A Method of Moments Estimator for Random Effect Multivariate Meta‐Analysis

Summary Meta‐analysis is a powerful approach to combine evidence from multiple studies to make inference about one or more parameters of interest, such as regression coefficients. The validity of the fixed effect model meta‐analysis depends on the underlying assumption that all studies in the meta‐analysis share the same effect size. In the presence of heterogeneity, the fixed effect model incorrectly ignores the between‐study variance and may yield false positive results. The random effect model takes into account both within‐study and between‐study variances. It is more conservative than the fixed effect model and should be favored in the presence of heterogeneity. In this paper, we develop a noniterative method of moments estimator for the between‐study covariance matrix in the random effect model multivariate meta‐analysis. To our knowledge, it is the first such method of moments estimator in the matrix form. We show that our estimator is a multivariate extension of DerSimonian and Laird’s univariate method of moments estimator, and it is invariant to linear transformations. In the simulation study, our method performs well when compared to existing random effect model multivariate meta‐analysis approaches. We also apply our method in the analysis of a real data example.     

The genetic variance but not the genetic covariance of life‐history traits changes towards the north in a time‐constrained insect

Seasonal time constraints are usually stronger at higher than lower latitudes and can exert strong selection on life‐history traits and the correlations among these traits. To predict the response of life‐history traits to environmental change along a latitudinal gradient, information must be obtained about genetic variance in traits and also genetic correlation between traits, that is the genetic variance‐covariance matrix, G . Here, we estimated G for key life‐history traits in an obligate univoltine damselfly that faces seasonal time constraints. We exposed populations to simulated native temperatures and photoperiods and common garden environmental conditions in a laboratory set‐up. Despite differences in genetic variance in these traits between populations (lower variance at northern latitudes), there was no evidence for latitude‐specific covariance of the life‐history traits. At simulated native conditions, all populations showed strong genetic and phenotypic correlations between traits that shaped growth and development. The variance–covariance matrix changed considerably when populations were exposed to common garden conditions compared with the simulated natural conditions, showing the importance of environmentally induced changes in multivariate genetic structure. Our results highlight the importance of estimating variance–covariance matrixes in environments that mimic selection pressures and not only trait variances or mean trait values in common garden conditions for understanding the trait evolution across populations and environments.     

Using aggregate data to estimate the standard error of a treatment–covariate interaction in an individual patient data meta‐analysis

Subgroup analyses are important to medical research because they shed light on the heterogeneity of treatment effectts. A treatment–covariate interaction in an individual patient data (IPD) meta‐analysis is the most reliable means to estimate how a subgroup factor modifies a treatment's effectiveness. However, owing to the challenges in collecting participant data, an approach based on aggregate data might be the only option. In these circumstances, it would be useful to assess the relative efficiency and power loss of a subgroup analysis without patient‐level data. We present methods that use aggregate data to estimate the standard error of an IPD meta‐analysis’ treatment–covariate interaction for regression models of a continuous or dichotomous patient outcome. Numerical studies indicate that the estimators have good accuracy. An application to a previously published meta‐regression illustrates the practical utility of the methodology.     

A matrix‐based method of moments for fitting the multivariate random effects model for meta‐analysis and meta‐regression

Multivariate meta‐analysis is becoming more commonly used. Methods for fitting the multivariate random effects model include maximum likelihood, restricted maximum likelihood, Bayesian estimation and multivariate generalisations of the standard univariate method of moments. Here, we provide a new multivariate method of moments for estimating the between‐study covariance matrix with the properties that (1) it allows for either complete or incomplete outcomes and (2) it allows for covariates through meta‐regression. Further, for complete data, it is invariant to linear transformations. Our method reduces to the usual univariate method of moments, proposed by DerSimonian and Laird, in a single dimension. We illustrate our method and compare it with some of the alternatives using a simulation study and a real example.     

A population‐averaged approach to diagnostic test meta‐analysis

The meta‐analysis of diagnostic accuracy studies is often of interest in screening programs for many diseases. The typical summary statistics for studies chosen for a diagnostic accuracy meta‐analysis are often two dimensional: sensitivities and specificities. The common statistical analysis approach for the meta‐analysis of diagnostic studies is based on the bivariate generalized linear‐mixed model (BGLMM), which has study‐specific interpretations. In this article, we present a population‐averaged (PA) model using generalized estimating equations (GEE) for making inference on mean specificity and sensitivity of a diagnostic test in the population represented by the meta‐analytic studies. We also derive the marginalized counterparts of the regression parameters from the BGLMM. We illustrate the proposed PA approach through two dataset examples and compare performance of estimators of the marginal regression parameters from the PA model with those of the marginalized regression parameters from the BGLMM through Monte Carlo simulation studies. Overall, both marginalized BGLMM and GEE with sandwich standard errors maintained nominal 95% confidence interval coverage levels for mean specificity and mean sensitivity in meta‐analysis of 25 of more studies even under misspecification of the covariance structure of the bivariate positive test counts for diseased and nondiseased subjects.     

