24
Where do we find instruments?
• Possibilities for IVs:
– Randomisation-by-baseline variable
interactions.
– Randomisation involving more than one active
treatment – i.e. to interventions specifically targeted
at particular intermediate variables/mediators.
– Randomisation-by-trial (multiple trials).
– Genetic markers (Mendelian Randomisation) used
together with randomisation.
25
Confounded Mediation Diagram
dX
U – the unmeasured confounders
U
Mediator
α
Randomisation
γ
β
Outcomes
If treatment is randomised then assumption of no confounding
of treatment and other variables (outcomes) is justified.
dY
26
Mediation Diagram with instruments
dX
U – the unmeasured confounders
Randomisation*Covariates
U
Mediator
α
Randomisation
γ
β
Outcomes
dY
Covariates
27
Multiple Instruments
• Here, treatment by covariates interactions represent
instrumental variables.
• Assumptions:
1. The interactions are significant in the first stage
regression (individually and joint F-test).
2. The only effect of the interactions on outcome is
through the mediator, and not a direct effect. This
is a very strong assumption
3. No other unmeasured confounders between the
interactions and outcome.
28
Summary so far…
• The analysis of mediation is more complex than it first
seems because of potential unmeasured confounding
(mediators are endogenous).
• We use moderators of the relationship between
randomisation and the mediator (i.e. the baseline by
randomisation interactions) as instruments.
• The analysis of mediation by instrumental variables
requires additional assumptions. Primarily, that these
covariates are not moderators of the randomisation on
outcome relationship (no direct effect).
• We illustrate these points on two examples now…
29
Plan for session 3
• Quick review of instrumental variables from Ian’s talk.
• Why do we use instrumental variables?
• Where do we find instrumental variables?
• Examples:
– PROSPECT mediator example
– SoCRATES S+A*S model.
• Designing trials with instruments in mind.
30
Example: PROSPECT
• PROSPECT (Prevention of Suicide in Primary Care Elderly:
Collaborative Trial) was a multi-site prospective,
randomised trial designed to evaluate the impact of a
primary care-based intervention on reducing major risk
factors (including depression) for suicide in elderly
depressed primary care patients.
• The two conditions were either:
– (a) an intervention based on treatment guidelines
tailored for the elderly with care management,
– (b) treatment as usual.
• An intermediate outcome in the PROSPECT trial was
whether the trial participant adhered to antidepressant
medication during the period following allocation of the
intervention.
See Bruce et al, JAMA (2004); Ten Have et al, Biometrics (2007); Bellamy et al, Clinical Trials (2007); Lynch et al, Health Services and Outcome Research Methodology
(2008). Thanks to Tom Ten Have for use of the data.
• The question here is whether changes in medication
31
Example: PROSPECT - question of interest
Randomisation*Covariates
Antidepressant
Use
Depression
Score
Randomisation
Covariates
32
Example: PROSPECT - summary stats
Site 1
Control
N=53
Site 2
Intervention
N=53
Site 3
Control
N=57
Intervention
Intervention
N=54
Control
N=42
N=38
Baseline characteristics: number (%)
Antidepressant
Use
22 (41.5)
18 (34.0)
25 (43.9)
25 (46.3)
25 (59.5)
21 (55.3)
Previous
medication
27 (50.9)
24 (45.3)
25 (43.9)
28 (51.9)
29 (69.1)
20 (52.6)
Suicidal ideation
9 (17.0)
13 (24.5)
12 (21.1)
18 (33.3)
13 (31.0)
16 (42.1)
45 (83.3)
30 (71.4)
34 (89.5)
Post-randomisation adherence to antidepressant medication: number (%)
Adherence
20 (37.7)
44 (83.0)
19 (33.3)
Hamilton Depression Rating Scores (HRDS): mean (SD)
Baseline HDRS
16.5 (5.3)
