How informative priors can help model heterogeneity Network meta-analyses in a sparse evidence bases April 11, 2016

Background Sparse networks are relatively common when dealing with network meta-analyses (NMA) For example, biologics trials can be very expensive leading to few trials describing each comparison When a network is too sparse it may become infeasible to conduct random-effects (RE) NMA If conducted, credible intervals can “explode” Random effects estimates not significant when the pairwise meta-analysis is 2

Psoriatic arthritis example As a motivating example, consider a network of biologics for the treatment of psoriatic arthritis (PsA) The principal outcome was the American College of Rheumatology responder criteria (ACR) 20 (20 percent improvement in tender or swollen joint counts as well as 20 percent improvement in three of the other five criteria) Also of interest was Psoriasis Area and Severity Index (PASI) 50 (reduction in PASI score of at least 50%) 3

placebo Etanercept 25mg biw ACR 20 at 12 weeks – Biologics-naïve Ustekinumab 90mg RAPID-PSA ADEPT Genovese 2007 PSUMMIT1 PSUMMIT2 PSUMMIT1 PSUMMIT2 PSUMMIT1 PSUMMIT2 Mease 2004 GO-REVEAL Golimumab 100mg Golimumab 50mg Ustekinumab 45mg Adalimumab 40mg Certolizumab 400 mg

Relative risk of ACR 20 at 12 weeks 5 Placebo 0.29 (0.22, 0.53) 0.40 (0.23, 1.04) 0.29 (0.20, 0.64) 0.26 (0.19, 0.52) 0.49 (0.30, 1.12) 0.51 (0.30, 1.18) 3.41 (1.88, 4.64) ADA (0.63, 3.43) 0.99 (0.52, 2.13) 0.90 (0.48, 1.73) 0.89 (0.49, 1.78) 1.68 (0.78, 3.79) 1.74 (0.82, 3.95) 2.51 (0.96, 4.26) 0.74 (0.29, 1.60) CZP 0.73 (0.27, 1.83) 0.66 (0.25, 1.52) 0.66 (0.25, 1.49) 1.24 (0.43, 3.16) 1.28 (0.45, 3.26) 3.45 (1.56, 5.05) 1.01 (0.47, 1.92) 1.37 (0.55, 3.69) ETN25BIW 0.91 (0.42, 1.84) 0.90 (0.41, 1.81) 1.69 (0.71, 3.87) 1.75 (0.73, 4.18) 3.80 (1.92, 5.29) 1.11 (0.58, 2.08) 1.51 (0.66, 3.95) 1.10 (0.54, 2.38) GOL (0.62, 1.56) 1.87 (0.82, 4.35) 1.93 (0.85, 4.44) 3.84 (1.94, 5.32) 1.12 (0.56, 2.05) 1.52 (0.67, 3.93) 1.11 (0.55, 2.42) 1.01 (0.64, 1.60) GOL (0.85, 4.41) 1.94 (0.86, 4.45) 2.03 (0.89, 3.36) 0.59 (0.26, 1.28) 0.81 (0.32, 2.31) 0.59 (0.26, 1.42) 0.53 (0.23, 1.22) 0.53 (0.23, 1.18) UST (0.54, 1.99) 1.96 (0.85, 3.28) 0.58 (0.25, 1.22) 0.78 (0.31, 2.22) 0.57 (0.24, 1.36) 0.52 (0.23, 1.18) 0.51 (0.22, 1.16) 0.97 (0.50, 1.84) UST90 Pair wise MA RRs: 2.03 ( ) and 1.97 ( )

PASI 50 at 12 weeks – Biologics-naïve placebo Adalimumab 40mg ADEPT SPIRIT-P1 Infliximab 5mg/kg IMPACT2 GO-REVEAL Golimumab 50mg Golimumab 100mg

Relative risk of PASI 50 at 12 weeks 7 Placebo 0.18 (0.09, ) 0.15 (0.08, ) 0.17 (0.09, ) 0.13 (0.08, ) 5.51 (0.01, 11.14) ADA (0.00, ) 0.98 (0.00, ) 0.73 (0.00, ) 6.84 (0.00, 11.87) 1.18 (0.00, ) GOL (0.00, ) 0.89 (0.00, ) 5.75 (0.00, 11.62) 1.02 (0.00, ) 0.89 (0.00, ) GOL (0.00, ) 7.91 (0.00, 12.32) 1.38 (0.00, ) 1.12 (0.00, ) 1.32 (0.00, ) INF5

