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I I I I I I I I 1 I 1 I I I BAYESIAN META-ANALYSIS, WITII APPLICATION TO STUDIES OF ETS AND LUNG CANCER Richard L. Tweedie, D.J. Scott, B.J. Biggerstaff and K.L. Mengersen Department of Statistics Colorado State University Fort Collins, Colorado, USA Abstract Meta-analysis enables researchers to combine the results of several studies to assess the information they provide as a whole. It has been used to give a systematic overview of many areas in which data on a possible association between an exposure and an outcome have been collected in a number of studies but where the overall picture remains obscure, both as to the existence or size of the effect. This paper outlines some innovations in meta-analysis, based on using Markov chain Monte Carlo (MCMC) techniques for implementing Bayesian hierarchical models, and compares these with a more well-known random effects (RE) model. The new techniques allow different aspects of variation to be incorporated into descriptions of the association, and in particular enable us to better quantify differences between studies. We apply both the classical and Bayesian methods to the current collection of studies of the association between incidence of lung cancer in female never-smokers and exposure to environmental tobacco smoke (ETS), both in the home through spousal smoking and in the workplace. We demonstrate that, compared with the RE model, the Bayesian methods (a) allow more detailed modelling of study heterogeneity to be incorporated; (b) are relatively robust against a wide choice of specification of such information; (c) allow for more detailed and satisfactory statements to be made, not only about the overall risk but about the individual studies, on the basis of the combined information. For the workplace exposure data set, the Bayesian methods give a somewhat lower overall estimate of relative risk of lung cancer associated with ETS, indicating the care that needs to be taken in using point estimates based on any one method of analysis. On the larger spousal data set the methods give similar answers. We also consider some of the other concerns with meta-analysis, such as consistency between different geographic areas (such as Asia and the United States), and show that Bayesian methods allow us to take into account the overall picture, thus improving the ability to estimate accurately in the subgroups; and publication bias, which we find with the spousal exposure data may lead to an inflated excess risk. I

I 1. Introduction In recent years there has been an enormous increase in the use of meta-analysis in many medical areas in order to obtain overall evaluations of association in areas where individual studies are equivocal [61]. With this has come a large number of discussion papers which assess the benefits, drawbacks and problems of these techniques (see for example [57, 19, 11, 58, 71]). Some of the most well-documented concerns are about the way in which data can be combined if the collection of studies is not homogeneous by design but is based on a variety of differently structured epidemiological cohort or case control studies [28, 51]. Some of these concerns are matters of judgment, and relate to such issues as differing aims of studies or differing study quality including control of confounders; others relate to the underlying variability in the information presented, and different statistical approaches have been developed to attempt to quantify this objectively. In the epidemiological literature a standard method of combining estimates of interest is via a "random effects" model, which attempts to allow for inter-study variation, perhaps due to uncontrolled covariates [58]. This has been argued to be preferable to an earlier "fixed effects" model which essentially assumes that any heterogeneity between studies is purely random (cf. [86, 81]) and hence is not modelled explicitly. The random effects model can be analyzed both in a frequentist or a Bayesian framework [58]. In the latter context it extends logically to hierarchical models such as those recently proposed by DuMouchel [16, 15] or Carlin [10]. In order to differentiate between the models we shall refer to the frequentist random effects model as the "RE model" and the hierarchical Bayes model, which is also formally a random effects model, as the Bayesian model. Details of these are given in the