18.1 (6.2)
17.3 (5.3)
19.9 (6.4)
18.6 (6.3)
18.7 (5.9)
4 month HDRS
13.4 (8.1)
12.0 (7.8)
14.1 (8.6)
12.1 (7.3)
13.0 (8.5)
10.0 (6.9)
33
PROSPECT data – Stata describe
. describe
Contains data from P:\SMinMR paper\Prospect.dta
obs:
297
vars:
8
11 Sep 2009 16:01
size:
20,196 (99.9% of memory free)
-------------------------------------------------------------------------------------------storage display
value
variable name
type
format
label
variable label
-------------------------------------------------------------------------------------------cad1
double %10.0g
Anti-depressant use at baseline visit
hdrs0
double %10.0g
Hamilton depression score at baseline visit
ssix01
double %10.0g
Suicide ideation at baseline visit
scr01
double %10.0g
Past medication use at baseline visit
hdrs4
double %10.0g
Hamilton depression score at 4 month visit
site
double %10.0g
Location of practices
interven
double %10.0g
Randomized assignment to intervention
Amedx
double %10.0g
Adherence to prescribed anti-depressant
medication
--------------------------------------------------------------------------------------------
34
PROSPECT data – Stata ivregress
. xi: ivregress 2sls hdrs4 hdrs0 cad1 ssix01 scr01 i.site i.interven (amedx = i.interven*hdrs0
i.interven*cad1 i.interven*ssix01 i.interven*scr01 i.interven*i.site), first
First-stage regressions
--------------------
Number of obs
=
296
F( 13,
282) =
21.71
Prob > F
=
0.0000
R-squared
=
0.5002
Adj R-squared
=
0.4772
Root MSE
=
0.3465
-----------------------------------------------------------------------------amedx |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hdrs0 |
.0065731
.0051473
1.28
0.203
-.0035588
.0167051
cad1 |
.166495
.0254223
6.55
0.000
.1164533
.2165366
ssix01 | -.0475454
.0721387
-0.66
0.510
-.1895441
.0944533
scr01 |
.2530611
.0746616
3.39
0.001
.1060962
.4000259
_Isite_2 |
-.018463
.0664307
-0.28
0.781
-.149226
.1123
_Isite_3 |
.1969925
.0734302
2.68
0.008
.0524516
.3415334
_Iinterven_1 |
.7825965
.1398924
5.59
0.000
.5072307
1.057962
_IintXhdrs~1 |
-.003633
.0071484
-0.51
0.612
-.0177041
.010438
_IintXcad1_1 |
-.118277
.0341169
-3.47
0.001
-.1854331
-.0511209
_IintXssix~1 |
.0504564
.0967541
0.52
0.602
-.1399956
.2409083
_IintXscr0~1 | -.2627584
.1029091
-2.55
0.011
-.4653259
-.0601909
_IintXsit_~2 | -.0099335
.095321
-0.10
0.917
-.1975645
.1776975
_IintXsit_~3 | -.1681695
.1054282
-1.60
0.112
-.3756956
.0393566
_cons | -.0465641
.0996531
-0.47
0.641
-.2427223
.1495942
------------------------------------------------------------------------------
35
PROSPECT data – Stata ivregress
Instrumental variables (2SLS) regression
Number of obs =
296
Wald chi2(8) = 102.68
Prob > chi2
= 0.0000
R-squared
= 0.2582
Root MSE
= 6.8425
-----------------------------------------------------------------------------hdrs4 |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------amedx |
-1.95302
2.672201
-0.73
0.465
-7.190438
3.284397
hdrs0 |
.6226062
.070337
8.85
0.000
.4847482
.7604642
cad1 | -.0654087
.4304821
-0.15
0.879
-.9091381
.7783208
ssix01 |
1.251204
.9399736
1.33
0.183
-.5911102
3.093518
scr01 |
1.585044
1.074312
1.48
0.140
-.5205695
3.690658
_Isite_2 | -.4971475
.9469522
-0.52
0.600
-2.35314
1.358845
_Isite_3 | -2.046048
1.08319
-1.89
0.059
-4.169062
.0769655
_Iinterven_1 | -2.375598
1.328982
-1.79
0.074
-4.980353
.2291584
_cons |
3.344043
1.467043
2.28
0.023
.4686928
6.219394
-----------------------------------------------------------------------------Instrumented: amedx
Instruments:
hdrs0 cad1 ssix01 scr01 _Isite_2 _Isite_3 _Iinterven_1
_IintXhdrs0_1 _IintXcad1_1 _IintXssix0_1 _IintXscr01_1
_IintXsit_1_2 _IintXsit_1_3
36
Example: PROSPECT - results
Using all baseline variables as covariates in an ANCOVA.
