Estimation and Adjustment of Bias in Randomised Evidence Using Mixed Treatment Comparison Meta-analysis

Sofia Dias, NJ Welton, AE Ades

with Valeria Marinho, Georgia Salanti, Julian Higgins

Avon RSS, May 2010

Department of Community Based Medicine

2

Overview

• Motivation• Treatment networks and MTC

• Adjusting for Bias in Mixed Treatment Comparisons Meta-analysis (MTC)• The MTC model• Example: Fluoride dataset• Probability of bias model

• Results and Conclusions

3

Mixed Treatment Comparisons

• Often more than two treatments for a given condition• Network of trials comparing different interventions

for a condition• Direct and indirect evidence available on treatment effects

• Because of the network structure, there is enough information to estimate and adjust for bias within the network

• For bias adjustment, there is no need to rely on exchangeability assumption between meta-analyses in different fields

4

Example: The Fluoride Data

• 6 different interventions for preventing dental caries in children and adolescents

1. No Treatment

2. Placebo

3. Fluoride in Toothpaste

4. Fluoride in Rinse

5. Fluoride in Gel

6. Fluoride in Varnish

• From 6 Cochrane Reviews*

Active Treatments

*Marinho et al., 2002; 2003; 2004 (Cochrane Library)

5

Network and Number of trials

Pl

NT

G

RV

T

691

31

133

1

4

9

46

31

4

1

• 130 trials • eight 3-arm trials • one 4-arm trial

• 150 pairwise comparisons

6

Introduction to MTC

1. Six treatments 1,2,3,4,5,6

2. Take treatment 1 (No Treatment) as reference

3. Then the treatment effects d1k of all other treatments relative to 1 are the basic parameters

4. Given them priors:

d1,2, d1,3,…, d1,6~ N(0,1002)

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Functional parameters in MTC• The remaining contrasts are functional parameters

d2,3 = d1,3 – d1,2

d2,4 = d1,4 – d1,2

…

d4,6 = d1,6 – d1,4

d5,6 = d1,6 – d1,5

• Any information on functional parameters tells us indirectly about basic parameters

• Either FE or RE model satisfying these conditions

CONSISTENCY assumptions

1 2 3

Notation• Data

i = 1,…,130 study index

k = 1, 2, 3,…,6 treatment index

rik – number of caries occurring in trial i, treatment k, during the trial follow-up period

Eik – exposure time in arm k of trial i

(in person years)

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Fluoride: Poisson MTC RE model

rate at which events occur in arm k of trial i

~ Poisson( )ik ik ikr E

Exposure time in person years

1

21, 1,~ ,

ik iik t tN d d

MTC consistency equations

21,

2

~ (0,10)

~ (0,100 ) 2,...,6

~ (0,100 ) 1,...,130

j

i

U

d N j

N i

Priors

1log( )ik i ik kI

i = 1,…,130

10

MTC results: LHR relative to No Treatment

Pl

T

R

G

V

-.8 -.6 -.4 -.2 0

Residual deviance is 278.6 (270 data points)

11

Posterior mean of residual deviances for each point

20 40 60 80 100 120

01

23

45

6

study number

MT

C

102

42

42

63

Check how evidence is combined in the network

• Poor fit can indicate inconsistency in the network

• For each pair, separate direct evidence from indirect evidence implied by the rest of the network*

• Can see how evidence is combined in the network to give overall MTC estimate

• Helpful to locate pairs of comparisons where there may be problems

12*Dias et al., Stats in Med. 2010

LHR for Placebo v Toothpaste

Bayesian p-value = 0.32

-1.0 -0.5 0.0 0.5

05

10

15

(Pl,T) is split

log-hazard ratio

De

nsi

ty

Direct

Indirect

MTC

13

LHR for Placebo v Varnish

-1.0 -0.5 0.0 0.5

05

10

15

(Pl,V) is split

log-hazard ratio

De

nsi

ty

DirectIndirect

MTC

Bayesian p-value = 0.04

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LHR for Rinse v Varnish

-1.0 -0.5 0.0 0.5

05

10

15

(R,V) is split

log-hazard ratio

De

nsi

ty

DirectIndirect

MTC

Bayesian p-value = 0.02

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BIAS MODELS

But we have additional information on the risk of bias of all included studies

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TreatmentsNo of

studies

Allocation

concealmentBlinding

NT P T R G V adequateunclea

rinadequate Double

Single

?

1 0 1 0 0 1 0

4 1 3 0 3 1 0

3 0 3 0 1 0 2

1 0 1 0 1 0 0

3 0 2 1 0 2 1

9 0 5 4 0 6 3

4 0 3 1 0 3 1

61 8 46 7 61 0 0

25 2 20 3 22 0 3

9 0 6 3 9 0 0

3 0 3 0 3 0 0

1 0 1 0 1 0 0

1 0 0 1 0 1 0

4 0 3 1 2 2 0

1 0 1 0 0 1 0

Total 130 11 98 21 103 17 10

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MTC RE model with bias

1log( )ik i k i kiik X I

1

21, 1,~ ,

ik iik t tN d d

MTC consistency equations

21,

2

~ (0,10)

~ (0,100 ) 2,...,6

~ (0,100 ) 1,...,130

k

i

U

d N k

N i

Priors

1 if study at risk of bias

0 otherwiseiX

~ ( , )ik iN b

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MTC Bias Model

• Assume non-zero mean bias, bi = b ≠ 0, in comparisons of NT or Pl with Active treatments

• For Active-Active comparisons assume mean bias is zero

• Expect bias to increase size of treatment effect: b < 0

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Fluoride: Risk of Bias indicators• Allocation concealment

• Best empirical evidence of bias• But… 98/130 studies ‘unclear’• Only 11/130 studies ‘adequate’• Some comparisons have no adequately concealed

trials

• Blinding also available to inform risk of bias status• Used “Any bias” as a composite indicator of bias:

54/130 studies at risk of bias.