Appendix. Interpretations of the two types of statistical approach are different but the context should make the interpretations clear. Two advantages of the Bayesian approach are its greater flexibility in utilizing other (often prior) information or relationships, and the ability to make useful probability statements on the basis of all information. Moreover,new Markov chain Monte Carlo (MCMC) methods now allow analysis of models based on very general formulations of such prior information, which were previously thought to result in mathematical expressions too complex to be solved. Through their use a wider range of inferences can be made in a straightforward way [2], as we demonstrate here. In the Bayesian meta-analysis context, we will use MCMC to analyze such hierarchical models, without the need to approximate the solutions. Although we do not pursue them here, we note that there are alternative approaches to combining epidemiological studies, also using MCMC methods: a logistic-r model with additional unknown covariates is proposed in [3], and methods of multiple comparisons, proposed in the NRC Report [58, pp. 149-158] for detecting nonequivalence between populations, can also be approached through MCMC [55]. In this paper the Bayesian methods are not used, in the main, to describe "prior information" in any strong sense. Rather, one can view the models as describing in more detail the way in which the -2- I I I I I I I I I I I

I 2. Data and Analysis of Studies on Exposure to ETS 2.1 Data Comparability and Bias Meta-analysis is designed to enable combination of results from studies which are comparable in outcome and exposure. The interpretation of comparability is a subjective and often difficult one. In order to paint an honest picture of the aims and applicability of any meta-analysis, we must first define the relevant measures of outcome and exposure with which we are concerned. The clinical outcome assessed in all of the ETS studies is death from "lung cancer." Several concentrate on or are dominated by one specific form of this disease (e.g., adenocarcinoma), and although some studies give data for different types of cancer, many others do not make such distinctions. Here we choose to combine RR estimates for all lung cancer types, but we are aware that the overall RR estimate may be based on individual RR's associated with quite different diseases in different studies. In order to identify studies with comparable exposures we primarily restrict the meta-analysis to the subset of all ETS studies of adults asserted to be never-smokers, with exposure to spousal smoking or workplace smoking the declared type of exposure to ETS. However, the relevant data are unavailable in a few "spousal" studies, and for these the restrictions are relaxed slightly to include other household exposure or long-time nonsmokers; see Lee [47] and the EPA Report [18] for further details. In choosing which studies to combine, we also need to consider the plausibility of comparing different subpopulations. Two obvious questions are whether there are gender or geographic differences. In accord with the practice in most individual studies and other meta-analyses of these data, we have analyzed males and females separately, and it is the latter that we report here. For males exposed to ETS in the workplace there is a comparable analysis in [4]. The geographic question seems more appropriately studied through a sensitivity analysis as in [18] and we do so in Section 3.3. It is also crucial in meta-analysis to attempt to collect all studies relevant to the relationship in question [27i. This involves collecting at least all published studies, if possible, and testing for the potential existence and influence of unpublished or uncollected studies. There is an insufficient number of studies of workplace exposure to decide if there might be missing infonnation due to publication bias. In contrast, for the spousal exposure studies detailed in the next section, it is possible to investigate completeness using funnel plots (see [511), and in Figure 1 of [56] there is a clear indication of the absence of small studies with negative (perhaps nonsignificant) estimates of effect. It does appear from this that there is indeed bias towards publication of raised relative risks, with perhaps 6-10 or so small but negative studies expected but not absent: this may impact on our overall results, and we comment on this in Section 4. Overall, our experiences with collating these data strongly reinforce those of Felson [19] and Chalmers [11]: data extraction and location is a nontrivial exercise, there are considerable problems in locating studies and relevant data within them, and there are many subjective decisions about data collection and analysis which need to be explicitly documented. We attempt to do this in the following sections. -4- I I I I I I I I I I I I I I

