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Quantifying the heterogeneity is an important issue in meta-analysis, and among the existing measures, the $I^2$ statistic is most commonly used. In this paper, we first illustrate with a simple example that the $I^2$ statistic is heavily dependent on the study sample sizes, mainly because it is used to quantify the heterogeneity between the observed effect sizes. To reduce the influence of sample sizes, we introduce an alternative measure that aims to directly measure the heterogeneity between the study populations involved in the meta-analysis. We further propose a new estimator, namely the $I_A^2$ statistic, to estimate the newly defined measure of heterogeneity. For practical implementation, the exact formulas of the $I_A^2$ statistic are also derived under two common scenarios with the effect size as the mean difference (MD) or the standardized mean difference (SMD). Simulations and real data analysis demonstrate that the $I_A^2$ statistic provides an asymptotically unbiased estimator for the absolute heterogeneity between the study populations, and it is also independent of the study sample sizes as expected. To conclude, our newly defined $I_A^2$ statistic can be used as a supplemental measure of heterogeneity to monitor the situations where the study effect sizes are indeed similar with little biological difference. In such scenario, the fixed-effect model can be appropriate; nevertheless, when the sample sizes are sufficiently large, the $I^2$ statistic may still increase to 1 and subsequently suggest the random-effects model for meta-analysis., Comment: 40 pages, 7 figures and 3 tables

Difference-based methods have been attracting increasing attention in nonparametric regression, in particular for estimating the residual variance.To implement the estimation, one needs to choose an appropriate difference sequence, mainly between {\em the optimal difference sequence} and {\em the ordinary difference sequence}. The difference sequence selection is a fundamental problem in nonparametric regression, and it remains a controversial issue for over three decades. In this paper, we propose to tackle this challenging issue from a very unique perspective, namely by introducing a new difference sequence called {\em the optimal-$k$ difference sequence}. The new difference sequence not only provides a better balance between the bias-variance trade-off, but also dramatically enlarges the existing family of difference sequences that includes the optimal and ordinary difference sequences as two important special cases. We further demonstrate, by both theoretical and numerical studies, that the optimal-$k$ difference sequence has been pushing the boundaries of our knowledge in difference-based methods in nonparametric regression, and it always performs the best in practical situations.

We consider a wireless network scenario applicable to metropolitan areas with developed public transport networks and high commute demands, where the mobile user equipments (UEs) move along fixed and predetermined trajectories and request to associate with millimeter-wave (mmWave) base stations (BSs). An effective and efficient algorithm, called the Sequence Q-learning Algorithm (SQA), is proposed to maximize the long-run average transmission rate of the network, which is an NP-hard problem. Furthermore, the SQA tackles the complexity issue by only allowing possible re-associations (handover of a UE from one BS to another) at a discrete set of decision epochs and has polynomial time complexity. This feature of the SQA also restricts too frequent handovers, which are considered highly undesirable in mmWave networks. Moreover, we demonstrate by extensive numerical results that the SQA can significantly outperform the benchmark algorithms proposed in existing research by taking all UEs' future trajectories and possible decisions into account at every decision epoch.

Quantifying the heterogeneity is an important issue in meta-analysis, and among the existing measures, the $I^2$ statistic is the most commonly used measure in the literature. In this paper, we show that the $I^2$ statistic was, in fact, defined as problematic or even completely wrong from the very beginning. To confirm this statement, we first present a motivating example to show that the $I^2$ statistic is heavily dependent on the study sample sizes, and consequently it may yield contradictory results for the amount of heterogeneity. Moreover, by drawing a connection between ANOVA and meta-analysis, the $I^2$ statistic is shown to have, mistakenly, applied the sampling errors of the estimators rather than the variances of the study populations. Inspired by this, we introduce an Intrinsic measure for Quantifying the heterogeneity in meta-analysis, and meanwhile study its statistical properties to clarify why it is superior to the existing measures. We further propose an optimal estimator, referred to as the IQ statistic, for the new measure of heterogeneity that can be readily applied in meta-analysis. Simulations and real data analysis demonstrate that the IQ statistic provides a nearly unbiased estimate of the true heterogeneity and it is also independent of the study sample sizes., Comment: With a move comprehensive version with the new title "An alternative measure for quantifying the heterogeneity in meta-analysis", this old version is no longer most suitable to be posted in the arXiv. We hence will submit the new version with a new title as arXiv:2403.16706 and withdraw this outdated version. Thank you very much for your kind consideration

