A non-parametric procedure for evaluating treatment effect in the meta-analysis of survival data

This paper addresses the problem of combining information from independent clinical trials which compare survival distributions of two treatment groups. Current meta-analytic methods which take censoring into account are often not feasible for meta-analyses which synthesize summarized results in published (or unpublished) references, as these methods require information usually not reported. The paper presents methodology which uses the log(-log) survival function difference, (i.e. log(-logS2(t))-log(-logS1(t)), as the contrast index to represent the multiplicative treatment effect on survival in independent trials. This article shows by the second mean value theorem for integrals that this contrast index, denoted as theta, is interpretable as a weighted average on a natural logarithmic scale of hazard ratios within the interval [0,t] in a trial. When the within-trial proportional hazards assumption is true, theta is the logarithm of the proportionality constant for the common hazard ratio for the interval considered within the trial. In this situation, an important advantage of using theta as a contrast index in the proposed methodology is that the estimation of theta is not affected by length of follow-up time. Other commonly used indices such as the odds ratio, risk ratio and risk differences do not have this invariance property under the proportional hazard model, since their estimation may be affected by length of follow-up time as a technical artefact. Thus, the proposed methodology obviates problems which often occur in survival meta-analysis because trials do not report survival at the same length of follow-up time. Even when the within-trial proportional hazards assumption is not realistic, the proposed methodology has the capability of testing a global null hypothesis of no multiplicative treatment effect on the survival distributions of two groups for all studies. A discussion of weighting schemes for meta-analysis is provided, in particular, a weighting scheme based on effective sample sizes is suggested for the meta-analysis of time-to-event data which involves censoring. A medical example illustrating the methodology is given. A simulation investigation suggested that the methodology performs well in the presence of moderate censoring.     

Aggregate data meta-analysis with time-to-event outcomes

In a meta-analysis of randomized controlled trials with time-to-event outcomes, an aggregate data approach may be required for some or all included studies. Variation in the reporting of survival analyses in journals suggests that no single method for extracting the log(hazard ratio) estimate will suffice. Methods are described which improve upon a previously proposed method for estimating the log(HR) from survival curves. These methods extend to life-tables. In the situation where the treatment effect varies over time and the trials in the meta-analysis have different lengths of follow-up, heterogeneity may be evident. In order to assess whether the hazard ratio changes with time, several tests are proposed and compared. A cohort study comparing life expectancy of males and females with cerebral palsy and a systematic review of five trials comparing two anti-epileptic drugs, carbamazepine and sodium valproate, are used for illustration.     

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.     

The rising relative risk of mortality for singles: meta-analysis and meta-regression

Never-married persons (singles) constitute a growing demographic group; yet, the magnitude of the all-cause relative mortality risk for nonelderly singles is not known and important moderating factors have not been explored. The authors used meta-analysis to examine 641 risk estimates from 95 publications that provided data on more than 500 million persons. The comparison group consisted of currently married individuals. The mean hazard ratio for mortality was 1.24 (95% confidence interval: 1.19, 1.30) among multivariate-adjusted hazard ratios with a high subjective quality rating. Meta-regressions showed that hazard ratios have been modestly increasing over time for both genders, but have done so somewhat more rapidly for women. The results also showed that the hazard ratio decreased with age and that study quality has an important relation to hazard ratio magnitude.     

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.     

Smooth estimation of survival functions and hazard ratios from interval-censored data using Bayesian penalized B-splines

We discuss the use of Bayesian P-spline and of the composite link model to estimate survival functions and hazard ratios from interval-censored data. If one further assumes proportionality of the hazards, the proposed strategy provides a smoothed estimate of the baseline hazard along with estimates of global covariate effects. The frequentist properties of our Bayesian estimators are assessed by an extensive simulation study. We further illustrate the methodology by two examples showing that the proportionality of the hazards might also be found inappropriate from interval-censored data.     