Comparison of One‐Step and Two‐Step Meta‐Analysis Models Using Individual Patient Data

The problem of combining information from separate trials is a key consideration when performing a meta‐analysis or planning a multicentre trial. Although there is a considerable journal literature on meta‐analysis based on individual patient data (IPD), i.e. a one‐step IPD meta‐analysis, versus analysis based on summary data, i.e. a two‐step IPD meta‐analysis, recent articles in the medical literature indicate that there is still confusion and uncertainty as to the validity of an analysis based on aggregate data. In this study, we address one of the central statistical issues by considering the estimation of a linear function of the mean, based on linear models for summary data and for IPD. The summary data from a trial is assumed to comprise the best linear unbiased estimator, or maximum likelihood estimator of the parameter, along with its covariance matrix. The setup, which allows for the presence of random effects and covariates in the model, is quite general and includes many of the commonly employed models, for example, linear models with fixed treatment effects and fixed or random trial effects. For this general model, we derive a condition under which the one‐step and two‐step IPD meta‐analysis estimators coincide, extending earlier work considerably. The implications of this result for the specific models mentioned above are illustrated in detail, both theoretically and in terms of two real data sets, and the roles of balance and heterogeneity are highlighted. Our analysis also shows that when covariates are present, which is typically the case, the two estimators coincide only under extra simplifying assumptions, which are somewhat unrealistic in practice.     

On Estimating the Relationship between Longitudinal Measurements and Time‐to‐Event Data Using a Simple Two‐Stage Procedure

Summary Ye, Lin, and Taylor (2008, Biometrics 64 , 1238–1246) proposed a joint model for longitudinal measurements and time‐to‐event data in which the longitudinal measurements are modeled with a semiparametric mixed model to allow for the complex patterns in longitudinal biomarker data. They proposed a two‐stage regression calibration approach that is simpler to implement than a joint modeling approach. In the first stage of their approach, the mixed model is fit without regard to the time‐to‐event data. In the second stage, the posterior expectation of an individual's random effects from the mixed‐model are included as covariates in a Cox model. Although Ye et al. (2008) acknowledged that their regression calibration approach may cause a bias due to the problem of informative dropout and measurement error, they argued that the bias is small relative to alternative methods. In this article, we show that this bias may be substantial. We show how to alleviate much of this bias with an alternative regression calibration approach that can be applied for both discrete and continuous time‐to‐event data. Through simulations, the proposed approach is shown to have substantially less bias than the regression calibration approach proposed by Ye et al. (2008) . In agreement with the methodology proposed by Ye et al. (2008) , an advantage of our proposed approach over joint modeling is that it can be implemented with standard statistical software and does not require complex estimation techniques.     

Test‐compatible confidence intervals for adaptive two‐stage single‐arm designs with binary endpoint

Inference after two‐stage single‐arm designs with binary endpoint is challenging due to the nonunique ordering of the sampling space in multistage designs. We illustrate the problem of specifying test‐compatible confidence intervals for designs with nonconstant second‐stage sample size and present two approaches that guarantee confidence intervals consistent with the test decision. Firstly, we extend the well‐known Clopper–Pearson approach of inverting a family of two‐sided hypothesis tests from the group‐sequential case to designs with fully adaptive sample size. Test compatibility is achieved by using a sample space ordering that is derived from a test‐compatible estimator. The resulting confidence intervals tend to be conservative but assure the nominal coverage probability. In order to assess the possibility of further improving these confidence intervals, we pursue a direct optimization approach minimizing the mean width of the confidence intervals. While the latter approach produces more stable coverage probabilities, it is also slightly anti‐conservative and yields only negligible improvements in mean width. We conclude that the Clopper–Pearson‐type confidence intervals based on a test‐compatible estimator are the best choice if the nominal coverage probability is not to be undershot and compatibility of test decision and confidence interval is to be preserved.     

