# Pooling and Meta-analysis. Tony O Hagan

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1 Pooling and Meta-analysis Tony O Hagan

2 Pooling Synthesising prior information from several experts 2

3 Multiple experts The case of multiple experts is important When elicitation is used to provide expert input to a decision problem with substantial consequences, we generally want to use the skill of as many experts as possible However, they are not separate pieces of evidence The experts typically base their judgements on the same or similar information It is better to treat them as different attempts to formulate the same information The prior information The question then arises of how to pool their judgements into a single prior density 3

4 Aggregating expert judgements Two approaches Aggregate the distributions Elicit a distribution from each expert separately Combine them using a suitable formula Called mathematical aggregation or pooling Aggregate the experts Get the experts together and elicit a single distribution Called behavioural aggregation Neither is without problems 4

5 Opinion pooling Mathematical aggregation methods combine several distributions into one Notation: N experts Their distributions are f i (x), for i = 1, 2,..., N Combined distribution f 0 (x) 5

6 Linear opinion pool The simplest way to combine the experts probability distributions is to average them 0.4 f 0 (x) = Σ i f i (x) / N Blue and green lines = two expert distributions Red line = pooled distribution y More generally, we can give weights w i to the experts x f 0 (x) = Σ i w i f i (x) / Σ i w i To reflect the experts different levels of expertise 6

7 Multiplicative opinion pool If instead we average the logarithms of the experts probability distributions this amounts to multiplying f 0 (x) = (Π i f i (x) ) 1/N Then scale the result so that it integrates to 1 Blue and green lines = two expert distributions Red line = pooled distribution More generally, we can give weights w i to the experts f 0 (x) = (Π i f i (x) w i ) 1/Σ iwi Then scale the result so that it integrates to 1 To reflect the experts different levels of expertise y x 7

8 Practical comparison Linear Each expert could be right Pooled distribution covers the range of their beliefs y 0.4 Multiplicative Both experts are right Pooled distribution based on the intersection of beliefs y x x 8

9 Theoretical comparison Linear Consistent when you simplify Pr 0 (X > 0) is average of Pr i (X > 0) But not when you add information f 0 (x X > 0) is not the average of f i (x X > 0) Multiplicative Consistent when you add information f 0 (x X > 0) is the logaverage of f i (x X > 0) But not when you simplify Pr 0 (X > 0) is not the logaverage of Pr i (X > 0) There is no way to pool without losing some desirable properties 9

10 Behavioural aggregation The alternative to mathematical aggregation Get the experts in a room together and ask them to elicit a single distribution Advantages Opportunity to share knowledge Avoids arbitrary choice of pooling rule Allows more subtle forms of aggregation y 0.4 y ? x x 10

11 But... More psychological hazards Group dynamic dominant/reticent experts Tendency to end up more confident Block votes Requires careful management What to do if they can t agree? End up with two or more composite distributions Need to apply mathematical pooling to these But this is rare in practice 11

12 Meta-analysis Synthesising indirect evidence 12

13 Basic normal meta-analysis The canonical context is to synthesise evidence from a set of clinical trials, all testing the same drug/treatment These are usually found from a systematic review of the literature In general, suppose we have found N trials Denote evidence from trial t by E t Could in principle be individual patient data More usually it comprises summary statistics Consider the case where E t = (z t, s t ) Where z t is estimated treatment effect and s t its standard error And we assume normally distributed estimate Lots of other situations arise We want to synthesise to make inference about true effect θ 13

14 A simple model Assume z t ~ N(θ, s t2 ) Meaning z t has a normal distribution with mean the true effect θ and with variance s t 2 These are now multiple data items as in Session 1 Combine with prior information Usually formulated as weak prior Derive posterior inferences about θ However, this may be too simple Experience shows trial estimates are often too varied Each trial has different features Recruitment criteria, trial conditions So each may be estimating a slightly different θ 14

15 Model with trial random effects A more sophisticated model is z t ~ N(θ + ε t, s t2 ) Where ε t is an effect due to the specific features of trial t And assume ε t ~ N(0, τ 2 ) Where τ 2 is the variance of the trial effects, an unknown parameter Complete the Bayesian formulation with (weak) priors for θ, τ 2 But genuine prior information about τ 2 would be useful if available Equivalent expressions of this model z t ~ N(θ, s t 2 + τ 2 ) Multiple data items but with enhanced variance z t ~ N(μ t, s t2 ), μ t ~ N(θ, τ 2 ) A more complex Bayesian analysis with extra parameters μ t The original expression is similar with extra parameters ε t Both allow estimation of the individual trial effects 15

