Relative efficiency of using summary versus individual data in random-effects meta-analysis.

Meta-analysis is a statistical methodology for combining information from diverse sources so that a more reliable and efficient conclusion can be reached. It can be conducted by either synthesizing study-level summary statistics or drawing inference from an overarching model for individual participant data (IPD) if available. The latter is often viewed as the "gold standard." For random-effects models, however, it remains not fully understood whether the use of IPD indeed gains efficiency over summary statistics. In this paper, we examine the relative efficiency of the two methods under a general likelihood inference setting. We show theoretically and numerically that summary-statistics-based analysis is at most as efficient as IPD analysis, provided that the random effects follow the Gaussian distribution, and maximum likelihood estimation is used to obtain summary statistics. More specifically, (i) the two methods are equivalent in an asymptotic sense; and (ii) summary-statistics-based inference can incur an appreciable loss of efficiency if the sample sizes are not sufficiently large. Our results are established under the assumption that the between-study heterogeneity parameter remains constant regardless of the sample sizes, which is different from a previous study. Our findings are confirmed by the analyses of simulated data sets and a real-world study of alcohol interventions.

Meta-analysis framework for exact inferences with application to the analysis of rare events.

The usefulness of meta-analysis has been recognized in the evaluation of drug safety, as a single trial usually yields few adverse events and offers limited information. For rare events, conventional meta-analysis methods may yield an invalid inference, as they often rely on large sample theories and require empirical corrections for zero events. These problems motivate research in developing exact methods, including Tian et al.'s method of combining confidence intervals (2009, Biostatistics, 10, 275-281) and Liu et al.'s method of combining p-value functions (2014, JASA, 109, 1450-1465). This article shows that these two exact methods can be unified under the framework of combining confidence distributions (CDs). Furthermore, we show that the CD method generalizes Tian et al.'s method in several aspects. Given that the CD framework also subsumes the Mantel-Haenszel and Peto methods, we conclude that the CD method offers a general framework for meta-analysis of rare events. We illustrate the CD framework using two real data sets collected for the safety analysis of diabetes drugs.

Efficient network meta-analysis: a confidence distribution approach.

Network meta-analysis synthesizes several studies of multiple treatment comparisons to simultaneously provide inference for all treatments in the network. It can often strengthen inference on pairwise comparisons by borrowing evidence from other comparisons in the network. Current network meta-analysis approaches are derived from either conventional pairwise meta-analysis or hierarchical Bayesian methods. This paper introduces a new approach for network meta-analysis by combining confidence distributions (CDs). Instead of combining point estimators from individual studies in the conventional approach, the new approach combines CDs which contain richer information than point estimators and thus achieves greater efficiency in its inference. The proposed CD approach can e ciently integrate all studies in the network and provide inference for all treatments even when individual studies contain only comparisons of subsets of the treatments. Through numerical studies with real and simulated data sets, the proposed approach is shown to outperform or at least equal the traditional pairwise meta-analysis and a commonly used Bayesian hierarchical model. Although the Bayesian approach may yield comparable results with a suitably chosen prior, it is highly sensitive to the choice of priors (especially the prior of the between-trial covariance structure), which is often subjective. The CD approach is a general frequentist approach and is prior-free. Moreover, it can always provide a proper inference for all the treatment effects regardless of the between-trial covariance structure.

Incorporating heterogeneous parent-child environmental effects in biometrical genetic models.

Heritability is an important index used to measure genetic effects in epidemiologic studies. For estimating heritability, researchers have proposed various statistical models to accommodate different study settings. These models include the ACE or ADE models for classical twin studies and the ACDE model for extended twin family studies. Researchers have shown that the ACDE model is less biased in heritability estimation, because of the utilization of the data of two generations. However, this model simply assumes that all family members, including twins and their parents, are exposed to the same environmental factors. This assumption may not be reasonable in many cases. In this paper, we propose a novel biometrical genetic model that can incorporate heterogeneous parent-child environmental effects. This model enables us to identify the heterogeneity of heritability, if it exists, between the two generations. This advantage is numerically demonstrated in our simulation studies. We apply our proposed model to the anterior chamber depth data from the Guangzhou Twin Eye Study in China. The analysis result reveals significant heterogeneity of heritability between the twin cohort and the parent cohort.

A new goodness-of-fit measure for probit models: Surrogate R2.