QUICK REVIEW 8

Bayesian parameter estimation Pr(  |Y)  Pr(Y|  ) x Pr(  ) “Likelihood” Suppose we wish to draw inference on , a parameter or set of parameters of interest (e.g., treatment effects), therefore  = parameter of interest Y represents the observed data We begin with a “prior” probability distribution for the parameters Typically a non-informative prior (blank slate) Full Bayesian framework allows for external information to inform our prior belief “Prior” Then use the data to determine the likelihood “How likely parameter values are given the observed data” The basis of frequentist statistics Parameter estimation is based on the posterior distribution Bayesian thinking: given my prior knowledge and data likelihood, what is the probability of a parameter estimate being true “Posterior” NMA can be conducted in either the frequentist of Bayesian framework Majority of NMA are conducted in the Bayesian framework

Arm-based fixed & random effects network meta-analysis model Random effects Fixed effects 10

PROBLEM AND SOLUTION 11

Likelihood Probability Prior Posterior Can result in unrealistically wide 95% credible intervals Between-trial variance (  2 ) Posterior heterogeneity distributions

Informative priors If there is too little information to overcome vague, non- informative priors, what are we to do? Informative priors can be used to integrate scientific knowledge external to the evidence base Turner et al reviewed 14,886 meta-analyses to help inform the distribution of τ when working with binary data This information can be used to construct informative priors on the heterogeneity variance parameter How it is done: Use a log-Normal prior on the heterogeneity variance τ 2 with mean and precision based on decades of scientific work 13

Informative heterogeneity priors 10

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Changes to the BUGS/JAGS model Effectively, this implies changing only two lines to most NMA code From To Why a log-Normal distribution? Best fitting distribution among a variety of candidates 16

PsA example revisited 17 Placebo 0.18 (0.11, 0.33) 0.14 (0.09, 0.30) 0.17 (0.10, 0.42) 0.12 (0.08, 0.20) 5.60 (2.99, 8.98) ADA (0.43, 1.68) 0.97 (0.49, 2.36) 0.69 (0.38, 1.16) 6.93 (3.39, 10.71) 1.23 (0.60, 2.33) GOL (0.74, 2.25) 0.85 (0.44, 1.40) 5.76 (2.40, 9.58) 1.03 (0.42, 2.05) 0.84 (0.44, 1.36) GOL (0.31, 1.24) 8.18 (4.98, 12.00) 1.46 (0.86, 2.62) 1.17 (0.71, 2.29) 1.41 (0.80, 3.25) INF5 PlaceboADA40CZPETN25BIWGOL100GOL50 UST (1.20, 3.05) 0.59 (0.34, 1.02) 0.81 (0.42, 1.63) 0.59 (0.33, 1.09) 0.54 (0.30, 0.96) 0.53 (0.30, 0.94) UST (1.12, 2.98) 0.58 (0.32, 0.99) 0.78 (0.40, 1.58) 0.57 (0.31, 1.06) 0.52 (0.28, 0.93) 0.52 (0.28, 0.92) Relative risk of ACR 20 Relative risk of PASI 50

Application to other data Dichotomous data are very popular A more recent study has conducted the same exercise for continuous data modeled with a Normal likelihood It requires that the data first be transformed as standardized mean differences And be back-transformed after completing the NMA As of yet, there is no such evidence for models based on Poisson or Multinomial likelihoods 18

Placebo LAMALABA+LAMA ICS+LABA+LAM A LAMA+PDE-4 LABA PDE-4LABA+PDE-4 ICS+LA BA ICS 3 trials Pair wise MA RR: 0.84 ( ) Network MA RR: 0.85 ( ) Example - COPD

A B C A B C Scenario #1 Scenario #2 I 2 =25% I 2 =35%I 2 =45% I 2 =5% I 2 =35%I 2 =65% How else might this be used? Most NMA assumes the degree of heterogeneity is equal in each comparison k=3 k=4k=5 k=12 If we relax this assumption, we may have to borrow strength estimation power from somewhere else

Conclusions Sparse networks and heterogeneity are two distinct concepts Informative priors can be used to ensure the correct model is used (i.e., random-effects) when it is otherwise infeasible Use of informative heterogeneity priors are becoming widely endorsed (used within NICE, CADTH, and academic experts) There are limitations to the use of informative priors: It can only be used with Binomial and Normal likelihoods It can only be used when all treatment comparisons fall within the same category 21

22 Thank you!