ITT effect:
-3.15 (0.82)
Small but statistically significant effect
Direct effect
γ (s.e.)
Analytical method
Standard regression -2.66 (0.93)
(Baron & Kenny)
Indirect effect
β (s.e.)
-1.24 (1.09)
37
Example: PROSPECT - results
Direct effect
γ (s.e.)
Analytical method
IV (ivreg)
-2.38 (1.35)
IV (treatreg - ml)
-2.34 (1.27)
G-estimation*
-2.58 (1.27)
Indirect effect
β (s.e.)
-1.95 (2.71)
-2.05 (2.49)
-1.43 (2.34)
Conclusion
Allowing for hidden confounding appears to have had little
effect, except to increase the SE of the estimate.
38
*From Ten Have et al, Biometrics (2007)
PROSPECT data – ivregress postestimation
. estat firststage
First-stage regressions
--------------------
Number of obs
=
296
F( 13,
282) =
21.71
Prob > F
=
0.0000
R-squared
=
0.5002
Adj R-squared
=
0.4772
Root MSE
=
0.3465
-----------------------------------------------------------------------------amedx |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------hdrs0 |
.0065731
.0051473
1.28
0.203
-.0035588
.0167051
cad1 |
.166495
.0254223
6.55
0.000
.1164533
.2165366
ssix01 | -.0475454
.0721387
-0.66
0.510
-.1895441
.0944533
scr01 |
.2530611
.0746616
3.39
0.001
.1060962
.4000259
_Isite_2 |
-.018463
.0664307
-0.28
0.781
-.149226
.1123
_Isite_3 |
.1969925
.0734302
2.68
0.008
.0524516
.3415334
_Iinterven_1 |
.7825965
.1398924
5.59
0.000
.5072307
1.057962
_IintXhdrs~1 |
-.003633
.0071484
-0.51
0.612
-.0177041
.010438
_IintXcad1_1 |
-.118277
.0341169
-3.47
0.001
-.1854331
-.0511209
_IintXssix~1 |
.0504564
.0967541
0.52
0.602
-.1399956
.2409083
_IintXscr0~1 | -.2627584
.1029091
-2.55
0.011
-.4653259
-.0601909
_IintXsit_~2 | -.0099335
.095321
-0.10
0.917
-.1975645
.1776975
_IintXsit_~3 | -.1681695
.1054282
-1.60
0.112
-.3756956
.0393566
_cons | -.0465641
.0996531
-0.47
0.641
-.2427223
.1495942
------------------------------------------------------------------------------
39
PROSPECT data – ivregress postestimation
(no endogenous regressors)
( 1) _IintXhdrs0_1 = 0
( 2) _IintXcad1_1 = 0
( 3) _IintXssix0_1 = 0
( 4) _IintXscr01_1 = 0
( 5) _IintXsit_1_2 = 0
( 6) _IintXsit_1_3 = 0
F(
6,
282) =
9.10
Prob > F =
0.0000
First-stage regression summary statistics
-------------------------------------------------------------------------|
Adjusted
Partial
Variable |
R-sq.
R-sq.
R-sq.
F(6,282)
Prob > F
-------------+-----------------------------------------------------------amedx | 0.5002
0.4772
0.1622
9.10057
0.0000
-------------------------------------------------------------------------Minimum eigenvalue statistic = 9.10057
Critical Values
# of endogenous regressors:
1
Ho: Instruments are weak
# of excluded instruments:
6
--------------------------------------------------------------------|
5%
10%
20%
30%
2SLS relative bias
| 19.28
11.12
6.76
5.15
-----------------------------------+--------------------------------|
10%
15%
20%
25%
2SLS Size of nominal 5% Wald test | 29.18
16.23
11.72
9.38
LIML Size of nominal 5% Wald test |
4.45
3.34
2.87
2.61
---------------------------------------------------------------------
40
Instrumental Variables in SPSS
Analyse – Regression –
2-stage Least Squares
Generate interactions as
additional variables using
compute
41
Instrumental Variables in SPSS
Outcome
Covariates and
endogenous variable
(mediator)
Covariates and
instruments
42
Example: 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.
