NETWORK META-ANALYSIS: A NOVEL APPROACH BASED ON A HIERARCHICAL DATA STRUCTURE

INTRODUCTION Meta-analysis is a powerful tool to cumulate and summarize the knowledge in a research field through statistical instruments, and to identify the overall measure of a treatment’s effect by combining several study-specific results. However, it is a controversial tool, because even small violations of certain rules can lead to misleading conclusions. Pooling data through meta-analysis can create problems, such as non linear correlations, multifactorial rather than unifactorial effects, limited coverage, or inhomogeneous data that fails to connect with the hypothesis. When head-to-head treatment comparisons are not available or conclusive, the limitations of standard (i.e. pairwise) meta-analyses can be overcome by network meta-analyses (NMA) which can provide estimates of treatment efficacy or safety of multiple treatment regimens. Different treatment strategies are analyzed by statistical inference methods rather than simply summing up trials that evaluated the same intervention compared to another intervention, standard care, or placebo. If a first trial compares drug A to drug B, showing that drug A is significantly superior to drug B, and a second trial investigates the same or a similar patient population comparing drug B versus drug C (demonstrating that drug B is equivalent to drug C), NMA may allow to infer that drug A is also potentially superior to drug C for this given patient population, even though there was no direct test of drug A against drug C. CONTENTS In this thesis we provided and discussed methods to overcome the limits of standard (univariate) meta-analysis, focusing on the ability to cope with multiple treatments and to deal with correlated data where correlation can derive from multiple endpoints, time-varying responses or from clustered observation. In the first chapter we explore the principal steps (from writing a prospective protocol of analysis to results’ interpretation) in order to minimize the risk of conducting a mediocre meta-analysis and to support researchers to accurately evaluate the published findings. The second chapter represents an overview of conceptual and practical issues of a network meta-analysis. We start from general considerations on network meta-analysis to specifically appraise how to collect study data, structure the analytical network, and specify the requirements for different models and parameter interpretations. Specifically, we outline the key steps, from literature search to sensitivity analysis, necessary to perform a valid network meta-analysis on binomial data. In the third party of this work, we focus our attention on data which can be analyzed with a binomial model applying the Bayesian hierarchical approach and using Markov Chain Monte Carlo approach. We also apply this analytical approach to a case study on the beneficial effects of anesthetic agents in order to further clarify the statistical details of the models, diagnostics, and computations. We presented a practical guide with the actual WinBUGS and SAS codes to allow transparency and ease of replication of all steps that are required when carrying out such quantitative syntheses. In the fourth chapter we propose an alternative frequentist approach to estimate consistency and inconsistency models for a network meta-analysis. We discuss the multilevel network meta-analysis which include a three-level data structure: subjects within studies at the first level, studies within study designs at the second level and design configuration at the third level. We discuss multilevel modeling which may be carried out within widely available statistical programs such as SAS software, and we compare the results of a published Bayesian network meta-analysis on a binary endpoint which examines the effect on mortality of desflurane, isoflurane, sevoflurane, and total intravenous anaesthetics at the longest follow-up available. In the final chapter we compare the Bayesian and the novel frequentist-multilevel approach in performing network meta-analysis on publicly available data and we investigate the descriptive characteristics that may contribute to decrease or increase the potential difference between the estimates derived from the two approaches. The two approaches were compared in terms of the difference between the pooled estimates or their standardized values, and of the Euclidean distance. BAYESIAN NETWORK META-ANALYSIS Suppose that J trials provide mixed comparisons among K treatments and that a is the trial-specific reference treatment. The random effect model is defined by: yja= β0+eja for j=1,2,...,J; a=1,2,…,K-1 yjk= β0j+δj,ak+ejk for j=1,2,...,J; a=2,3,…,K; b<k where β0j is the absolute effect in the trial j for the reference treatment a, while σ2j,ak ~N(dak, σ2j,ak) are the trial-specific relative effects of the treatment group k compared with the reference treatment a. The Bayesian approach requires that the prior distributions are specified for the unknown parameter β0j, δj,ak, and σ2j,ak. It has been suggested to use vague priors for β0j and σ2j,ak, such as N(0,σ2) with