Multivariate meta-analysis: modelling the heterogeneity mixing apples and oranges; dangerous or delicious?

Meta-analysis may be broadly defined as the quantitative review and synthesis of the results of related but independent studies. For the simple case where meta-analysis concerns only one outcome measure in each study, the statistical methods are well established now. However, in many practical situations there are several outcome measures presented in the individual studies included in a meta- analysis. In that case analyzing each outcome measure separately in a univariate manner is often suboptimal, and analyzing all outcome measures jointly using multivariate methods is indicated. For many situations with multivariate outcome, appropriate meta-analytic methods are still lacking or underdeveloped. The work presented in this thesis aims at the development of statistical methods that are suited to analyse meta-analytic data with a multivariate nature in a right and optimal way. With our proposed methods one can answer more comprehensive research questions than with the standard univariate methods that are usually used in practice. Our explicit aim is that our suggested statistical methods are relatively easy to use for most researchers. Chapter 1 is a general introduction to the topic of this thesis. In this chapter some basic terms from the field of meta-analysis are explained and an outline of the thesis is given. The first situation of a meta-analysis with multivariate outcome in this thesis (Chapter 2) is the analysis of the relationship between treatment effect and baseline risk. A straightforward way of assessing this relationship is to compute the ordinary weighted least squares (WLS) regression line of the treatment effects estimated from the different trials on the estimated so called baseline risks observed in the control groups. This conventional method has potential pitfalls and has been seriously criticised. We propose another method based on a bivariate meta-analysis. Although we did most of the analyses using the BUGS implementation of Markov Chain Monte Carlo (MCMC) numerical integration techniques, we also show for one of the examples how it can be carried out with a general linear mixed model in SAS Proc Mixed. The advantage of using BUGS is that an exact measurement error model can be specified. On the other hand, in practice it is easier to use the procedure P! roc Mixed of SAS. In Chapter 3 we discuss the general linear mixed model as a natural and convenient framework for meta-analysis. It is thoroughly pointed out how many existing metaanalysis methods can be carried out using Proc Mixed of SAS, one of the most important statistical packages. Thus far ad hoc programs had to be used. We discuss several methods to analyse univariate as well as bivariate outcomes in meta-analysis and meta-regression analysis. Several extensions of the models are presented, like exact likelihood, non-normal mixtures and multiple endpoints. All methods are illustrated by a clinical meta-analysis example for which the complete syntax needed for the software program SAS is given. In Chapter 4 we discuss a meta-analysis of the effect of surgery (endarterectomy) compared to conservative treatment on the short and long term risk of stroke in patients with increased risk of stroke. Three summary measures per trial are available, which we jointly meta-analyse with a general linear mixed model. As far as we know this is the first published example of a multivariate random effects metaanalysis combining more than two outcomes simultaneously. We demonstrate the advantages of the multivariate analysis upon the univariate analyses where only one outcome measure at a time is measured. The multivariate approach reveals the relations between the different outcomes and gives simple expressions for estimation of derived treatment effect parameters such as the cumulative survival probability ratio as a function of follow-up duration. Besides, the results of the multivariate approach enable us also to estimate the relation of the different treatment effect parameters and the underlying risk. We fit the trivariate model in the standard general linear mixed model program of SAS using approximate likelihood. In a few special cases an exact likelihood approach is possible as well. For the multivariate case we fit the model using Bayesian methods to specify a binomial distribution for the number of post-operative events and a Poisson distribution for the events on long term in both treatment groups. The results of the approximate and exact likelihood approach are very similar. Another application of multivariate meta-analysis in this thesis (Chapter 5) is the meta-analysis of ROC curve data. We consider the situation where per study on pair of estimated sensitivity and specificity is available. Meta-analysis of ROC-curve data is usually done with fixed effects models. Despite some random effects models have been published to execute a meta-analysis of ROC-curve data, these models are not often used in practice. Therefore we propose a more straightforward modelling technique for multivariate random effects meta-analysis of ROC-curve data, which can be fitted with standard software. The sensitivities and specificities of the diagnostic studies were analysed simultaneously using a two- dimensional random effects model. We show that different choices could be made to characterise the estimated bivariate normal distribution by a regression line or a so- called summary ROC curve. Under an extra assumption the model also provides individual study specific ROC curves. When a random intercept model is used to get individual study specific ROC curves, all study specific curves are parallel around the summary ROC curve. We have shown that it is also possible to fit a random slope next to a random intercept, even when there is only one point per study. With the random intercept and slope model, the study specific ROC curves are not necessary parallel to the summary ROC curve anymore. The general linear mixed model is also suited for meta-analysis of survival curve data (Chapter 6), where clinical trials each present several survival percentages and their standard errors during a follow-up period. In practice the follow-up times and the number of follow-up times are different among studies. To tackle this problem, investigators often reduce the survival curve to one or some fixed points in time, e.g. the five-years survival rate. Then the data can be analysed with the standard univariate random (or fixed) effects model for each of the chosen time points separately. However, doing separate analyses for each point in time and thus carrying out many meta-analyses, is inefficient and could lead to inappropriate conclusions. Better methods have been proposed, but all of them are fixed effects methods. One of the most recommended methods is the one of Dear. Dear proposed a general linear model with survival estimate as dependent variable and follow-up time, treatment and study as categorical covariates. The parameters are estimated by generalised weighted least squares (GLS). In this thesis we generalise the GLS-method of Dear towards a multivariate random effects model, which could be applied on data with an arbitrary number of survival estimates and spacing of times between them per curve, possibly different between studies. This enables the meta-analyst to analyse all available data as provided in the publications, without need to inter- or extrapolate to fixed times. The method fits in principle in the framework of the general linear mixed model. However, it has to be adapted in this case, because the correlations between the different survival estimates of the same curve have to be estimated as well. Finally, in Chapter 7, the main findings of this thesis are considered. In addition we discuss some limitations and make recommendations for future research.

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