Probit models are used extensively for inferential purposes in the social sciences as discrete data are prevalent in a vast body of social studies. Among many accompanying model inference problems, a critical question remains unsettled: how to develop a goodness-of-fit measure that resembles the ordinary least square (OLS) R2 used for linear models. Such a measure has long been sought to achieve 'comparability' of different empirical models across multiple samples addressing similar social questions. To this end, we propose a novel R2 measure for probit models using the notion of surrogacy - simulating a continuous variable S as a surrogate of the original discrete response (Liu & Zhang, Journal of the American Statistical Association, 113, 845 and 2018). The proposed R2 is the proportion of the variance of the surrogate response explained by explanatory variables through a linear model, and we call it a surrogate R2 . This paper shows both theoretically and numerically that the surrogate R2 approximates the OLS R2 based on the latent continuous variable, preserves the interpretation of explained variation, and maintains monotonicity between nested models. As no other pseudo R2 , McKelvey and Zavoina's and McFadden's included, can meet all the three criteria simultaneously, our measure fills this crucial void in probit model inference.

Residuals and Diagnostics for Ordinal Regression Models: A Surrogate Approach.

Ordinal outcomes are common in scientific research and everyday practice, and we often rely on regression models to make inference. A long-standing problem with such regression analyses is the lack of effective diagnostic tools for validating model assumptions. The difficulty arises from the fact that an ordinal variable has discrete values that are labeled with, but not, numerical values. The values merely represent ordered categories. In this paper, we propose a surrogate approach to defining residuals for an ordinal outcome Y. The idea is to define a continuous variable S as a "surrogate" of Y and then obtain residuals based on S. For the general class of cumulative link regression models, we study the residual's theoretical and graphical properties. We show that the residual has null properties similar to those of the common residuals for continuous outcomes. Our numerical studies demonstrate that the residual has power to detect misspecification with respect to 1) mean structures; 2) link functions; 3) heteroscedasticity; 4) proportionality; and 5) mixed populations. The proposed residual also enables us to develop numeric measures for goodness-of-fit using classical distance notions. Our results suggest that compared to a previously defined residual, our residual can reveal deeper insights into model diagnostics. We stress that this work focuses on residual analysis, rather than hypothesis testing. The latter has limited utility as it only provides a single p-value, whereas our residual can reveal what components of the model are misspecified and advise how to make improvements.

Modeling Hybrid Traits for Comorbidity and Genetic Studies of Alcohol and Nicotine Co-Dependence.

We propose a novel multivariate model for analyzing hybrid traits and identifying genetic factors for comorbid conditions. Comorbidity is a common phenomenon in mental health in which an individual suffers from multiple disorders simultaneously. For example, in the Study of Addiction: Genetics and Environment (SAGE), alcohol and nicotine addiction were recorded through multiple assessments that we refer to as hybrid traits. Statistical inference for studying the genetic basis of hybrid traits has not been well-developed. Recent rank-based methods have been utilized for conducting association analyses of hybrid traits but do not inform the strength or direction of effects. To overcome this limitation, a parametric modeling framework is imperative. Although such parametric frameworks have been proposed in theory, they are neither well-developed nor extensively used in practice due to their reliance on complicated likelihood functions that have high computational complexity. Many existing parametric frameworks tend to instead use pseudo-likelihoods to reduce computational burdens. Here, we develop a model fitting algorithm for the full likelihood. Our extensive simulation studies demonstrate that inference based on the full likelihood can control the type-I error rate, and gains power and improves the effect size estimation when compared with several existing methods for hybrid models. These advantages remain even if the distribution of the latent variables is misspecified. After analyzing the SAGE data, we identify three genetic variants (rs7672861, rs958331, rs879330) that are significantly associated with the comorbidity of alcohol and nicotine addiction at the chromosome-wide level. Moreover, our approach has greater power in this analysis than several existing methods for hybrid traits.Although the analysis of the SAGE data motivated us to develop the model, it can be broadly applied to analyze any hybrid responses.

Multivariate Meta-Analysis of Heterogeneous Studies Using Only Summary Statistics: Efficiency and Robustness.

