SESSION 2Mediation and moderation of treatment

effects Andrew Pickles

Methods of explanatory analysis for psychological treatment

trials workshop

Methodology Research Group

Funded by:MRC Methodology Grant G0600555

MHRN Methodology Research Group

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Moderators & Mediators

• Moderator is a variable that modifies the form or strength of the relation between an independent and a dependent variable.

• Mediator is a variable that is intermediate in the causal sequence relating an independent variable to a dependent variable.

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Moderators in RCTs

• Moderators are baseline characteristics that influence the effect of treatment, or the effect of treatment allocation (on intermediate or final outcomes).

• They are pre-randomisation effect-modifiers.

• Examples: sex, age, previous history of mental illness, insight, treatment centre, therapist characteristics, genes etc.

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Typical local example

Creed et al., Psychosomatic Medicine 67:490–499 (2005)

Figure 2. SF36 scores by abuse categories at baseline and follow-up (treated patients only)

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Testing for Moderation

• A moderator variable is typically a baseline variable (e.g. not-abused, abused)

• Makes treatment effect greater in one group than another (moderator may or may not have an additional direct effect on outcome). It is a source of treatment effect heterogeneity

• A classic error is to claim moderation when treatment effect is significant effect in one group and not significant in another. Is simply a recipe for increasing Type I (false positive) error rate

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Interaction & Synergy

• Need to show significant interaction with treatment on outcome

• But on what scale?– Can find that interaction significant on one scale but is not

significant if outcome variable is transformed. Choice of scale requires both statistical and clinical considerations.

– If outcome binary then usual test is for interaction on the log-odds scale

• Some argue that main effects on log-odds scale already suggests synergy

– e.g. if the base outcome rate is low and the treatment and moderator each increase outcome by 100% then the two together increase the outcome rate not by 200% but by 300% even without an interaction

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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: Analysed as two conditions

Control condition: Treatment as Usual (TAU),

Treatment condition: TAU plus psychological, either CBT + TAU or SC + TAU.

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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).

• 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.

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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). .

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SoCRATES - Summary Statistics

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

Lewis et al, BJP (2002); Tarrier et al, BJP (2004); Dunn & Bentall, Stats in Medicine (2007).

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Socrates positive symptoms:basic analysis

xi: regress enpstot psubtota rgrp

Source | SS df MS Number of obs = 225-------------+------------------------------ F( 2, 222) = 14.80 Model | 792.779676 2 396.389838 Prob > F = 0.0000 Residual | 5945.22032 222 26.7802717 R-squared = 0.1177-------------+------------------------------ Adj R-squared = 0.1097 Total | 6738 224 30.0803571 Root MSE = 5.175

------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- psubtota | .34999 .0785569 4.46 0.000 .1951774 .5048026 rgrp | -2.240193 .7425587 -3.02 0.003 -3.703559 -.7768275 _cons | 6.986856 1.954491 3.57 0.000 3.135127 10.83859------------------------------------------------------------------------------

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Socrates positive symptoms:including main effects of centre

xi: regress enpstot psubtota i.centre rgrp

------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- psubtota | .1710413 .0847491 2.02 0.045 .0040172 .3380653 _Icentre_2 | 1.679312 .8588526 1.96 0.052 -.0133193 3.371944 _Icentre_3 | -2.857869 .823287 -3.47 0.001 -4.480408 -1.235331 rgrp | -2.158757 .7039854 -3.07 0.002 -3.546176 -.7713389 _cons | 11.42025 2.038804 5.60 0.000 7.402161 15.43833------------------------------------------------------------------------------

testparm _Icen*

( 1) _Icentre_2 = 0 ( 2) _Icentre_3 = 0

F( 2, 220) = 13.56 Prob > F = 0.0000

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Socrates positive symptoms:treatment effect by centre