Inference for meta‐analysis with a suspected temporal trend

There is sometimes a clear evidence of a strong secular trend in the treatment effect of studies included in a meta‐analysis. In such cases, estimating the present‐day treatment effect by meta‐regression is both reasonable and straightforward. We however consider the more common situation where a secular trend is suspected, but is not strongly statistically significant. Typically, this lack of significance is due to the small number of studies included in the analysis, so that a meta‐regression could give wild point estimates. We introduce an empirical Bayes meta‐analysis methodology, which shrinks the secular trend toward zero. This has the effect that treatment effects are adjusted for trend, but where the evidence from data is weak, wild results are not obtained. We explore several frequentist approaches and a fully Bayesian method is also implemented. A measure of trend analogous to I² is described, and exact significance tests for trend are given. Our preferred method is one based on penalized or h‐likelihood, which is computationally simple, and allows invariance of predictions to the (arbitrary) choice of time origin. We suggest that a trendless standard random effects meta‐analysis should routinely be supplemented with an h‐likelihood analysis as a sensitivity analysis.     

Environmental effects on the structure of the G‐matrix

Genetic correlations between traits determine the multivariate response to selection in the short term, and thereby play a causal role in evolutionary change. Although individual studies have documented environmentally induced changes in genetic correlations, the nature and extent of environmental effects on multivariate genetic architecture across species and environments remain largely uncharacterized. We reviewed the literature for estimates of the genetic variance–covariance ( G ) matrix in multiple environments, and compared differences in G between environments to the divergence in G between conspecific populations (measured in a common garden). We found that the predicted evolutionary trajectory differed as strongly between environments as it did between populations. Between‐environment differences in the underlying structure of G (total genetic variance and the relative magnitude and orientation of genetic correlations) were equal to or greater than between‐population differences. Neither environmental novelty, nor the difference in mean phenotype predicted these differences in G . Our results suggest that environmental effects on multivariate genetic architecture may be comparable to the divergence that accumulates over dozens or hundreds of generations between populations. We outline avenues of future research to address the limitations of existing data and characterize the extent to which lability in genetic correlations shapes evolution in changing environments.     

Multivariate meta‐analysis with an increasing number of parameters

Meta‐analysis can average estimates of multiple parameters, such as a treatment's effect on multiple outcomes, across studies. Univariate meta‐analysis (UVMA) considers each parameter individually, while multivariate meta‐analysis (MVMA) considers the parameters jointly and accounts for the correlation between their estimates. The performance of MVMA and UVMA has been extensively compared in scenarios with two parameters. Our objective is to compare the performance of MVMA and UVMA as the number of parameters, p, increases. Specifically, we show that (i) for fixed‐effect (FE) meta‐analysis, the benefit from using MVMA can substantially increase as p increases; (ii) for random effects (RE) meta‐analysis, the benefit from MVMA can increase as p increases, but the potential improvement is modest in the presence of high between‐study variability and the actual improvement is further reduced by the need to estimate an increasingly large between study covariance matrix; and (iii) when there is little to no between‐study variability, the loss of efficiency due to choosing RE MVMA over FE MVMA increases as p increases. We demonstrate these three features through theory, simulation, and a meta‐analysis of risk factors for non‐Hodgkin lymphoma.     

A Robust Test for Two‐Stage Design in Genome‐Wide Association Studies

Summary A two‐stage design is cost‐effective for genome‐wide association studies (GWAS) testing hundreds of thousands of single nucleotide polymorphisms (SNPs). In this design, each SNP is genotyped in stage 1 using a fraction of case–control samples. Top‐ranked SNPs are selected and genotyped in stage 2 using additional samples. A joint analysis, combining statistics from both stages, is applied in the second stage. Follow‐up studies can be regarded as a two‐stage design. Once some potential SNPs are identified, independent samples are further genotyped and analyzed separately or jointly with previous data to confirm the findings. When the underlying genetic model is known, an asymptotically optimal trend test (TT) can be used at each analysis. In practice, however, genetic models for SNPs with true associations are usually unknown. In this case, the existing methods for analysis of the two‐stage design and follow‐up studies are not robust across different genetic models. We propose a simple robust procedure with genetic model selection to the two‐stage GWAS. Our results show that, if the optimal TT has about 80% power when the genetic model is known, then the existing methods for analysis of the two‐stage design have minimum powers about 20% across the four common genetic models (when the true model is unknown), while our robust procedure has minimum powers about 70% across the same genetic models. The results can be also applied to follow‐up and replication studies with a joint analysis.     