Inflation of type I error rate in two statistical tests for the detection of publication bias in meta-analyses with binary outcomes

The use of meta-analysis to combine results of several trials is still increasing in the medical field. The validity of a meta-analysis may be affected by various sources of bias (for example, publication bias, language bias). Therefore, an analysis of bias should be an integral part of any systematic review. Statistical tests and graphical methods have been developed for this purpose. In this paper, two statistical tests for the detection of bias in meta-analysis were investigated in a simulation study. Binary outcome data, which are very common in medical applications, were considered and relative effect measures (odds ratios, relative risk) were used for pooling. Sample sizes were generated according to findings in a survey of eight German medical journals. Simulation results indicate an inflation of type I error rates for both tests when the data are sparse. Results get worse with increasing treatment effect and number of trials combined. Valid statistical tests for the detection of bias in meta-analysis with sparse data need to be developed.     

Meta-analysis of sex-specific genome-wide association studies

Despite the success of genome-wide association studies, much of the genetic contribution to complex human traits is still unexplained. One potential source of genetic variation that may contribute to this "missing heritability" is that which differs in magnitude and/or direction between males and females, which could result from sexual dimorphism in gene expression. Such sex-differentiated effects are common in model organisms, and are becoming increasingly evident in human complex traits through large-scale male- and female-specific meta-analyses. In this article, we review the methodology for meta-analysis of sex-specific genome-wide association studies, and propose a sex-differentiated test of association with quantitative or dichotomous traits, which allows for heterogeneity of allelic effects between males and females. We perform detailed simulations to compare the power of the proposed sex-differentiated meta-analysis with the more traditional "sex-combined" approach, which is ambivalent to gender. The results of this study highlight only a small loss in power for the sex-differentiated meta-analysis when the allelic effects of the causal variant are the same in males and females. However, over a range of models of heterogeneity in allelic effects between genders, our sex-differentiated meta-analysis strategy offers substantial gains in power, and thus has the potential to discover novel loci contributing effects to complex human traits with existing genome-wide association data.     

Vitamin C concentrations in plasma as a function of intake: a meta-analysis

The purpose of this study was to estimate the intake-plasma relationship for vitamin C by means of a meta-analysis. A MEDLINE search revealed 30 publications matching our inclusion criteria. We completed the set with 5 older papers and with one monograph. The proposed statistical model corrects for inconsistencies with regard to methodological differences between the various studies. Therefore, the contribution of a particular study to the estimation is independent of the number of data points. The estimations were performed for the complete data set as well as for different subgroups: "adult" aged 15-65 years, "elderly" aged 60-96 years, "nonsmokers" and "smokers". The 50th percentile of the plasma concentration for a daily vitamin C intake of 60 mg was 42.4 mumol/L. The corresponding values for the different subgroups were: "adult" 44.1 mumol/L, "elderly" 31.0 mumol/L, "nonsmokers" 42.4 mumol/L, and "smokers" 33.6 mumol/L. Thus, this meta-analysis confirms earlier results that the requirements of vitamin C is higher in "elderly" and "smokers" compared to "adult" and "nonsmokers" and it can be used for the estimation of the vitamin C intake in order to achieve a desired plasma level within a target population. In the general population the assumed optimal plasma concentration of 50 mumol/L, as proposed by a consensus conference, can be achieved by the intake of 100 mg per day, which is the new recommendation of the Austrian, German, and Swiss Nutrition Societies.     

What to add to nothing? Use and avoidance of continuity corrections in meta‐analysis of sparse data

OBJECTIVES: To compare the performance of different meta-analysis methods for pooling odds ratios when applied to sparse event data with emphasis on the use of continuity corrections. BACKGROUND: Meta-analysis of side effects from RCTs or risk factors for rare diseases in epidemiological studies frequently requires the synthesis of data with sparse event rates. Combining such data can be problematic when zero events exist in one or both arms of a study as continuity corrections are often needed, but, these can influence results and conclusions. METHODS: A simulation study was undertaken comparing several meta-analysis methods for combining odds ratios (using various classical and Bayesian methods of estimation) on sparse event data. Where required, the routine use of a constant and two alternative continuity corrections; one based on a function of the reciprocal of the opposite group arm size; and the other an empirical estimate of the pooled effect size from the remaining studies in the meta-analysis, were also compared. A number of meta-analysis scenarios were simulated and replicated 1000 times, varying the ratio of the study arm sizes. RESULTS: Mantel-Haenszel summary estimates using the alternative continuity correction factors gave the least biased results for all group size imbalances. Logistic regression was virtually unbiased for all scenarios and gave good coverage properties. The Peto method provided unbiased results for balanced treatment groups but bias increased with the ratio of the study arm sizes. The Bayesian fixed effect model provided good coverage for all group size imbalances. The two alternative continuity corrections outperformed the constant correction factor in nearly all situations. The inverse variance method performed consistently badly, irrespective of the continuity correction used. CONCLUSIONS: Many routinely used summary methods provide widely ranging estimates when applied to sparse data with high imbalance between the size of the studies' arms. A sensitivity analysis using several methods and continuity correction factors is advocated for routine practice.     