• 201 participants were allocated to one of three groups:
– Control: Treatment as Usual (TAU)
– Treatment: TAU plus psychological intervention,
either CBT + TAU or SC + TAU
– The two treatment groups are combined in our
analyses
• Outcome: psychotic symptoms score (PANSS) at 18
months
43
Example: SoCRATES - summary stats
Centre 1 - Liv
Centre 2 - Man
Centre 3 - Nott
Mean (SD)
Control
N=39
Treated
N=29
Control
N=35
Treated
N=49
Control
N=26
Treated
N=23
Baseline
PANSS
80.0
(12.36)
77.7
(13.93)
97.9
(16.6)
100.5
(16.3)
84.9
(14.91)
83.4
(10.84)
18 month
PANSS
69.5
(13.55)
50.2
(13.48)
73.2
(22.4)
74.4
(20.00)
54.5
(10.07)
49.1
(7.25)
CALPAS
-
5.73
(0.81)
-
5.07
(0.88)
-
5.15
(1.47)
Sessions
0
18.14
(3.60)
0
16.16
(4.58)
0
13.87
(4.95)
High Alliance:
N(%)
-
23
(79.3)
-
30
(61.2)
-
13
(56.5)
# of observed
18m PANSS
23
23
25
39
21
22
44
Lewis et al, BJP (2002); Tarrier et al, BJP (2004); Dunn & Bentall, Stats in Medicine (2007); Emsley, White and Dunn, Stats Methods in Medical Research (2009).
Confounded Dose-Response
dX
U
α
Randomisation
Sessions
Attended
β
Psychotic
Symptoms
dY
Are the effects of Randomisation on Sessions (α) and, more interestingly, the effects of
Sessions on Outcome (β), influenced by the strength of the therapeutic alliance?
45
The S + A*S model
• We want to estimate the joint effects of the strength of the
therapeutic alliance as measured by CALPAS (A) and
number of sessions attended (S).
• We postulate a structural model as follows:
E[Yi(1)-Yi(0)| Xi, Di(1)=s, Di(0)=0 & Ai=a]
βs*s + βsa*s*(a-7)
=
• No sessions implies no treatment effect.
• The effect of alliance is multiplicative so we only have an
interaction effect of alliance – no sessions = no alliance.
46
Dunn and Bentall, SiM (2007)
SoCRATES analysis results
Method
Instrumental variables
Standard regression (B&K)
βs (se)
βsa (se)
-2.40 (0.70) -1.28 (0.48)
-0.95 (0.22) -0.39 (0.11)
Note: A has been rescaled so that maximum=0.
When A=0 (i.e. maximum alliance)
the slope for effect of Sessions is -2.40
When A=-7 (i.e. minimum alliance)
the slope is -2.40 + 7*1.28 = +6.56
This suggests that when alliance is very poor attending more
sessions makes the outcome worse!
47
SoCRATES – S + A*S using regress
. regress pant18
sessions s_a pantot logdup c1 c2 yearsed
Source |
SS
df
MS
-------------+-----------------------------Model | 24414.5544
7 3487.79349
Residual | 32051.4194
145 221.044272
-------------+-----------------------------Total | 56465.9739
152
371.48667
Number of obs
F( 7,
145)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
153
15.78
0.0000
0.4324
0.4050
14.868
-----------------------------------------------------------------------------pant18 |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------sessions | -.9459469
.2209236
-4.28
0.000
-1.382593
-.5093003
s_a | -.3866447
.1117784
-3.46
0.001
-.6075702
-.1657192
pantot |
.3843765
.087454
4.40
0.000
.2115272
.5572259
logdup |
2.331363
2.398488
0.97
0.333
-2.409152
7.071878
c1 |
4.322976
3.48805
1.24
0.217
-2.571014
11.21697
c2 | -11.96141
3.292382
-3.63
0.000
-18.46867
-5.454147
yearsed | -1.110149
.5318061
-2.09
0.039
-2.161242
-.0590559
_cons |
43.94059
11.21352
3.92
0.000
21.77752
66.10366
------------------------------------------------------------------------------
48
SoCRATES – S + A*S using ivregress
. ivregress 2sls pant18 pantot logdup c1 c2 yearsed (sessions s_a = group
c2gp yrgp pgp)
