Investigating trial and treatment heterogeneity in an individual patient data meta-analysis of survival data by means of the penalized maximum likelihood approach

In a meta-analysis combining survival data from different clinical trials, an important issue is the possible heterogeneity between trials. Such intertrial variation can not only be explained by heterogeneity of treatment effects across trials but also by heterogeneity of their baseline risk. In addition, one might examine the relationship between magnitude of the treatment effect and the underlying risk of the patients in the different trials. Such a scenario can be accounted for by using additive random effects in the Cox model, with a random trial effect and a random treatment-by-trial interaction. We propose to use this kind of model with a general correlation structure for the random effects and to estimate parameters and hazard function using a semi-parametric penalized marginal likelihood method (maximum penalized likelihood estimators). This approach gives smoothed estimates of the hazard function, which represents incidence in epidemiology. The idea for the approach in this paper comes from the study of heterogeneity in a large meta-analysis of randomized trials in patients with head and neck cancers (meta-analysis of chemotherapy in head and neck cancers) and the effect of adding chemotherapy to locoregional treatment. The simulation study and the application demonstrate that the proposed approach yields satisfactory results and they illustrate the need to use a flexible variance-covariance structure for the random effects.     

One-stage individual participant data meta-analysis models for continuous and binary outcomes: Comparison of treatment coding options and estimation methods

A one-stage individual participant data (IPD) meta-analysis synthesizes IPD from multiple studies using a general or generalized linear mixed model. This produces summary results (eg, about treatment effect) in a single step, whilst accounting for clustering of participants within studies (via a stratified study intercept, or random study intercepts) and between-study heterogeneity (via random treatment effects). We use simulation to evaluate the performance of restricted maximum likelihood (REML) and maximum likelihood (ML) estimation of one-stage IPD meta-analysis models for synthesizing randomized trials with continuous or binary outcomes. Three key findings are identified. First, for ML or REML estimation of stratified intercept or random intercepts models, a t-distribution based approach generally improves coverage of confidence intervals for the summary treatment effect, compared with a z-based approach. Second, when using ML estimation of a one-stage model with a stratified intercept, the treatment variable should be coded using “study-specific centering” (ie, 1/0 minus the study-specific proportion of participants in the treatment group), as this reduces the bias in the between-study variance estimate (compared with 1/0 and other coding options). Third, REML estimation reduces downward bias in between-study variance estimates compared with ML estimation, and does not depend on the treatment variable coding; for binary outcomes, this requires REML estimation of the pseudo-likelihood, although this may not be stable in some situations (eg, when data are sparse). Two applied examples are used to illustrate the findings.     

Statistical estimation of parameters in a disease transmission model: analysis of a Cryptosporidium outbreak

Population dynamic models, commonly used tools in the study of epidemics and other complex population processes, are implicit non-linear mathematical equations. Inference based on such models can be difficult due to the problems associated with high dimensional parameters that may be non-identified and complex likelihood functions that are difficult to maximize. To address a problem of non-identifiability due to collinearity of parameter estimates in a mathematical model of the 1993 Milwaukee Cryptosporidium parvum outbreak, we examined the utility of a constrained profile likelihood approach. This method was used to study two parameters of interest from the mathematical model: (i). the rate of secondary transmission; (ii). the proportional increase in primary transmission due to water treatment failure. The estimated values of these parameters were shown to depend strongly on poorly understood aspects of Cryptosporidium epidemiology such as asymptomatic proportion and the population immune status. Our analysis demonstrated that the combination of a disease transmission model and a constrained profile likelihood procedure provides an effective approach for inference and estimation of important parameters regulating infectious disease outbreaks.     