Expected value of sample information calculations in medical decision modeling.

There has been an increasing interest in using expected value of information (EVI) theory in medical decision making, to identify the need for further research to reduce uncertainty in decision and as a tool for sensitivity analysis. Expected value of sample information (EVSI) has been proposed for determination of optimum sample size and allocation rates in randomized clinical trials. This article derives simple Monte Carlo, or nested Monte Carlo, methods that extend the use of EVSI calculations to medical decision applications with multiple sources of uncertainty, with particular attention to the form in which epidemiological data and research findings are structured. In particular, information on key decision parameters such as treatment efficacy are invariably available on measures of relative efficacy such as risk differences or odds ratios, but not on model parameters themselves. In addition, estimates of model parameters and of relative effect measures in the literature may be heterogeneous, reflecting additional sources of variation besides statistical sampling error. The authors describe Monte Carlo procedures for calculating EVSI for probability, rate, or continuous variable parameters in multi parameter decision models and approximate methods for relative measures such as risk differences, odds ratios, risk ratios, and hazard ratios. Where prior evidence is based on a random effects meta-analysis, the authors describe different ESVI calculations, one relevant for decisions concerning a specific patient group and the other for decisions concerning the entire population of patient groups. They also consider EVSI methods for new studies intended to update information on both baseline treatment efficacy and the relative efficacy of 2 treatments. Although there are restrictions regarding models with prior correlation between parameters, these methods can be applied to the majority of probabilistic decision models. Illustrative worked examples of EVSI calculations are given in an appendix.     

Meta-analysis combining parallel and cross-over clinical trials. III: The issue of carry-over

In meta-analysis combining results from parallel and cross-over trials, there is a risk of bias originating from the carry-over effect in cross-over trials. When pooling treatment effects estimated from parallel trials and two-period two-treatment cross-over trials, meta-analytic estimators of treatment effect can be obtained from the combination of parallel trial results either with cross-over trial results based on data of the first period only or with cross-over trial results analysed with data from both periods. Taking data from the first cross-over period protects against carry-over but gives less efficient treatment estimators and may lead to selection bias. This study evaluates in terms of variance reduction and mean square error the cost of calculating meta-analysis estimates with data from the first period instead of data from the two cross-over periods. If the information on cross-over sequence is available, we recommend performing two combined design meta-analyses, one using the first cross-over period data and one based on data from both cross-over periods. To investigate simultaneously the statistical significance of these two estimators as well as the carry-over at meta-analysis level, a method based on a multivariate analysis of the meta-analytic treatment effect and carry-over estimates is proposed.     

Nonproportional Hazards in Network Meta-Analysis: Efficient Strategies for Model Building and Analysis

ObjectivesTo develop efficient approaches for fitting network meta-analysis (NMA) models with time-varying hazard ratios (such as fractional polynomials and piecewise constant models) to allow practitioners to investigate a broad range of models rapidly and to achieve a more robust and comprehensive model selection strategy.MethodsWe reformulated the fractional polynomial and piecewise constant NMA models using analysis of variance–like parameterization. With this approach, both models are expressed as generalized linear models (GLMs) with time-varying covariates. Such models can be fitted efficiently with standard frequentist techniques. We applied our approach to the example data from the study by Jansen et al, in which fractional polynomial NMA models were introduced.ResultsFitting frequentist fixed-effect NMAs for a large initial set of candidate models took less than 1 second with standard GLM routines. This allowed for model selection from a large range of hazard ratio structures by comparing a set of criteria including Akaike information criterion/Bayesian information criterion, visual inspection of goodness-of-fit, and long-term extrapolations. The “best” models were then refitted in a Bayesian framework. Estimates agreed very closely.ConclusionsNMA models with time-varying hazard ratios can be explored efficiently with a stepwise approach. A frequentist fixed-effect framework enables rapid exploration of different models. The best model can then be assessed further in a Bayesian framework to capture and propagate uncertainty for decision-making.     