lgp c1gp
First-stage regressions
-----------------------
Number of obs
=
153
F( 11,
141) =
78.68
Prob > F
=
0.0000
R-squared
=
0.8599
Model for sessions
Adj R-squared
=
0.8490
Root MSE
=
3.3588
-----------------------------------------------------------------------------sessions |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------pantot |
1.71e-14
.0310634
0.00
1.000
-.0614103
.0614103
logdup |
2.46e-13
.858628
0.00
1.000
-1.697449
1.697449
c1 | -3.59e-13
1.125814
-0.00
1.000
-2.225657
2.225657
c2 |
4.70e-14
1.022741
0.00
1.000
-2.021889
2.021889
yearsed |
1.17e-13
.1929797
0.00
1.000
-.3815077
.3815077
group |
16.09465
5.201659
3.09
0.002
5.811326
26.37798
lgp |
.1800265
1.104039
0.16
0.871
-2.002583
2.362636
c1gp | -1.281224
1.574428
-0.81
0.417
-4.39376
1.831312
c2gp | -3.772746
1.471898
-2.56
0.011
-6.682588
-.8629052
yrgp |
.1835663
.2475856
0.74
0.460
-.3058935
.6730261
pgp | -.0104563
.0407688
-0.26
0.798
-.0910534
.0701407
_cons | -3.05e-12
4.115125
-0.00
1.000
-8.135319
8.135319
------------------------------------------------------------------------------
49
SoCRATES – S + A*S using ivregress
Model for
sessions*alliance
Number of obs
F( 11,
141)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
153
16.59
0.0000
0.5641
0.5301
12.0225
-----------------------------------------------------------------------------s_a |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------pantot | -1.89e-14
.1111878
-0.00
1.000
-.2198106
.2198106
logdup | -1.89e-13
3.073353
-0.00
1.000
-6.075809
6.075809
c1 |
3.31e-13
4.029712
0.00
1.000
-7.966465
7.966465
c2 | -3.78e-14
3.660775
-0.00
1.000
-7.237101
7.237101
yearsed | -1.00e-13
.6907472
-0.00
1.000
-1.36556
1.36556
group |
-16.2085
18.6187
-0.87
0.385
-53.0164
20.59939
lgp | -6.186983
3.951771
-1.57
0.120
-13.99936
1.625398
c1gp | -11.44637
5.635471
-2.03
0.044
-22.58731
-.3054279
c2gp | -4.923988
5.268477
-0.93
0.352
-15.33941
5.49143
yrgp | -.1321276
.8862022
-0.15
0.882
-1.884089
1.619833
pgp |
.0765408
.1459268
0.52
0.601
-.2119464
.3650281
_cons |
2.96e-12
14.72958
0.00
1.000
-29.11937
29.11937
------------------------------------------------------------------------------
50
SoCRATES – S + A*S using ivregress
Instrumental variables (2SLS) regression
Number of obs
Wald chi2(7)
Prob > chi2
R-squared
Root MSE
=
=
=
=
=
153
83.17
0.0000
0.1795
17.401
-----------------------------------------------------------------------------pant18 |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------sessions | -2.401159
.6776074
-3.54
0.000
-3.729245
-1.073073
s_a | -1.281461
.4380021
-2.93
0.003
-2.139929
-.4229929
pantot |
.3864756
.1024045
3.77
0.000
.1857664
.5871848
logdup | -.2044085
3.091853
-0.07
0.947
-6.264329
5.855512
c1 |
-1.21612
4.868577
-0.25
0.803
-10.75836
8.326116
c2 | -16.32291
4.324444
-3.77
0.000
-24.79866
-7.847155
yearsed | -.9923864
.6258703
-1.59
0.113
-2.21907
.2342968
_cons |
49.26983
13.27743
3.71
0.000
23.24655
75.29311
-----------------------------------------------------------------------------Instrumented: sessions s_a
Instruments:
pantot logdup c1 c2 yearsed group lgp c1gp c2gp yrgp pgp
51
Plan for session 3
• Quick review of instrumental variables from Ian’s talk.
• Why do we use instrumental variables?
• Where do we find instrumental variables?
• Examples:
– PROSPECT mediator example
– SoCRATES S+A*S model.
• Designing trials with instruments in mind.