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Probability of Bias Model• Any study with unclear allocation

concealment has a probability p of being at risk of bias

• Adequately concealed trials are not at risk of bias

• Inadequately concealed trials are at risk of bias• Use only allocation concealment as bias

indicator• Bias terms identifiable in this rich network

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Probability of Bias Model

1ik i ik ik i kX I

1 if allocation concealment is inadequate

if allocation concealment is unclear

0 if allocation concealment is adequatei iX B

~ Bernoulli( ) and ~ (1,1)iB p p Beta

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Comparing Model Fit

ResDev* pD DICBetween trial heterogeneity

MTC with no bias adjustment 278.6 259.3 537.9 0.22 (0.19, 0.26)

Bias adjustment

AnyBias 277.6 257.9 535.5 0.15 (0.12, 0.18)

Probability of bias 274.6 253.0 527.6 0.12 (0.10, 0.15)

* Compare with 270 data points

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Posterior mean of residual deviances for each point: MTC and Prob of bias models

0 1 2 3 4 5 6

01

23

45

6

MTC

Pro

ba

b o

f bia

s

10242

4263

Study 42: Placebo v Toothpaste (1 of 69 trials)Allocation concealment unclearStudy 63: No Treat v Varnish (1 of 4 trials)Allocation concealment unclear and not “double blind”Study 102: Placebo v Varnish (1 of 3 trials)Allocation concealment unclear

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Treatment effects relative to No Treatment (LHR)Unadjusted MTC (solid) and Probability of Bias model (dashed)

Pl

T

R

G

V

-.8 -.6 -.4 -.2 0

Varnish effects

• Cochrane Review to assess efficacy of Fluoride Varnish (Marinho et al, 2004)

• Noted that the small number and poor methodological quality of varnish trials might be overestimating the true effect of this intervention.

• The results of the bias-adjusted analysis support this hypothesis.

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Which treatment is best?Unadjusted MTC Bias-adjusted MTC

Probability Best (%) Rank

Probability Best (%) Rank

No Treatment 0 6 0 6

Placebo 0 5 0 5

Toothpaste 3.6 2.9 9.3 2.7

Rinse 4.1 2.8 53.8 1.6

Gel 3.7 3.2 12.4 2.9

Varnish 88.5 1.2 24.6 2.8

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Results: Probability of Bias• Bias

• posterior mean = -0.19, CrI (-0.36, -0.02)

• posterior sd = 0.40, CrI (0.29, 0.55)

• Trials with unclear allocation concealment are at risk of bias with probability p• Posterior mean of p = 0.13

• Model identified 5 trials (with unclear allocation concealment) as having a high probability of bias

Prob of bias for studies with unclear allocation concealment

0 20 40 60 80 100 120

0.0

0.2

0.4

0.6

0.8

1.0

Study

Pro

po

rtio

n

1721 42 63 102 142143144145146147148149150151

o – unclear allocation concealment+ – unclear allocation concealment and single blind∆ – unclear allocation concealment and unclear blinding status

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Other findings

• Between trial heterogeneity in treatment effects reduced in bias-adjusted model

• Model with Active-Active bias was also fitted with similar results: Active-Active bias had posterior mean of zero• But assumptions on direction of bias…• Assumed bias would favour the newest treatment

(also the most intensive)

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Conclusions• Bias estimation and adjustment possible within

MTC because there is a degree of redundancy in the network

• Assumption that study specific biases are exchangeable within the network• Uses only internal evidence• Weaker than required from using external evidence

• Ideas extend to multiple bias indicators• But will need a very rich evidence structure

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Consequences for Decision Modelling

• Uses only internal evidence • May be more acceptable to patient groups,

pharmaceutical industry…• Risk of bias indicator chosen based on empirical

research• Results may change if different bias indicators

chosenAgain:• Assessment of model fit & sensitivity analysis

crucial if decisions based on these models are to have credence

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References• Our website: http://bristol.ac.uk/cobm/research/mpes

• Dias S, Welton NJ, Marinho VCC, Salanti G, Higgins JPT and Ades AE (2010) Estimation and adjustment of Bias in randomised evidence using Mixed Treatment Comparison Meta-analysis. Journal of the Royal Statistical Society A, to appear Vol 173 issue 4 (available online).

• Dias S, Welton NJ, Caldwell DM and Ades AE (2010) Checking consistency in mixed treatment comparison meta-analysis. Statistics in Medicine, 29, 945-955.

• Schulz KF, Chalmers I, Hayes RJ and Altman DG (1995) Empirical Evidence of Bias. Dimensions of Methodological Quality Associated With Estimates of Treatment Effects in Controlled Trials. JAMA, 273, 408-412.