非随机化医学研究中风险比的一种估计方法

目的提出一种适用于非随机化医学研究的,结合倾向指数与非参数生存分析估计风险比的方法.方法首先对倾向指数进行估计,然后对倾向指数分布分层以消除比较两组间协变量分布的不均衡.其次对分层样本用非参数生存分析的方法估计两组间发病或死亡的风险比.最后比较本法与常用的Cox模型方法并探讨其适用性.结果将本法应用于一项评价某降血脂新药效果的4期临床试验数据后显示:(1)对倾向指数分布分层后基本上消除了由于随机分组方案失败导致的新药组与传统药物组之间协变量分布的不均衡性,使得非参数生存分析方法得以应用;(2)由本法得到的新药效果的估计-风险比与由Cox模型得到的结果基本一致.结论对于非随机化医学研究,结合倾向指数进行非参数生存分析是一种新的可选择的统计方法.     

Alternatives to the hazard ratio in summarizing efficacy in time-to-event studies: an example from influenza trials

Estimates of the efficacy of new medicines are key to the investigation of their clinical effectiveness. The most widely recommended approach to summarizing time-to-event data from clinical trials is to use a hazard ratio. When the proportional hazards assumption is questionable, a hazard ratio depends on the length of patient follow-up. Hazard ratios do not directly translate into differences in times to events and therefore can present difficulties in interpretation. This paper describes an area where summary by hazard ratio would seem unsuitable and explores alternative estimates of efficacy. In particular, the difference in median time to event between treatments can provide a useful and consistent measure of efficacy. Methods of calculating confidence intervals for differences in medians for censored time-to-event will be described. Accelerated failure time models provide a useful alternative approach to proportional hazards modelling. Estimates of the ratio of the median time to event between treatments are directly available from these models. One of the reasons given for summarizing time-to-event studies by a hazard ratio is to facilitate meta-analyses. The bootstrap estimate of standard error for difference in median in each trial can provide a method for combining results based on summary statistics.     

Using multiple timescale models for the evaluation of a time‐dependent treatment

In survival analysis, the absolute measure of cumulative risk provided by the Kaplan‐Meier estimator is still the most used quantity for its easy calculation and direct interpretability. However, for describing the survival after an intervention that may occur at different times from baseline observation, the Kaplan‐Meier estimator generally yields to biased results if intervention is considered as fixed at baseline. The main focus of the present paper is to extend the use of a multiple timescale model in the presence of a time dependent intervention. The aim is to obtain 1) an estimate of treatment effect in terms of hazard ratios by flexible modeling, 2) a valid prediction tool, i.e. estimate of prognosis for a patient who changes treatment later in time, and 3) an appropriate graphical representation of survival in the presence of a time dependent treatment change, accounting for different timescales. We will show the advantages of this approach on the comparison of chemotherapy versus transplant in children with high‐risk acute lymphoblastic leukemia in first remission. We considered a model with two timescales that accounts for the change in treatment at different times in the disease course. An alternative approach to survival estimates is also proposed which has some advantages over the traditional landmark approach: it uses all the data available to plot survival from the date of remission, it avoids the arbitrary choice of a landmark time and explicitly models the change in hazard due to transplant. Copyright © 2015 John Wiley & Sons, Ltd.     