52
Instrumental Variables in observational
studies
• There are numerous examples of instruments in the
absence of randomisation:
–
–
–
–
–
Access to health care
Distance to hospital
Genes (known as Mendelian randomisation)
Proxy measures of genes (product intolerance)
Physician’s preference (ask, or use proportion of
patients on treatment)
53
Designing trials with IVs in mind
• Thinking back to some of the possibilities for IVs we
introduced earlier with design considerations:
– Randomisation-by-baseline variable interactions.
Can we measure any extra baseline variables?
– Randomisation involving more than one active
treatment – i.e. to interventions specifically targeted
at particular intermediate variables/mediators.
More complicated designs/parallel trials
– Randomisation-by-trial (multiple trials).
Meta-regression approaches (new MRC grant)
– Genetic markers (Mendelian Randomisation) used
together with randomisation.
Need to measure genotype in patients
54
Example: Series of parallel trials
Mediator 1
Trial 1
Common
Outcome
Randomisation 1
Mediator 2
Trial 2
Common
Outcome
Randomisation 2
Mediator 3
Trial 3
Randomisation 3
Common
Outcome
55
Example: measuring additional variables
e.g. Measure of
patient’s interaction
with other
individuals: Care
coordinator, family
members, etc.
e.g. Patient
characteristics
which could
influence ability to
form alliance:
personality
disorders, etc.
Putative mediator is a measure
of the therapist/patient
interaction or relationship
Similar
Baseline
measures
Randomisation
Therapeutic
Alliance
Outcomes
56
Short small group discussion
• We will work in small groups again.
• We are thinking about designing psychological
treatment trials in order to answer some of the
explanatory questions discussed in this session?
• When considering the following potential mediators:
– How would we accurately measure the mediator?
– What additional baseline variables might we be able
to collect which would help in the causal/IV analysis?
– What problems could you foresee in the collection of
this information?
– How might you justify the need to collect this
information to funders of the trials who would prefer
to keep it “large and simple”?
57
Potential mediators for discussion
What are the participant’s beliefs?
Does psychotherapy change attributions (beliefs),
which, in turn, lead to better outcome?
What is the concomitant medication?
Does psychotherapy improve compliance with
medication which, in turn, leads to better outcome?
What is the concomitant substance abuse?
Does psychotherapy reduce substance use, which in
turn leads to improvements in psychotic symptoms?
58
References – Mediation & Effect Moderation
in Psychological Treatment Trials
Methodology for IV methods with mediation:
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 (available online).
Maracy M & Dunn G (2009). Estimating dose-response
effects in psychological treatment trials: the role of
instrumental variables. Statistical Methods in Medical
Research. In press (available online).
Dunn G & Bentall R (2007). Modelling treatment-effect
heterogeneity in randomized controlled trials of complex
interventions (psychological treatments). Statistics in
Medicine 26, 4719-4745.
Website with downloads:
59
http://www.medicine.manchester.ac.uk/healthmethod
ology/research/biostatistics/
Some Further Reading
Ten Have TR, Joffe MM, Lynch KG, Brown GK, Maisto SA & Beck AT (2007). Causal
mediation analyses with rank preserving models. Biometrics 63, 926-934.
Gallop R, Small DS, Lin JY, Elliot MR, Joffe MM & Ten Have TR (2009). Mediation
analysis with principal stratification. Statistics in Medicine 28, 1108-1130.
Bellamy SL, Lin JY & Ten Have TR (2007). An introduction to causal modelling in
clinical trials. Clinical Trials 4, 58-73.
Lynch K, Cary M, Gallop R, Ten Have TR (2008). Causal mediation analyses for
randomized trials. Health Services & Outcomes Research Methodology 8, 57-76.
Albert JM (2008). Mediation analysis via potential outcomes models. Statistics in
Medicine 27, 1282-1304.
Jo B (2008). Causal inference in randomized experiments with mediational processes.
Psychological Methods 13, 314-336.
Gennetian LA, Morris PA, Bos JM & Bloom HS (2005). Constructing instrumental
variables from experimental data to explore how treatments produce effects. In:
Bloom HS, editor. Learning More From Social Experiments: Evolving Analytic
Approaches. New York: Russell Sage Foundation; pp. 75-114.
MacKinnon DP (2008). Introduction to Statistical Mediation Analysis. New York: Taylor
& Francis Group.
60