Meta-analysis has been widely used to synthesize evidence from multiple studies for common hypotheses or parameters of interest. However, it has not yet been fully developed for incorporating heterogeneous studies, which arise often in applications due to different study designs, populations or outcomes. For heterogeneous studies, the parameter of interest may not be estimable for certain studies, and in such a case, these studies are typically excluded from conventional meta-analysis. The exclusion of part of the studies can lead to a non-negligible loss of information. This paper introduces a metaanalysis for heterogeneous studies by combining the confidence density functions derived from the summary statistics of individual studies, hence referred to as the CD approach. It includes all the studies in the analysis and makes use of all information, direct as well as indirect. Under a general likelihood inference framework, this new approach is shown to have several desirable properties, including: i) it is asymptotically as efficient as the maximum likelihood approach using individual participant data (IPD) from all studies; ii) unlike the IPD analysis, it suffices to use summary statistics to carry out the CD approach. Individual-level data are not required; and iii) it is robust against misspecification of the working covariance structure of the parameter estimates. Besides its own theoretical significance, the last property also substantially broadens the applicability of the CD approach. All the properties of the CD approach are further confirmed by data simulated from a randomized clinical trials setting as well as by real data on aircraft landing performance. Overall, one obtains an unifying approach for combining summary statistics, subsuming many of the existing meta-analysis methods as special cases.

Temporal relationship between vitamin D status and parathyroid hormone in the United States.

BACKGROUND: Interpretation of parathyroid hormone (iPTH) requires knowledge of vitamin D status that is influenced by season. OBJECTIVE: Characterize the temporal relationship between 25-hydroxyvitamin D3 levels [25(OH)D3] and intact iPTH for several seasons, by gender and latitude in the U.S. and relate 25-hydrovitamin D2 [25(OH)D2] levels with PTH levels and total 25(OH)D levels. METHOD: We retrospectively determined population weekly-mean concentrations of unpaired [25(OH)D2 and 25(OH)D3] and iPTH using 3.8 million laboratory results of adults. The 25(OH)D3 and iPTH distributions were normalized and the means fit with a sinusoidal function for both gender and latitudes: North >40, Central 32-40 and South <32 degrees. We analyzed PTH and total 25(OH)D separately in samples with detectable 25(OH)D2 (≥4 ng/mL). FINDINGS: Seasonal variation was observed for all genders and latitudes. 25(OH)D3 peaks occurred in September and troughs in March. iPTH levels showed an inverted pattern of peaks and troughs relative to 25(OH)D3, with a delay of 4 weeks. Vitamin D deficiency and insufficiency was common (33% <20 ng/mL; 60% <30 ng/mL) as was elevated iPTH levels (33%>65 pg/mL). The percentage of patients deficient in 25(OH)D3 seasonally varied from 21% to 48% and the percentage with elevated iPTH reciprocally varied from 28% to 38%. Patients with detectable 25(OH)D2 had higher PTH levels and 57% of the samples with a total 25(OH)D > 50 ng/mL had detectable 25(OH)D2. INTERPRETATION: 25(OH)D3 and iPTH levels vary in a sinusoidal pattern throughout the year, even in vitamin D2 treated patients; 25(OH)D3, being higher in the summer and lower in the winter months, with iPTH showing the reverse pattern. A large percentage of the tested population showed vitamin D deficiency and secondary hyperparathyroidism. These observations held across three latitudinal regions, both genders, multiple-years, and in the presence or absence of detectable 25(OH)D2, and thus are applicable for patient care.

Exact meta-analysis approach for discrete data and its application to 2 × 2 tables with rare events.

This paper proposes a general exact meta-analysis approach for synthesizing inferences from multiple studies of discrete data. The approach combines the p-value functions (also known as significance functions) associated with the exact tests from individual studies. It encompasses a broad class of exact meta-analysis methods, as it permits broad choices for the combining elements, such as tests used in individual studies, and any parameter of interest. The approach yields statements that explicitly account for the impact of individual studies on the overall inference, in terms of efficiency/power and the type I error rate. Those statements also give rises to empirical methods for further enhancing the combined inference. Although the proposed approach is for general discrete settings, for convenience, it is illustrated throughout using the setting of meta-analysis of multiple 2 × 2 tables. In the context of rare events data, such as observing few, zero or zero total (i.e., zero events in both arms) outcomes in binomial trials or 2 × 2 tables, most existing meta-analysis methods rely on the large-sample approximations which may yield invalid inference. The commonly used corrections to zero outcomes in rare events data, aiming to improve numerical performance can also incur undesirable consequences. The proposed approach applies readily to any rare event setting, including even the zero total event studies without any artificial correction. While debates continue on whether or how zero total event studies should be incorporated in meta-analysis, the proposed approach has the advantage of automatically including those studies and thus making use of all available data. Through numerical studies in rare events settings, the proposed exact approach is shown to be efficient and, generally, outperform commonly used meta-analysis methods, including Mental-Haenszel and Peto methods.