Improved tests for a random effects meta-regression with a single covariate

The explanation of heterogeneity plays an important role in meta-analysis. The random effects meta-regression model allows the inclusion of trial-specific covariates which may explain a part of the heterogeneity. We examine the commonly used tests on the parameters in the random effects meta-regression with one covariate and propose some new test statistics based on an improved estimator of the variance of the parameter estimates. The approximation of the distribution of the newly proposed tests is based on some theoretical considerations. Moreover, the newly proposed tests can easily be extended to the case of more than one covariate. In a simulation study, we compare the tests with regard to their actual significance level and we consider the log relative risk as the parameter of interest. Our simulation study reflects the meta-analysis of the efficacy of a vaccine for the prevention of tuberculosis originally discussed in Berkey et al. The simulation study shows that the newly proposed tests are superior to the commonly used test in holding the nominal significance level.     

Estimating survival probabilities by exposure levels: utilizing vital statistics and complex survey data with mortality follow‐up

We present a two‐step approach for estimating hazard rates and, consequently, survival probabilities, by levels of general categorical exposure. The resulting estimator utilizes three sources of data: vital statistics data and census data are used at the first step to estimate the overall hazard rate for a given combination of gender and age group, and cohort data constructed from a nationally representative complex survey with linked mortality records, are used at the second step to divide the overall hazard rate by exposure levels. We present an explicit expression for the resulting estimator and consider two methods for variance estimation that account for complex multistage sample design: (1) the leaving‐one‐out jackknife method, and (2) the Taylor linearization method, which provides an analytic formula for the variance estimator. The methods are illustrated with smoking and all‐cause mortality data from the US National Health Interview Survey Linked Mortality Files, and the proposed estimator is compared with a previously studied crude hazard rate estimator that uses survey data only. The advantages of a two‐step approach and possible extensions of the proposed estimator are discussed. Copyright © 2015 John Wiley & Sons, Ltd.     

Visualizing covariates in proportional hazards model

We present a graphical method called the rank‐hazard plot that visualizes the relative importance of covariates in a proportional hazards model. The key idea is to rank the covariate values and plot the relative hazard as a function of ranks scaled to interval [0, 1]. The relative hazard is plotted with respect to the reference hazard, which can be, for example, the hazard related to the median of the covariate. Transformation to scaled ranks allows plotting of covariates measured in different units in the same graph, which helps in the interpretation of the epidemiological relevance of the covariates. Rank‐hazard plots show the difference of hazards between the extremes of the covariate values present in the data and can be used as a tool to check if the proportional hazards assumption leads to reasonable estimates for individuals with extreme covariate values. Alternative covariate definitions or different transformations applied to covariates can be also compared using rank‐hazard plots. We apply rank‐hazard plots to the data from the FINRISK study where population‐based cohorts have been followed up for events of cardiovascular diseases and compare the relative importance of the covariates cholesterol, smoking, blood pressure and body mass index. The data from the Study to Understand Prognoses Preferences Outcomes and Risks of Treatment (SUPPORT) are used to visualize nonlinear covariate effects. The proposed graphics work in other regression models with different interpretations of the y‐axis. Copyright © 2009 John Wiley & Sons, Ltd.     

Body mass index and body girths as predictors of mortality in black and white men.

Anthropometric measurements collected from black and white men in the 1960 (n = 946) and 1963 (n = 456) examinations of the Charleston Heart Study cohort (Charleston County, South Carolina) were examined as predictors of all cause and coronary heart disease mortality. Anthropometric measurements included body mass index, chest girth (at the third intercostal space), abdominal girth (at the umbilicus) and midarm circumference. Vital status of 98 percent of the cohort was determined through 1988. Body mass index was not associated with mortality in the white men; however, it was predictive of all cause and coronary heart disease mortality in the black men. Analyses conducted separately in the lower and upper range of body mass index in black men showed the adjusted relative hazard at the 50th versus the 10th percentile of body mass index was 0.54 for all cause mortality, but was not significant for coronary heart disease mortality; whereas the adjusted relative hazard for the 90th relative to the 50th percentile was 1.7 for coronary heart disease deaths, but not significant for deaths from all causes. The circumference measurements were not predictive of all cause or coronary heart disease mortality in the white men. In the black men, the adjusted relative hazard ratios for all cause mortality for the 85th relative to the 15th percentiles were 0.22 for midarm circumference and 2.0 for abdominal circumference.