xi:xi: bysort centre : regress enpstot psubtota rgrp

-> centre = 1------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- psubtota | .1686252 .1880974 0.90 0.373 -.2066189 .5438693 rgrp | -3.439661 1.348812 -2.55 0.013 -6.130467 -.7488547 _cons | 12.34583 4.501713 2.74 0.008 3.365161 21.3265-------------------------------------------------------------------------------> centre = 2------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- psubtota | .0768268 .1548842 0.50 0.621 -.2314624 .385116 rgrp | -1.785964 1.448293 -1.23 0.221 -4.668719 1.096791 _cons | 15.31862 4.007697 3.82 0.000 7.341504 23.29575-------------------------------------------------------------------------------> centre = 3------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- psubtota | .3170135 .086804 3.65 0.001 .1437987 .4902283 rgrp | -.7935641 .7000109 -1.13 0.261 -2.190414 .6032859 _cons | 4.4636 2.042415 2.19 0.032 .3880247 8.539176------------------------------------------------------------------------------

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Socrates positive symptoms:moderation/heterogeneity?

xi: regress enpstot psubtota i.centre*rgrp

------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- psubtota | .1685492 .0866722 1.94 0.053 -.0022736 .339372 _Icentre_2 | .6983508 1.398679 0.50 0.618 -2.058313 3.455015 _Icentre_3 | -4.532945 1.481842 -3.06 0.002 -7.453517 -1.612374 rgrp | -3.439764 1.245653 -2.76 0.006 -5.894829 -.9846987_IcenXrgrp_2 | 1.458311 1.720016 0.85 0.397 -1.931679 4.848301_IcenXrgrp_3 | 2.418837 1.779205 1.36 0.175 -1.087808 5.925483 _cons | 12.3476 2.257408 5.47 0.000 7.898459 16.79674------------------------------------------------------------------------------

testparm _IcenX*

( 1) _IcenXrgrp_2 = 0 ( 2) _IcenXrgrp_3 = 0

F( 2, 218) = 0.94 Prob > F = 0.3912

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Mediators in Randomised Clinical Trials (RCTs)

• Mediators are intermediate outcomes on the causal pathway between allocation to or receipt of treatment and final outcome.

• By definition, in an RCT, they are measured after randomisation.

• Treatment effect may be fully or partially explained by a given mediator. Possible for a given mediator to serve the role of surrogate outcome.

• Possibility of multiple mediators (multiple pathways) and interactions between mediators.

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Post-randomisation effect modifiers

• Intermediate outcomes that influence either (a) the effects of treatment/treatment allocation on other intermediate outcomes (mediators) or (b) the effects of the other intermediate outcomes on the final outcome.

• Candidates: amount of treatment (sessions attended), treatment fidelity, therapeutic alliance.

• Distinction between these variables and mediators not

obvious.

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Examples

Compliance with allocated treatment Does the participant turn up for any therapy?

How many sessions does she attend?

Fidelity of therapyHow 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 relationshipWhat is the strength of the therapeutic alliance?

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Examples (cont.)

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?

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?

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The Mediation Industry

Baron RM & Kenny DA (1986). The moderator-mediator variable distinction in social psychological research: conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology 51, 1173-1182.

As of 16th September 2009: 12,292 citations!

Assumptions are very rarely stated, let alone their validity discussed.

One suspects that the majority of investigators are oblivious of the implications.

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Number ofsessions

Mediator M

PsychoticSymptomsDependent

Y

Randomised toPsych treatmentIndependent X

e3 e2

Regression eqns used to assess mediation

Y=d1+cX+e1

Y=d2+c’X+bM+e2

M=d3+aX+e3

total effect=c mediated effect= ab or (c-c’) (in simple linear models these should be equal if estimated on same sample)

a

b

c’

A Naïve Look at mediation: the B&K framework

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Testing for Mediation

Estimate of mediated effect =

Confidence interval +/- 1.96*seab

Estimate of seab = sqrt( seb2

+ sea2)

Bootstrap resampling better (allows for asymmetry)

Test of mediation (1) if 0 within CI

(2) z-test for /seab

ab̂^

a2^ b2^

ab̂^

ab̂^

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Baron & Kenny Steps: naïve mediation

• Effect of X on Y (c) must be significant

• Effect of X on M (a) must be significant

• Effect of M on X (b) must be significant

• When controlling for M, the direct effect of X on Y (c’) must be non-significant