I I I I I I I I I I I I , I I r I I 2.2 Spousal and workplace exposure to ETS Table 1 lists all studies known to us, through Medline and Cancerlink searches and reference to published reviews [18, 47, 491, which provide data relevant to a meta-analysis of the association between ETS and lung cancer in nonsmoking adults, using spousal smoking as the primary measure of exposure. This currently comprises 40 studies of which 3 are unpublished theses, and we given details of location and sex studied. 0 ~ i CO -5- tW ~ N I

I 1 I I I I I I I I I I I I I I I Table 2. Unadjusted RR, Bayesian shrinkage estimates and adjusted RR with 958 CI for female nonsmokers exposed to ETS through spousal smoking 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 1.52 1.82 0.96 0.80 0.75 2.07 1.09 1.26 1.19 1.23 2.16 N/A 2.34 2.55 N/A 2.27 0.90 0.79 1.55 1.55 1.65 2.01 1.03 0.74 1.66 1.03 1.08 1.06 N/A 1.26 2.08 0.75 1.41 1.20 0.79 N/A 2.44 1.17 1.39 1.89 (0.87-2.63) (0.45-7.36) (0.77-1.20) (0.34-1.90) (0.43-1.30) (0.81-5.25) (0.64-1.85) (1.04-1.54) (0.82-1.73 (0.81-1.87) (1.08-4.29) N/A (0.81-6.75) (0.74-8.78) N/A (0.75-6.82) (0.46-1.76) (0.25-2.45) (0.87-2.83) (0.90-2.67) (1.16-2.35) (1.09-3.72) (0.41-2.55) (0.32-1.69) (0.73-3.78) (0.61-1.74) (0.64-1.82) (0.74-1.52) N/A (0.57-2.81) (1.20-3.59) (0.47-1.20) (0.54-3.67) (0.48-3.01) (0.62-1.02) N/A (0.58-10.22) (0.85-1.61) (0.97-1.98) (0.22-16.23) (0.85-2.n) 1.31 (0.33-8.90) 1.27 (0.77-1.21) 1.03 (0.32-2.21) 1.15 (0.42-1.35) 1.06 (0.75-6.06) 1.33 (0.62-1.93) 1.19 (1.14-1.40) 1.25 (0.80-1.77) 1.21 (0.80-1.92) 1.23 (1.03-4.56) 1.41 N/A (0.76-8.59) 1.34 (0.67-11.91) 1.32 N/A (0.68-8.28) 1.32 (0.44-1.88) 1.15 (0.22-2.83) 1.18 (0.83-2.92) 1.32 (0.86-2.77) 1.32 (1.144.39) 1.43 (1.04-3.92) 1.41 (0.38-2.88) 1.20 (0.31-1.92) 1.13 (0.68-4.14) 1.30 (0.59-1.80) 1.16 (0.62-1.89) 1.18 (0.73-1.55) 1.14 N/A (0.54-3.16) 1.24 (1.16-3.76) 1.45 (0.46-1.23) 1.02 (0.49-4.20) 1.25 (0-39-3.40) 1.23 (0.61-1.02) 0.92 N/A (0.38-12.55) 1.30 (0.84-1.64) 1.20 (0.96-2.04) 1.30 (0.21-8.96) 1.25 (0.96-1.83) 1.50 (0.9-2) (0.86-1.88) 1.68 (0.39-2.97) (0.84-1.23) 1.0 (0.8-1.2) (0.79-1.59) (0.73-1.41) (0.93-1.98) (0.86-1.60) (1.06-1-49) 1.29 (1.04-1.60) (0.92-1.57) About 1.4 (0.93-1.60) (0.99-2.06) N/A 1.18 (0.47-2.99) (0.92-1.98) 2.20 (0.8-6.6) (0.91-1.99) 2.25 (0.8-8.8) N/A 0.93 (0.55-1.57)* (0.91-1.99) (0.80-1.57) (0.79-1.68) (0.96-1.84) 2.11 (1.09-4.08) (0.96-1.81) 1.64 (0.87-3.09) (1.09-1.87) (1.01-2.01) (0.82-1.69) 1.00 (0.37-2.71) (0.75-1.57) (0.91-1.89) (0.83-1.54) 1.20 (0.7-2.1) (0.86-1.56) (0.88-1.46) 1.13 (0.78-1.63) N/A 1.6 (0.8-3.0) (0.87-1.73) About 1.5 (1.05-2.09) (0.72-1.34) (0.86-1.80) 1.20 (0.50-3.30) (0.84-1.76) (0.72-1.14) 0.7 (0.6-0.9) N/A (0.88-1.96) 2.02 (0.48-8.56) (0.94-1.50) 1.18 (1.01-1.68) 1.45 (1.02-2.08) (0.83-1.89) 2.41 (0.45-12.83)* Overall 1.20 (1.07-1.34) 1.22 (1-08-1.37) Results for sexes combined I

I I I I I I I I I I I 1 I I I I We have omitted from our workplace meta-analyses published studies concerning occupation- specific environments such as passenger cabins in commercial airlines [60, 31] or the food service industry [66]; and also the study of Brownson et al [7) which relates to smokers and nonsmokers and only considers specific high lung-cancer risk occupations, since these are sufficiently different in design to violate the applicability of our models [58]. We have also only included studies for which the exposure is solely in the workplace, excluding those (Lam and Cheng [41] and Svensson et al [70]) which give relative risks for lung cancer when ETS is measured through exposure "at home or at work" or "at home and at work." 2.3 Exact and Approxitnate Analyses of Individual Studies Typically, studies report results either in "crude" or "unadjusted" from, as 2 x 2 tables, or as "reported" results, which may be adjusted in covariates as described by the individual authors. Ideally one would wish to construct a model with complete control of such covariates (eg [83]). Most often, however, the required information is not available in published epidemiological papers. Instead meta-analysis must be performed only on the basis of summary statistics. These statistical quantities of interest in the individual studies, which are later combined in our meta-analyses, are the point estimates and associated confidence intervals (CIs) of the relative risk (RR) of outcome in a population with some defined exposure (either spousal or workplace ETS in our examples), compared with outcome for an unexposed population. In Tables 2 and 3 we first provide analyses of the unadjusted data for the spousal and workplace studies respectively. A more detailed description of the methodology we use is relegated to the Appendix, and here we describe the notation and quantities needed to interpret these tables. We use the following notation throughout: we suppose that we have k studies, and that RRi = observed estimate of relative risk in study i Yi = log RRi, true log relative risk in study i, an appropriate estimate of (Var[Yi])'1. In the traditional setting for epidemiological studies, the empirical odds ratio provides a point estimate of the true relative risk for each study, and we use this throughout in this paper. Tables 2 and 3 also contain estimates of the individual parameters Bi and corresponding confidence intervals based on logit approximations to the variance, with an assumption of normality of the Yt which is known to be reasonable, at least for large individual sample sizes. As seen in these Tables we find that, compared with an exact method (also discussed in the Appendix) the logit method gives CIs that are perhaps 5-10 % too short; but for our purposes we will accept this level of accuracy -9- I

here. (In [4] these methods are also compared with the results generated by Mantel-Haenszel methods [5, p. 141], [64], which are found to be typically less accurate again.) Even without 2 x 2 tabulations, reported results may be combined provided all the confidence intervals are also reported, through deriving a Normal-based variance estimate for the log relative risk estimate. This is the case for many of the studies in Tables 2 and 3. Note, however, that the different factors for which adjustment was made in each of the studies render it more difficult to be sure that like is being compared with like in such an analysis. In Table 2 we see that 28 of the 36 relevant studies with female respondents reported an increase in the unadjusted relative risk of lung cancer associated with spousal ETS exposure, with just 5 of these significantly different from 1.0 at the 95 % level. (Because we use both frequentist and Bayesian methods, it will be convenient to define the phrase "significantly different from 1.0" to cover either the situation in which there is a constructed 95 % confidence interval which does not cover 1.0, or a Bayesian 95% credible interval which does not cover 1.0: the context should make it clear which is meant.) In Table 3 we see that only 4 of the 9 relevant studies with female respondents reported an increase in the unadjusted relative risk of lung cancer associated with spousal ETS exposure, with just one of these significantly different from 1.0 (as indicated from the exact CI). Thus, as stated above, both of these collections of studies are certainly such that a simple interpretation is difficult and in which heterogeneity may well be a problem that both the RE and the Bayesian analyses can help to overcome. 2.4 Random Effects and Bayesian Approaches to Meta-analysis The RE model for meta-analysis is a natural starting point to describe a Bayesian methodology for meta-analysis. As described more formally in the Appendix, in the RE method we assume that there is a true underlying log RR over all studies, denoted µ, and that the observed log relative risks Yi for each study are from a distribution governed by qu,,antities Bi and ai2 which represent the true RR and within-study variability of study i, and a quanti~r~ which provides a measure of the between- or across- study variability. In the special case in which = 0, indicating homogeneity between studies, this RE model reduces to the well-known fixed effects model (see [86, 81] and others). In this non-Bayesian paradigm, µ, a and r are presumed fixed, and the Os are random variables with mean µ. In a general hierarchical Bayesian scheme [16], a12 and 72 are also random