For clinical studies with continuous outcomes, when the data are potentially skewed, researchers may choose to report the whole or part of the five-number summary (the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation. In the recent literature, it is often suggested to transform the five-number summary back to the sample mean and standard deviation, which can be subsequently used in a meta-analysis. However, if a study contains skewed data, this transformation and hence the conclusions from the meta-analysis are unreliable. Therefore, we introduce a novel method for detecting the skewness of data using only the five-number summary and the sample size, and meanwhile propose a new flow chart to handle the skewed studies in a different manner. We further show by simulations that our skewness tests are able to control the type I error rates and provide good statistical power, followed by a simulated meta-analysis and a real data example that illustrate the usefulness of our new method in meta-analysis and evidence-based medicine., Comment: 40 pages, 10 figures, 7 tables

As a classic parameter from the binomial distribution, the binomial proportion has been well studied in the literature owing to its wide range of applications. In contrast, the reciprocal of the binomial proportion, also known as the inverse proportion, is often overlooked, even though it also plays an important role in various fields including clinical studies and random sampling. The maximum likelihood estimator of the inverse proportion suffers from the zero-event problem, and to overcome it, alternative methods have been developed in the literature. Nevertheless, there is little work addressing the optimality of the existing estimators, as well as their practical performance comparison. Inspired by this, we propose to further advance the literature by developing an optimal estimator for the inverse proportion in a family of shrinkage estimators. We further derive the explicit and approximate formulas for the optimal shrinkage parameter under different settings. Simulation studies show that the performance of our new estimator performs better than, or as well as, the existing competitors in most practical settings. Finally, to illustrate the usefulness of our new method, we also revisit a recent meta-analysis on COVID-19 data for assessing the relative risks of physical distancing on the infection of coronavirus, in which six out of seven studies encounter the zero-event problem.

For high-dimensional small sample size data, Hotelling's T2 test is not applicable for testing mean vectors due to the singularity problem in the sample covariance matrix. To overcome the problem, there are three main approaches in the literature. Note, however, that each of the existing approaches may have serious limitations and only works well in certain situations. Inspired by this, we propose a pairwise Hotelling method for testing high-dimensional mean vectors, which, in essence, provides a good balance between the existing approaches. To effectively utilize the correlation information, we construct the new test statistics as the summation of Hotelling's test statistics for the covariate pairs with strong correlations and the squared $t$ statistics for the individual covariates that have little correlation with others. We further derive the asymptotic null distributions and power functions for the proposed Hotelling tests under some regularity conditions. Numerical results show that our new tests are able to control the type I error rates, and can achieve a higher statistical power compared to existing methods, especially when the covariates are highly correlated. Two real data examples are also analyzed and they both demonstrate the efficacy of our pairwise Hotelling tests., Comment: 66 pages and 6 figures and 3 tables

When reporting the results of clinical studies, some researchers may choose the five-number summary (including the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation, particularly for skewed data. For these studies, when included in a meta-analysis, it is often desired to convert the five-number summary back to the sample mean and standard deviation. For this purpose, several methods have been proposed in the recent literature and they are increasingly used nowadays. In this paper, we propose to further advance the literature by developing a smoothly weighted estimator for the sample standard deviation that fully utilizes the sample size information. For ease of implementation, we also derive an approximation formula for the optimal weight, as well as a shortcut formula for the sample standard deviation. Numerical results show that our new estimator provides a more accurate estimate for normal data and also performs favorably for non-normal data. Together with the optimal sample mean estimator in Luo et al., our new methods have dramatically improved the existing methods for data transformation, and they are capable to serve as "rules of thumb" in meta-analysis for studies reported with the five-number summary. Finally for practical use, an Excel spreadsheet and an online calculator are also provided for implementing our optimal estimators., Comment: 30 pages and 4 figures and 2 tables. arXiv admin note: substantial text overlap with arXiv:1801.01267