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naïve mediation

xi:regress nosess rgrp psubtota i.centre------------------------------------------------------------------------------ nosess | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- rgrp | 13.82383 .5893788 23.45 0.000 12.66366 14.98401 psubtota | .1047549 .0649339 1.61 0.108 -.0230656 .2325754 _Icentre_2 | -1.387014 .7189374 -1.93 0.055 -2.802223 .0281941 _Icentre_3 | -2.87773 .7188629 -4.00 0.000 -4.292792 -1.462668 _cons | -1.210907 1.551379 -0.78 0.436 -4.264754 1.84294------------------------------------------------------------------------------

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naïve mediation

xi:regress enpstot nosess rgrp psubtota i.centre------------------------------------------------------------------------------ enpstot | Coef. Std. Err. t P>|t| [95% Conf. Interval]-------------+---------------------------------------------------------------- nosess | .0417879 .0795974 0.52 0.600 -.1151377 .1987135 rgrp | -2.81782 1.345742 -2.09 0.037 -5.470936 -.1647028 psubtota | .1606631 .0871823 1.84 0.067 -.011216 .3325421 _Icentre_2 | 1.926335 .9083031 2.12 0.035 .1356243 3.717046 _Icentre_3 | -2.54384 .9285473 -2.74 0.007 -4.374462 -.7132184 _cons | 11.47856 2.103109 5.46 0.000 7.332299 15.62482------------------------------------------------------------------------------

A=13.80 (0.59) , B=0.042 (0.08): A times B =0.58 (1.10)

Sobel estimate of standard error sqrt(13.82*0.082+0.0422*0.592)=1.10

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Stata code for naïve mediation: bootstrap 1

global model1 “nosess rgrp psubtota i.centre"global model2 “enpstota nosess rgrp psubtota i.centre"

program mediate, rclassversion 8xi:regress $model1matrix a=e(b)xi:regress $model2matrix b=e(b)return scalar mediate=a[1,1]*b[1,1]end

bootstrap mediate product=r(mediate), reps(100) dots

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Stata code for naïve mediation: bootstrap 2

bootstrap mediate product=r(mediate), reps(100) dots

command: mediatestatistic: product = r(mediate).................................................................................

...................

Bootstrap statistics Number of obs = 213 Replications = 100

------------------------------------------------------------------------------Variable | Reps Observed Bias Std. Err. [95% Conf. Interval]-------------+---------------------------------------------------------------- product | 100 .5776688 -.1432935 1.057901 -1.521436 2.676773 (N) | -1.682636 2.333766 (P) | -1.682636 2.333766 (BC)------------------------------------------------------------------------------Note: N = normal P = percentile BC = bias-corrected

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Mediation and measurement error: mediation or direct and indirect effects in SEM (Mplus)

• Testing for and estimating mediation can be susceptible to measurement error bias

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Direct and Indirect: Longitudinal and sleeper effects

y1 y2 y3 y4a b c

• y1 directly influences y2 through path a

• y1 only indirectly influences y3 through y2 on paths a and b

• In a longitudinal study if y1 influences y3 directly (i.e. not through y2) this is a ‘sleeper effect’

• This structure of restricting effects to those from the previous occasion is known as first order autorgression (AR1)

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Longitudinal Ability Data% correct at ages 6,7, 9 and 11

STANDARD DEVIATIONS6.374 7.319 7.796 10.386

CORRELATION MATRIX10.809 10.806 0.850 10.765 0.831 0.867 1

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AR1 Model: ability1.inp

TITLE: Ability autoregressive model

DATA: FILE IS D:\courses\mplus\ability.dat;

TYPE IS CORRELATION STDEVIATIONS;

NOBSERVATIONS=204;

VARIABLE: NAMES ARE y1-y4;

USEVARIABLES ARE y1-y4;

MODEL: y2 on y1;

y3 on y2;

y4 on y3;

OUTPUT: SAMPSTAT STANDARDIZED RESIDUAL;

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Indirect effects: Ability1b.inp