variables with (in our case) a x2 distribution, and these X2 distributions are in turn governed by parameters (degrees of freedom) dfQ and dfr which indicate how well the variance structures are assumed to be known. I I I I I I I I I The distributions of these quantities are specified a priori according to the application. It is standard practice to assume a "flat" or "uninformative" prior to µ as we do below, as even with a small " N  the combined data become relatively informative about the location of the effect-size number of studies, prior distribution" [10, p. 146]. The imposition of distributions on 0, a2 and 72 enables us to describe O 00 much more explicitly any underlying variability in the way the study outcomes are distributed. i -4 ~ W -10- CJ N ~ V , ~

I I I I t I I I I I I I I I In this formulation, the posterior distributions become quite complicated, leading DuMouchel [16] to make some (reasonable) approximations to normality for computational convenience. In contrast, in this paper we use simulation methods (specifically the Gibbs samples through the software package BUGS [68]) to carry out the analysis. As previously mentioned, these algorithms provide powerful computational tools for Bayesian analysis and release the user from restrictive assumptions about the distribution of the data and of prior information [2]. In the ETS case, for example, although the model (2) in the Appendix was considered appropriate, approximations to the posterior distribution were not needed, although comparison of our results here to those in [4] show that the Normal approximations of DuMouchel [16] are in fact very effective in this case. The Bayesian method, as implemented through MCMC software, also enables us to make inferences about the posterior probability that the overall relative risk is above 1.0, enabling more exact inferences to be made and thus more effectively enabling the meta-analysis to achieve one of its overall goals. It is equally possible to quantify statements such as P {overall US mean > 1} using this method, which is not a simple task in the RE models. In this paper we will show that the use of this more flexible description of the way in which relative risks are spread across studies can lead to small but possibly important differences in the overall conclusions made, and that these conclusions are essentially independent of how the prior distributions are chosen, so that in fact it is the data that are driving the conclusions. 3. Results 3.1 Analysis based on unadjusted relative risks The results of meta-analyses under both the RE and Bayesian paradigms are given as the "Overall" values at the bottoms of Tables 2 and 3. In the second-last column of Tables 2 and 3 we also give the estimates for the individual studies after "shrinkage" towards the overall mean through "borrowing strength" from the totality of studies. Note that these estimates have much tighter credible intervals than the original study estimates, since they are based on a combination of individual and overall study information. In Table 2, the overall Bayesian posterior mean estimate for spousal studies (1.22) is slightly higher than that of the logit-based RE model (1.20), although they are very much within each other's CI. For the spousal exposure studies we find P{µ > 1} = 0.9996, significant at well above the 5% level with this data. The values appear robust to some change in model choice. Under the Bayesian model, there were negligible changes to posterior distributions when input values for prior distributions were changed. Only when the degrees of freedom associated with the distributions of u2 and T2 or the entries in the matrix controlling the between studies variance were set at extreme and unreasonable values were there any real changes to posterior estimates. Other changes in prior specifications produced no effect at all. N O Co -L -4 00 I