Likelihood approaches for proportional likelihood ratio model with right‐censored data

Regression methods for survival data with right censoring have been extensively studied under semiparametric transformation models such as the Cox regression model and the proportional odds model. However, their practical application could be limited because of possible violation of model assumption or lack of ready interpretation for the regression coefficients in some cases. As an alternative, in this paper, the proportional likelihood ratio model introduced by Luo and Tsai is extended to flexibly model the relationship between survival outcome and covariates. This model has a natural connection with many important semiparametric models such as generalized linear model and density ratio model and is closely related to biased sampling problems. Compared with the semiparametric transformation model, the proportional likelihood ratio model is appealing and practical in many ways because of its model flexibility and quite direct clinical interpretation. We present two likelihood approaches for the estimation and inference on the target regression parameters under independent and dependent censoring assumptions. Based on a conditional likelihood approach using uncensored failure times, a numerically simple estimation procedure is developed by maximizing a pairwise pseudo‐likelihood. We also develop a full likelihood approach, and the most efficient maximum likelihood estimator is obtained by a profile likelihood. Simulation studies are conducted to assess the finite‐sample properties of the proposed estimators and compare the efficiency of the two likelihood approaches. An application to survival data for bone marrow transplantation patients of acute leukemia is provided to illustrate the proposed method and other approaches for handling non‐proportionality. The relative merits of these methods are discussed in concluding remarks. Copyright © 2014 John Wiley & Sons, Ltd.     

Semi-parametric modelling of the distribution of the baseline risk in meta-analysis

In meta-analysis of clinical trials, often meta-regression analyses are performed to explain the heterogeneity in treatment effects that usually exist between trials. A popular explanatory variable is the risk observed in the control group, the baseline risk. The relationship between the treatment effect and the baseline risk is investigated by fitting a linear model that allows randomness on the true baseline risk by assuming a normal distribution with unknown mean and variance. However, the normality assumption could be too strong to adequately describe the underlying distribution. Therefore, we developed a new semi-parametric method that relaxes the normality assumption to a more flexible and general distribution. We applied a penalized Gaussian mixture distribution to represent the baseline risk distribution. Furthermore, a bivariate hierarchical model is formulated in order to take into account the correlation between the baseline and treatment effect. To fit the proposed model, a penalized likelihood function is maximized by an Expectation Maximization (EM) algorithm. We illustrate our method on a number of simulated data sets and on a published meta-analysis data set.     

Investigating heterogeneity in an individual patient data meta-analysis of time to event outcomes

Differences across studies in terms of design features and methodology, clinical procedures, and patient characteristics, are factors that can contribute to variability in the treatment effect between studies in a meta-analysis (statistical heterogeneity). Regression modelling can be used to examine relationships between treatment effect and covariates with the aim of explaining the variability in terms of clinical, methodological, or other factors. Such an investigation can be undertaken using aggregate data or individual patient data. An aggregate data approach can be problematic as sufficient data are rarely available and translating aggregate effects to individual patients can often be misleading. An individual patient data approach, although usually more resource demanding, allows a more thorough investigation of potential sources of heterogeneity and enables a fuller analysis of time to event outcomes in meta-analysis. Hierarchical Cox regression models are used to identify and explore the evidence for heterogeneity in meta-analysis and examine the relationship between covariates and censored failure time data in this context. Alternative formulations of the model are possible and illustrated using individual patient data from a meta-analysis of five randomized controlled trials which compare two drugs for the treatment of epilepsy. The models are further applied to simulated data examples in which the degree of heterogeneity and magnitude of treatment effect are varied. The behaviour of each model in each situation is explored and compared.     

Higher-order likelihood inference in meta-analysis and meta-regression

This paper investigates the use of likelihood methods for meta-analysis, within the random-effects models framework. We show that likelihood inference relying on first-order approximations, while improving common meta-analysis techniques, can be prone to misleading results. This drawback is very evident in the case of small sample sizes, which are typical in meta-analysis. We alleviate the problem by exploiting the theory of higher-order asymptotics. In particular, we focus on a second-order adjustment to the log-likelihood ratio statistic. Simulation studies in meta-analysis and meta-regression show that higher-order likelihood inference provides much more accurate results than its first-order counterpart, while being of a computationally feasible form. We illustrate the application of the proposed approach on a real example.     