Genetic basis of between‐individual and within‐individual variance of docility

Between‐individual variation in phenotypes within a population is the basis of evolution. However, evolutionary and behavioural ecologists have mainly focused on estimating between‐individual variance in mean trait and neglected variation in within‐individual variance, or predictability of a trait. In fact, an important assumption of mixed‐effects models used to estimate between‐individual variance in mean traits is that within‐individual residual variance (predictability) is identical across individuals. Individual heterogeneity in the predictability of behaviours is a potentially important effect but rarely estimated and accounted for. We used 11 389 measures of docility behaviour from 1576 yellow‐bellied marmots (Marmota flaviventris) to estimate between‐individual variation in both mean docility and its predictability. We then implemented a double hierarchical animal model to decompose the variances of both mean trait and predictability into their environmental and genetic components. We found that individuals differed both in their docility and in their predictability of docility with a negative phenotypic covariance. We also found significant genetic variance for both mean docility and its predictability but no genetic covariance between the two. This analysis is one of the first to estimate the genetic basis of both mean trait and within‐individual variance in a wild population. Our results indicate that equal within‐individual variance should not be assumed. We demonstrate the evolutionary importance of the variation in the predictability of docility and illustrate potential bias in models ignoring variation in predictability. We conclude that the variability in the predictability of a trait should not be ignored, and present a coherent approach for its quantification.     

The importance of individual developmental variation in stage‐structured population models

Population stage structure is fundamental to ecology, and models of this structure have proven useful in many different systems. Many ecological variables other than stage, such as habitat type, site occupancy and metapopulation status are also modelled using transitions among discrete states. Transitions among life stages can be characterised by the distribution of time spent in each stage, including the mean and variance of each stage duration and within‐individual correlations among multiple stage durations. Three modelling traditions represent stage durations differently. Matrix models can be derived as a long‐run approximation from any distribution of stage durations, but they are often interpreted directly as a Markov model for stage transitions. Statistical stage‐duration distribution models accommodate the variation typical of cohort development data, but such realism has rarely been incorporated in population theory or statistical population models. Delay‐differential equation models include lags but no variation, except in limited cases. We synthesise these models in one framework and illustrate how individual variation and correlations in development can impact population growth. Furthermore, different development models can yield the same long‐term matrix transition rates but different sensitivities and elasticities. Finally, we discuss future directions for estimating realistic stage duration models from data.     

Measuring Sustainability Efficiency Using a Two‐Stage Data Envelopment Analysis Approach

In this article, we apply an additive two‐stage data envelopment analysis estimator on a panel of 20 countries with advanced economies for the time period 1990–2011 in order to create a composite sustainability efficiency index. We use a window‐based approach in order to study the countries over the years. The sustainability efficiency index is decomposed into production efficiency and eco‐efficiency indicators. The results reveal inequalities among the examined countries between the two stages. The eco‐efficiency stage is characterized by large inequalities among countries and significantly lower efficiency scores than the overall sustainability efficiency and the production efficiency. Finally, it is reported that a country's high production efficiency level does not ensure a high eco‐efficiency performance.     

Statistical Modelling of Physiological Heat Stress Response by Means of a Nonlinear Two‐Stage Model

In the context of medical or biological studies, very often parameters of interest are measured repeatedly over time under a given set of conditions. This results in a set of (often similarly shaped) time series. Then, the objective is the determination of the functional relationship between the parameter of interest and time on the one hand, and the analysis of the variation of this functional relationship between experiments, on the other hand. This may be done by means of a two‐stage model. The present work describes the theory of the two‐stage model and its application to the increase of human core temperature for a set of 678 experiments where the subjects were exposed to warm and hot environments. The data originating from 6 European research institutes, have been pooled into one database for the Heat Stress Project within the scope of the BIOMED 2 programme of the European Union. A nonlinear two‐stage model was applied, with a logistic function modelling the nonlinear time course of the core temperature, and with its parameters depending on air temperature, mean radiant temperature, air velocity, partial vapour pressure, clothing insulation, metabolic rate, gender, acclimatisation status and body surface area. We conclude that acclimatisation, clothing insulation, body surface area, air temperature, air velocity, partial vapour pressure, metabolic rate, and the difference between mean radiant temperature and air temperature play an important role for work in warm and hot environments. We show how our results can be used for the estimation of allowable exposure times for work in hot environments.     