I I I I I I I I I 1 I I I I I I (i) For workplace exposure the following two groups of datasets were considered separately: only those studies of females which provided unadjusted RR estimates: these results can thus be directly compared with those of Tables 2 and 3; (ii) all studies, combining both males and females, giving a result directly comparable with that of Lee [48, Table 5]. Results are presented in Table 4. For dataset (a), under either model the combined point estimate for the females exposed to workplace smoking is 1.10-1.11. The dataset (b) includes both genders and indicates that the male studies are somewhat different in the sense that now ~2 = 0.017. The overall estimate of 1.07-1.08 is only marginally higher than that of Lee [48, Table 5] as we should expect since they differ only by two studies. The Bayesian methodology also enables us to assert that the posterior probability that the overall underlying relative risk is greater than 1.0 is 0.83-0.84 in both these cases. Table 4. Meta-analyses of Results (Adjusted where Available) of Workplace Exposure Studies °- . . . ayeslan ' e - o e RR (95X:GI) . RF. (95Y CI) f .. ~ Stutl T e . - (a) Females (9 Studies) 1.10 (0.89-1.32) 1.11 (0.96-1.29) 0.005 (6) Combined (14 Stadies) 1.08 (0.92-1.26) 1.07 (0.93-1.24) 0.017 For exposure to spousal smoking, we consider a different approach to the adjusted results, and indicate the effect of combining the case-control and cohort studies. Under a fixed effects model this is not advisable due to the inherent differences in the methodology. Here we are able to take that into account. We analyze against the totality of studies, consisting of adjusted RRs where given as in Table 2, and unadjusted RRs for other studies, thus using the maximum number of 35 case control and 4 cohort studies for females. Table 5 gives the results of analyzing this dataset. Again we note that the Bayesian and the RE models give very similar answers. The inhomogeneity in the studies is supported by a value of ~2 = 0.052, although the inclusion of the cohort studies in this case actually decreases ~2 slightly. -13- IV O 00 s V 00 W N N 0 I

Table 5. Meta-Analysis of Results (Adjusted where Available) of Spousal Exposure Studies : > ayes. an e . e . - Stu T RR (95%CI) .. . : .. ..RR (95X CI) P2 e. ... _ ase ontro 1.22 (1.07-1.39) 1.22 (1.06-1.41) 0.061 Cohort 1.33 (1.03-1.78) 1.29 (1-02-1.64) 0 All 1.23 (1.09-1.39) 1.23 (1.08-1.39) 0.052 3.3 Choosing subgroups of studies Ensuring comparability an entail close examination of the data to identify appropriate subsets of studies for combination. As noted in the EPA Report [18], for this particular meta-analysis it may be sensible to consider the effect of grouping studies by geographic region, especially given the rather inexplicit nature of "exposure to ETS" and the way in which it might vary in different cultures. To illustrate the effect of geographic location we provide in Table 6 meta-analyses of two subgroups of studies of spousal smoking which may a priori be considered internally more homogeneous: Asian populations only (China, Japan, Hong Kong, Taiwan), comprising 15 case control and one cohort study; and U.S. studies only, comprising 11 case control and two cohort studies. Resulting overall unadjusted relative risks with logit variance estimates are again compared under both frequentist and Bayesian approaches. It can be seen that there are considerable differences between the two country groups with respect to both overall estimate and between-study heterogeneity (and that again there is 10%-15% difference between using RE and Bayesian methods). The relative risk estimate is significantly increased above 1.0 at the 5% level for the Asian studies, but for the U.S. studies we calculate the probability of the relative risk being above 1.0 as 0.92. Table 6. Meta-analysis of Asian and U.S. subgroups of studies of spousal exposure . . ayes an e - y Study T e RR (95X CI) .. . RR (95Y:CI) P2 stan tu 1es 1.25 (1.03-1.50) 1.25 (1.02-1.52) 0.067 U.S. Studies 1.13 (0.95-1.34) 1.11 (0.98-1.26) 0.003 One implication of the different RRs is that extrapolation of overall results to individual studies and from one country group to another may not be appropriate. It certainly highlights a need for further investigation of these differences, perhaps through a closer exploration of covariates or possible biases. Some recognized covariates in the association between lung cancer and exposure to ETS include diet and socioeconomic status. Possible biases include different underlying rates of lung cancer and misclassifica- tion of active smoking. There are many other breakups of the data that could be accomplished by the methods used here. For example, there has been considerable recent interest [54, 76, 18, 57] in accounting for the differing quality of studies in meta-analysis. The EPA Report [18] groups studies into four tiers based on a -14- I I I I I I I I I I I I I I I I I I