Advanced methods in meta-analysis: multivariate approach and meta-regression

This tutorial on advanced statistical methods for meta-analysis can be seen as a sequel to the recent Tutorial in Biostatistics on meta-analysis by Normand, which focused on elementary methods. Within the framework of the general linear mixed model using approximate likelihood, we discuss methods to analyse univariate as well as bivariate treatment effects in meta-analyses as well as meta-regression methods. Several extensions of the models are discussed, like exact likelihood, non-normal mixtures and multiple endpoints. We end with a discussion about the use of Bayesian methods in meta-analysis. All methods are illustrated by a meta-analysis concerning the efficacy of BCG vaccine against tuberculosis. All analyses that use approximate likelihood can be carried out by standard software. We demonstrate how the models can be fitted using SAS Proc Mixed.     

Flexibly modeling the baseline risk in meta‐analysis

This paper investigates a likelihood‐based approach in meta‐analysis of clinical trials involving the baseline risk as explanatory variable. The approach takes account of the errors affecting the measure of either the treatment effect or the baseline risk, while facing the potential misspecification of the baseline risk distribution. To this aim, we suggest to model the baseline risk through a flexible family of distributions represented by the skew‐normal. We describe how to carry out inference within this framework and evaluate the performance of the approach through simulation. The method is compared with the routine likelihood approach based on the restrictive normality assumption for the baseline risk distribution and with the weighted least‐squares regression. We apply the competing approaches to the analysis of two published datasets. Copyright © 2012 John Wiley & Sons, Ltd.     

Baseline risk as predictor of treatment benefit: three clinical meta-re-analyses

A relationship between baseline risk and treatment effect is increasingly investigated as a possible explanation of between-study heterogeneity in clinical trial meta-analysis. An approach that is still often applied in the medical literature is to plot the estimated treatment effects against the estimated measures of risk in the control groups (as a measure of baseline risk), and to compute the ordinary weighted least squares regression line. However, it has been pointed out by several authors that this approach can be seriously flawed. The main problem is that the observed treatment effect and baseline risk measures should be viewed as estimates rather than the true values. In recent years several methods have been proposed in the statistical literature to potentially deal with the measurement errors in the estimates. In this article we propose a vague priors Bayesian solution to the problem which can be carried out using the 'Bayesian inference using Gibbs sampling' (BUGS) implementation of Markov chain Monte Carlo numerical integration techniques. Different from other proposed methods, it uses the exact rather than an approximate likelihood, while it can handle many different treatment effect measures and baseline risk measures. The method differs from a recently proposed Bayesian method in that it explicitly models the distribution of the underlying baseline risks. We apply the method to three meta-analyses published in the medical literature and compare the results with the outcomes of the other recently proposed methods. In particular we compare our approach to McIntosh's method, for which we show how it can be carried out using standard statistical software. We conclude that our proposed method offers a very general and flexible solution to the problem, which can be carried out relatively easily with existing Bayesian analysis software. A confidence band for the underlying relationship between true effect measure and baseline risk and a confidence interval for the value of the baseline risk measure for which there is no treatment effect are easily obtained by-products of our approach.     

轮廓似然函数及其应用

目的介绍轮廓似然方法及其两个应用。方法拟合logistic回归模型,计算轮廓似然的置信区间,并与Wald置信区间进行比较;采用轮廓似然方法解决多余参数过多的模型拟合问题。结果轮廓似然可以解决偏态分布下Wald置信区间失效的问题;最大轮廓似然估计与最大似然估计的结果是一致的,而轮廓似然仅需要较弱的假设条件。结论参数呈非正态分布时,轮廓似然置信区间要优于Wald置信区间;轮廓似然作为最大似然估计的替代方法,可以解决最大似然无法计算或计算困难的问题,也可以提高模型的适用性。     