16 Numerical example The data here are made up, but intended to represent a realistic scenario 9 trials of widely varying sizes (and hence standard errors) Trial z t s t If all the trials are providing unbiased estimates of the same θ then for instance the difference between trial 6 and trials 4 and 9 is surprising 16

17 Numerical example (contd.) The simple model produces a normal posterior distribution for θ with mean and standard deviation The random effects model produces a posterior distribution for θ with mean and standard deviation The posterior mean of τ 2 is 0.52 The two results are quite different The value of τ 2 indicates that treatment effect varies widely with patient group Even when random effects appear small they should not be ignored 0,05 0,045 0,04 0,035 0,03 0,025 0,02 0,015 0,01 0, ,5 0 0,5 1 1,5 2 17

18 Data gaps It is very common to find that data concern a slightly different situation from the one of interest We are interested in a parameter θ, but the data inform us about a different parameter α α is related to θ, but we need to link the two in order to use the data to learn about θ I call this a data gap Or data discrepancy In the case of random effects meta-analysis Each trial provides information about a different parameter α t We build the link with θ by writing α t = θ + ε t Together with a distribution for the discrepancy ε t 18

19 Data gaps (contd.) There is usually very little information about discrepancies In this case just 9 trials to learn about the discrepancy variance Sometimes there is essentially no data to fill data gaps Hence the name Expert elicitation is then the only feasible way to fill them We ll encounter similar gaps and discrepancies elsewhere in this course 19

20 More complex meta-analyses Mixed treatment comparisons and meta-regression 20

21 Beyond simple treatment comparisons Basic meta-analysis combines data from several trials about a single treatment effect Usually a difference between the treatment of interest and a comparator An alternative treatment, placebo or standard care Often we don t find multiple trials all with the same comparator Some compared with placebo and some compared with active comparator Combination therapies Here are some simple examples of what is called Mixed Treatment Comparison 21

22 Different comparators We are interested in the effect of treatment A compared with treatment B We have some trials comparing A with C and some comparing B with C Model the effects in the second group as β + ε t Model the effects in the first group as θ + β + ε t Then θ represents the additional effect of A versus B Now suppose we also have a 3-arm trial comparing A, B and C Now need to model each arm of each trial separately For a treatment A arm we write the mean response as θ + β + η a + ε t Where η a is a random effect of arm a and ε t is a trial random effect as before 22

23 Combination therapies In cancer treatment, it is normal to give patients a combination of drugs Also found in other serious conditions E.g. patients who have had a stroke or heart attack We may want to estimate the effect of a combination which has never appeared yet in any trial Suppose in a treatment arm, patients receive drugs A, B and C Then the mean effect in that arm could be modelled as a sum of drug effects θ A + θ B + θ C plus random effects But this assumes independent effects, and some interaction effects may also be modelled Interactions in combinations that have not yet been tested represent a data gap for which there is no evidence 23

24 Introducing covariates These mixed treatment comparison models are particular cases of a more general meta-regression framework For each trial effect or treatment arm we may have some covariates that might explain trial differences Average patient age or disease severity Trial duration We can model the mean response on a treatment arm or the mean effect in a trial using conventional regression techniques E.g. θ + βs t + ε t Where s t is average severity in trial t and β is regression coefficient Mixed treatment comparisons involve 0-1 covariates 0 if treatment not used, 1 if it is used 24

25 Summary of session 2 Combining beliefs of several experts is a different kind of synthesis Mathematical aggregation is simple but involves an arbitrary choice of pooling rule Behavioural aggregation is potentially more powerful but also more complex SHELF system to help facilitators Meta-analysis is a powerful range of techniques for synthesising published data sources Widely used for synthesising evidence from clinical trials Including mixed treatment comparisons and meta-regression Can be adapted to many other contexts We frequently need to be aware of data gaps/discrepancies Gaps are often hard to fill and may require expert elicitation 25

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