variance equal to 0.001 or 0.0001. A uniform distribution, σ2j,ak ~Uniform(0,A), can be used as the prior for the standard deviation of the binomial model with logit link function. Furthermore, the use of the Inverse-Gamma distribution, 1/ σ2j,ak ~Gamma(ε,ε), is suitable when data is sparse, improving stability and convergence. Usually, the hyperparameter ε is set to a low value, such as 0.001. For the fixed effect model, which considers the variability across studies as exclusively due to random variation (σ2j,ak =0), the model will become: yja= β0+eja for j=1,2,...,J; a=1,2,…,K-1 yjk= β0j+dak+ejk for j=1,2,...,J; a=2,3,…,K; b<k where dak are the fixed effects of the treatment group k compared with the reference a. MULTILEVEL NETWORK META-ANALYSIS The fixed effect model is implemented by performing a population-averaged (also known as marginal or unconditional) model. We analyzed data using Generalized Estimating Equation model to consider correlated features within study. For simplicity, we assumed the same correlation between any two elements (specific-arm) of a cluster (study). The random effect model (cluster-specific, also known as subject-specific or conditional model) is implemented via the three-level random intercept model. Suppose N arms (at level 1) are nested within J studies (at level 2), with nj arms in study j, and K treatments and j studies are nested within L study-designs (at level 3). We indicate with yijl the response for arm i in study j in design l and with τijlk the factor variables used to parameterize the treatment effect (1 representing the reference treatment). The random intercept model for a response yijl is as follows: yijl= β0+ β1k τijlk + vl + u0jl + u1jτijlk + eijl for i=1,2,...,N; j=1,2,…,J; k=2,3,…K; l=1,2,…L where β0 is the response mean in the reference treatment 1, β1k represents the difference in the effect between treatment k and the reference treatment 1,vl, ujl and eijl are the residual terms, independent and identically distributed as a Normal Distribution. This model allows for the presence of heterogeneity of the treatment effects between studies (term u1jτijlk). CONCLUSIONS The results of a meta-analysis should be interpreted in the light of the various checks which can inform the readers of the likely reliability of the conclusions. The easiest way to compare two treatment arms is to look at the relative difference in the effect size estimate (i.e. weighted mean difference, relative risk, odds ratio) between the group of interest and the reference group, treating such effect size estimates as independent. However this ability to cope with multiple treatments implies that NMA provides naturally a more general framework to deal with correlated data where correlation can derive from multiple endpoints, time-varying responses or from clustered observation. Multilevel modeling approaches offer a valuable framework for carrying out NMA taking advantage of an existing hierarchical data structure. In this work we provided a comprehensive and detailed overview of the conceptual and practical issues involved in performing a and interpreting network meta-analysis on binomial data. We have discussed the general topics related to network meta-analysis, including how to collect study data, structure the network, and set assumptions about the network that lead to different models and interpretations of model parameters. We have strived to put together the most important topics (making available the major references) and we offer, for the first time, a thorough yet manageable guideline to conduct (from literature search to results interpretation) a rigorous network meta-analysis on binomial data, applying both the Bayesian and frequentist approaches. A drawback of the Bayesian approach is the complexity in model specification, which requires familiarity with the WinBUGS software and the MCMC methods. On the other hand, multilevel models essentially are suited for simpler regression structures where a single outcome variable depends on a few covariates, and therefore they do not allow to inspect the full range of relationships between variables. The multilevel approach, taking into account the clustering structure, provides correct estimates for standard errors, confidence intervals and tests which are generally more conservative than those stemming from the Bayesian approach and the traditional ones obtained by ignoring the presence of groups. We suggest to consider the arm-based data, instead of contrast-based ones, as input data structure. In fact, the use of arm-level summaries allows to adopt the exact likelihood of the data rather than its normal approximation and to not specify the variance-covariance matrix for each multi-arm trial to reflect the data correlation structure. In the network meta-analysis framework, Bayesian and frequentist approaches are expected to give approximately the same results because it is a common practice to use a non-informative priors in the Bayesian strategy. Indeed, our analyses revealed that there is no material difference in the pooled estimates obtained with the Bayesian and frequentist-multilevel approaches.