TITLE: Ability latent autoregressive modelDATA: FILE IS D:\courses\mplus\ability.dat; TYPE IS CORRELATION STDEVIATIONS; NOBSERVATIONS=204;VARIABLE: NAMES ARE y1-y4; USEVARIABLES ARE y1-y4;MODEL: y2 on y1; y3 on y2; y4 on y3;MODEL INDIRECT:

y4 IND y1; y3 IND y1;OUTPUT: STANDARDIZED; CINTERVAL;

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AR1 Model Output-1

Effects from Y1 to Y4 Total 0.971 0.076 12.814 0.971 0.596 Total indirect 0.971 0.076 12.814 0.971 0.596

Specific indirect Y4 Y3 Y2 Y1 0.971 0.076 12.814 0.971 0.596

Effects from Y1 to Y3 Total 0.841 0.056 14.956 0.841 0.688 Total indirect 0.841 0.056 14.956 0.841 0.688

Specific indirect Y3 Y2 Y1 0.841 0.056 14.956 0.841 0.688

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Autoregressive Output-1

Chi-Square Test of Model Fit Value 62.124 ! This fits Degrees of Freedom 3 ! Very badly P-Value 0.0000

ESTIMATED MODEL AND RESIDUALS (OBSERVED - ESTIMATED) Model Estimated Covariances/Correlations/Residual Correlations Y2 Y3 Y4 Y1 ________ ________ ________ ________ Y2 53.305 Y3 48.263 60.481 Y4 55.745 69.857 107.341 Y1 37.556 34.003 39.275 40.429

Residuals for Covariances/Correlations/Residual Correlations Y2 Y3 Y4 Y1 ________ ________ ________ ________ Y2 0.000 Y3 -0.001 -0.001 Y4 7.114 -0.001 -0.001 Y1 0.000 5.852 11.120 0.000

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Autoregressive Output-2

TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS Estimates S.E. Est./S.E. Std StdYXEffects from Y1 to Y4 Total 0.971 0.076 12.814 0.971 0.596 Total indirect 0.971 0.076 12.814 0.971 0.596 Specific indirect Y4 Y3 Y2 Y1 0.971 0.076 12.814 0.971 0.596Effects from Y1 to Y3 Total 0.841 0.056 14.956 0.841 0.688 Total indirect 0.841 0.056 14.956 0.841 0.688 Specific indirect Y3 Y2 Y1 0.841 0.056 14.956 0.841 0.688

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Simplex Model

f1 f2 f3 f4

y1 y2 y3 y4

Age 6 Age 7 Age 9 Age 11

V’s measured with errorAutoregressive F’s

Curiously, middle part of model is identified without restrictions, but the whole model is not identified without some restrictive assumptions e.g. measurement error and reliability constant with age

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Simplex Model: ability2.inpTITLE: Ability latent autoregressive modelDATA: FILE IS D:\courses\mplus\ability.dat; TYPE IS STDEVIATIONS CORRELATION; NOBSERVATIONS=204; VARIABLE: NAMES ARE y1-y4; USEVARIABLES ARE y1-y4;MODEL: f1 by y1 (1); f2 by y2 (1); f3 by y3 (1); f4 by y4 (1); y1 y2 y3 y4 (2); f2 on f1; f3 on f2; f4 on f3;MODEL INDIRECT: f3 IND f1; f4 IND f1;OUTPUT: STANDARDIZED;

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Simplex Model: ability2.out

TESTS OF MODEL FITChi-Square Test of Model Fit Value 1.440 Degrees of Freedom 2 P-Value 0.4835

TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS

Estimates S.E. Est./S.E. Std StdYXEffects from F1 to F3 Total 1.170 0.074 15.901 0.925 0.925 Total indirect 1.170 0.074 15.901 0.925 0.925 Specific indirect F3 F2 F1 1.170 0.074 15.901 0.925 0.925Effects from F1 to F4 Total 1.516 0.105 14.396 0.877 0.877 Total indirect 1.516 0.105 14.396 0.877 0.877

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Simplex Model: conclusion

• Conclusion.In the presence of measurement error in the mediator the mediated effect is underestimated (attenuated) and the residual “direct” effect over-estimated.

With multiple predictors (mediators) measurement error can result in decreased, increased and quite spurious effects being estimated.

• But still ignores possible confounding – to be addressed this afternoon