Self‐controlled case series method with smooth age effect

The self‐controlled case series method, commonly used to investigate potential associations between vaccines and adverse events, requires information on cases only and automatically controls all age‐independent multiplicative confounders while allowing for an age‐dependent baseline incidence. In the parametric version of the method, we modelled the age‐specific relative incidence by using a piecewise constant function, whereas in the semiparametric version, we left it unspecified. However, mis‐specification of age groups in the parametric version can lead to biassed estimates of exposure effect, and the semiparametric approach runs into computational problems when the number of cases in the study is moderately large. We, thus, propose to use a penalized likelihood approach where the age effect is modelled using splines. We use a linear combination of cubic M‐splines to approximate the age‐specific relative incidence and integrated splines for the cumulative relative incidence. We conducted a simulation study to evaluate the performance of the new approach and its efficiency relative to the parametric and semiparametric approaches. Results show that the new approach performs equivalently to the existing methods when the sample size is small and works well for large data sets. We applied the new spline‐based approach to data on febrile convulsions and paediatric vaccines. Copyright © 2013 John Wiley & Sons, Ltd.     

The binomial distribution of meta-analysis was preferred to model within-study variability

OBJECTIVE: When studies report proportions such as sensitivity or specificity, it is customary to meta-analyze them using the DerSimonian and Laird random effects model. This method approximates the within-study variability of the proportion by a normal distribution, which may lead to bias for several reasons. Alternatively an exact likelihood approach based on the binomial within-study distribution can be used. This method can easily be performed in standard statistical packages. We investigate the performance of the standard method and the alternative approach. STUDY DESIGN AND SETTING: We compare the two approaches through a simulation study, in terms of bias, mean-squared error, and coverage probabilities. We varied the size of the overall sensitivity or specificity, the between-studies variance, the within-study sample sizes, and the number of studies. The methods are illustrated using a published meta-analysis data set. RESULTS: The exact likelihood approach performs always better than the approximate approach and gives unbiased estimates. The coverage probability, in particular for the profile likelihood, is also reasonably acceptable. In contrast, the approximate approach gives huge bias with very poor coverage probability in many cases. CONCLUSION: The exact likelihood approach is the method of preference and should be used whenever feasible.     

Meta-analysis of multiple outcomes by regression with random effects

Earlier work showed how to perform fixed-effects meta-analysis of studies or trials when each provides results on more than one outcome per patient and these multiple outcomes are correlated. That fixed-effects generalized-least-squares approach analyzes the multiple outcomes jointly within a single model, and it can include covariates, such as duration of therapy or quality of trial, that may explain observed heterogeneity of results among the trials. Sometimes the covariates explain all the heterogeneity, and the fixed-effects regression model is appropriate. However, unexplained heterogeneity may often remain, even after taking into account known or suspected covariates. Because fixed-effects models do not make allowance for this remaining unexplained heterogeneity, the potential exists for bias in estimated coefficients, standard errors and p-values. We propose two random-effects approaches for the regression meta-analysis of multiple correlated outcomes. We compare their use with fixed-effects models and with separate-outcomes models in a meta-analysis of periodontal clinical trials. A simulation study shows the advantages of the random-effects approach. These methods also facilitate meta-analysis of trials that compare more than two treatments. © 1998 John Wiley & Sons, Ltd.     

A LIKELIHOOD APPROACH TO META-ANALYSIS WITH RANDOM EFFECTS

In a meta-analysis of a set of clinical trials, a crucial but problematic component is providing an estimate and confidence interval for the overall treatment effect θ. Since in the presence of heterogeneity a fixed effect approach yields an artificially narrow confidence interval for θ, the random effects method of DerSimonian and Laird, which incorporates a moment estimator of the between-trial components of variance σ²_B, has been advocated. With the additional distributional assumptions of normality, a confidence interval for θ may be obtained. However, this method does not provide a confidence interval for σ²_B, nor a confidence interval for θ which takes account of the fact that σ²_B has to be estimated from the data. We show how a likelihood based method can be used to overcome these problems, and use profile likelihoods to construct likelihood based confidence intervals. This approach yields an appropriately widened confidence interval compared with the standard random effects method. Examples of application to a published meta-analysis and a multicentre clinical trial are discussed. It is concluded that likelihood based methods are preferred to the standard method in undertaking random effects meta-analysis when the value of σ²_B has an important effect on the overall estimated treatment effect.     