The biology hidden inside residual within‐individual phenotypic variation

Phenotypes vary hierarchically among taxa and populations, among genotypes within populations, among individuals within genotypes, and also within individuals for repeatedly expressed, labile phenotypic traits. This hierarchy produces some fundamental challenges to clearly defining biological phenomena and constructing a consistent explanatory framework. We use a heuristic statistical model to explore two consequences of this hierarchy. First, although the variation existing among individuals within populations has long been of interest to evolutionary biologists, within‐individual variation has been much less emphasized. Within‐individual variance occurs when labile phenotypes (behaviour, physiology, and sometimes morphology) exhibit phenotypic plasticity or deviate from a norm‐of‐reaction within the same individual. A statistical partitioning of phenotypic variance leads us to explore an array of ideas about residual within‐individual variation. We use this approach to draw attention to additional processes that may influence within‐individual phenotypic variance, including interactions among environmental factors, ecological effects on the fitness consequences of plasticity, and various types of adaptive variance. Second, our framework for investigating variation in phenotypic variance reveals that interactions between levels of the hierarchy form the preconditions for the evolution of all types of plasticity, and we extend this idea to the residual level within individuals, where both adaptive plasticity in residuals and canalization‐like processes (stability) can evolve. With the statistical tools now available to examine heterogeneous residual variance, an array of novel questions linking phenotype to environment can be usefully addressed.     

Transgenerational effects benefit offspring across diverse environments: a meta‐analysis in plants and animals

The adaptive value of transgenerational effects (the ancestor environmental effects on offspring) in changing environments has received much attention in recent years, but the related empirical evidence remains equivocal. Here, we conducted a meta‐analysis summarising 139 experimental studies in plants and animals with 1170 effect sizes to investigate the generality of transgenerational effects across taxa, traits, and environmental contexts. It was found that transgenerational effects generally enhanced offspring performance in response to both stressful and benign conditions. The strongest effects are in annual plants and invertebrates, whereas vertebrates appear to benefit mostly under benign conditions, and perennial plants show hardly any transgenerational responses at all. These differences among taxonomic/life‐history groups possibly reflect that vertebrates can avoid stressful conditions through their mobility, and longer‐lived plants have alternative strategies. In addition to environmental contexts and taxonomic/life‐history groups, transgenerational effects also varied among traits and developmental stages of ancestors and offspring, but the effects were similarly strong across three generations of offspring. By way of a more comprehensive data set and a different effect size, our results differ from those of a recent meta‐analysis, suggesting that transgenerational effects are widespread, strong and persistent and can substantially impact the responses of plants and animals to changing environments.     

Time‐Varying Latent Effect Model for Longitudinal Data with Informative Observation Times

Summary In analysis of longitudinal data, it is not uncommon that observation times of repeated measurements are subject‐specific and correlated with underlying longitudinal outcomes. Taking account of the dependence between observation times and longitudinal outcomes is critical under these situations to assure the validity of statistical inference. In this article, we propose a flexible joint model for longitudinal data analysis in the presence of informative observation times. In particular, the new procedure considers the shared random‐effect model and assumes a time‐varying coefficient for the latent variable, allowing a flexible way of modeling longitudinal outcomes while adjusting their association with observation times. Estimating equations are developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug‐in method. One additional advantage of the procedure is that it provides a unified framework to test whether the effect of the latent variable is zero, constant, or time‐varying. Simulation studies show that the proposed approach is appropriate for practical use. An application to a bladder cancer data is also given to illustrate the methodology.     

Double‐wavelet transform for multisubject task‐induced functional magnetic resonance imaging data

The goal of this article is to model multisubject task‐induced functional magnetic resonance imaging (fMRI) response among predefined regions of interest (ROIs) of the human brain. Conventional approaches to fMRI analysis only take into account temporal correlations, but do not rigorously model the underlying spatial correlation due to the complexity of estimating and inverting the high dimensional spatio‐temporal covariance matrix. Other spatio‐temporal model approaches estimate the covariance matrix with the assumption of stationary time series, which is not always feasible. To address these limitations, we propose a double‐wavelet approach for modeling the spatio‐temporal brain process. Working with wavelet coefficients simplifies temporal and spatial covariance structure because under regularity conditions, wavelet coefficients are approximately uncorrelated. Different wavelet functions were used to capture different correlation structures in the spatio‐temporal model. The main advantages of the wavelet approach are that it is scalable and that it deals with nonstationarity in brain signals. Simulation studies showed that our method could reduce false‐positive and false‐negative rates by taking into account spatial and temporal correlations simultaneously. We also applied our method to fMRI data to study activation in prespecified ROIs in the prefontal cortex. Data analysis showed that the result using the double‐wavelet approach was more consistent than the conventional approach when sample size decreased.