A flexible family of transformation cure rate models

In this paper, we introduce a flexible family of cure rate models, mainly motivated by the biological derivation of the classical promotion time cure rate model and assuming that a metastasis‐competent tumor cell produces a detectable‐tumor mass only when a specific number of distinct biological factors affect the cell. Special cases of the new model are, among others, the promotion time (proportional hazards), the geometric (proportional odds), and the negative binomial cure rate model. In addition, our model generalizes specific families of transformation cure rate models and some well‐studied destructive cure rate models. Exact likelihood inference is carried out by the aid of the expectation?maximization algorithm; a profile likelihood approach is exploited for estimating the parameters of the model while model discrimination problem is analyzed by the aid of the likelihood ratio test. A simulation study demonstrates the accuracy of the proposed inferential method. Finally, as an illustration, we fit the proposed model to a cutaneous melanoma data‐set. Copyright © 2017 John Wiley & Sons, Ltd.     

We should not pool diagnostic likelihood ratios in systematic reviews

Some authors plead for the explicit use of diagnostic likelihood ratios to describe the accuracy of diagnostic tests. Likelihood ratios are also preferentially used by some journals, and, naturally, are also used in meta-analysis. Although likelihood ratios vary between zero and infinity, meta-analysis is complicated by the fact that not every combination in Re(+) is appropriate. The usual bivariate meta-analysis with a bivariate normal distribution can sometimes lead to positive probability mass at values that are not possible. We considered, therefore, three different statistical models that do not suffer from this drawback. All three approaches are so complicated that we advise to consider meta-analysis of sensitivity and specificity values instead of likelihood ratios.     

Nested frailty models using maximum penalized likelihood estimation

The frailty model is a random effect survival model, which allows for unobserved heterogeneity or for statistical dependence between observed survival data. The nested frailty model accounts for the hierarchical clustering of the data by including two nested random effects. Nested frailty models are particularly appropriate when data are clustered at several hierarchical levels naturally or by design. In such cases it is important to estimate the parameters of interest as accurately as possible by taking into account the hierarchical structure of the data. We present a maximum penalized likelihood estimation (MPnLE) to estimate non-parametrically a continuous hazard function in a nested gamma-frailty model with right-censored and left-truncated data. The estimators for the regression coefficients and the variance components of the random effects are obtained simultaneously. The simulation study demonstrates that this semi-parametric approach yields satisfactory results in this complex setting. In order to illustrate the MPnLE method and the nested frailty model, we present two applications. One is for modelling the effect of particulate air pollution on mortality in different areas with two levels of geographical regrouping. The other application is based on recurrent infection times of patients from different hospitals. We illustrate that using a shared frailty model instead of a nested frailty model with two levels of regrouping leads to inaccurate estimates, with an overestimation of the variance of the random effects. We show that even when the frailty effects are fairly small in magnitude, they are important since they alter the results in a systematic pattern.     

A multilevel model framework for meta-analysis of clinical trials with binary outcomes

In this paper we explore the potential of multilevel models for meta-analysis of trials with binary outcomes for both summary data, such as log-odds ratios, and individual patient data. Conventional fixed effect and random effects models are put into a multilevel model framework, which provides maximum likelihood or restricted maximum likelihood estimation. To exemplify the methods, we use the results from 22 trials to prevent respiratory tract infections; we also make comparisons with a second example data set comprising fewer trials. Within summary data methods, confidence intervals for the overall treatment effect and for the between-trial variance may be derived from likelihood based methods or a parametric bootstrap as well as from Wald methods; the bootstrap intervals are preferred because they relax the assumptions required by the other two methods. When modelling individual patient data, a bias corrected bootstrap may be used to provide unbiased estimation and correctly located confidence intervals; this method is particularly valuable for the between-trial variance. The trial effects may be modelled as either fixed or random within individual data models, and we discuss the corresponding assumptions and implications. If random trial effects are used, the covariance between these and the random treatment effects should be included; the resulting model is equivalent to a bivariate approach to meta-analysis. Having implemented these techniques, the flexibility of multilevel modelling may be exploited in facilitating extensions to standard meta-analysis methods.