According to Davey et al. (2011) with a total of 22,453 meta-analyses from the January 2008 Issue of the Cochrane Database of Systematic Reviews, the median number of studies included in each meta-analysis is only three. In other words, about a half or more of meta-analyses conducted in the literature include only two or three studies. While the common-effect model (also referred to as the fixed-effect model) may lead to misleading results when the heterogeneity among studies is large, the conclusions based on the random-effects model may also be unreliable when the number of studies is small. Alternatively, the fixed-effects model avoids the restrictive assumption in the common-effect model and the need to estimate the between-study variance in the random-effects model. We note, however, that the fixed-effects model is under appreciated and rarely used in practice until recently. In this paper, we compare all three models and demonstrate the usefulness of the fixed-effects model when the number of studies is small. In addition, we propose a new estimator for the unweighted average effect in the fixed-effects model. Simulations and real examples are also used to illustrate the benefits of the fixed-effects model and the new estimator., Comment: 29 pages, 6 figures

Meta-analysis is an important tool for combining results from multiple studies and has been widely used in evidence-based medicine for several decades. This paper reports, for the first time, an interesting and valuable paradox in random-effects meta-analysis that is likely to occur when the number of studies is small and/or the heterogeneity is large. With the incredible paradox, we hence advocate meta-analysts to be extremely cautious when interpreting the final results from the random-effects meta-analysis. And more importantly, with the unexpected dilemma in making decisions, the new paradox has raised an open question whether the current random-effects model is reasonable and tenable for meta-analysis, or it needs to be abandoned or further improved to some extent., Comment: 7 pages, 5 figures

Background: High-throughput techniques bring novel tools but also statistical challenges to genomic research. Identifying genes with differential expression between different species is an effective way to discover evolutionarily conserved transcriptional responses. To remove systematic variation between different species for a fair comparison, the normalization procedure serves as a crucial pre-processing step that adjusts for the varying sample sequencing depths and other confounding technical effects. Results: In this paper, we propose a scale based normalization (SCBN) method by taking into account the available knowledge of conserved orthologous genes and hypothesis testing framework. Considering the different gene lengths and unmapped genes between different species, we formulate the problem from the perspective of hypothesis testing and search for the optimal scaling factor that minimizes the deviation between the empirical and nominal type I errors. Conclusions: Simulation studies show that the proposed method performs significantly better than the existing competitor in a wide range of settings. An RNA-seq dataset of different species is also analyzed and it coincides with the conclusion that the proposed method outperforms the existing method. For practical applications, we have also developed an R package named "SCBN" and the software is available at http://www.bioconductor.org/packages/devel/bioc/html/SCBN.html.

In this paper, we propose a varying coefficient panel data model with unobservable multiple interactive fixed effects that are correlated with the regressors. We approximate each coefficient function by B-spline, and propose a robust nonlinear iteration scheme based on the least squares method to estimate the coefficient functions of interest. We also establish the asymptotic theory of the resulting estimators under certain regularity assumptions, including the consistency, the convergence rate and the asymptotic distribution. Furthermore, we develop a least squares dummy variable method to study an important special case of the proposed model: the varying coefficient panel data model with additive fixed effects. To construct the pointwise confidence intervals for the coefficient functions, a residual-based block bootstrap method is proposed to reduce the computational burden as well as to avoid the accumulative errors. Simulation studies and a real data analysis are also carried out to assess the performance of our proposed methods., Comment: 58 pages, 5 figures, 2 tables

As the most important tool to provide high-level evidence-based medicine, researchers can statistically summarize and combine data from multiple studies by conducting meta-analysis. In meta-analysis, mean differences are frequently used effect size measurements to deal with continuous data, such as the Cohen's d statistic and Hedges' g statistic values. To calculate the mean difference based effect sizes, the sample mean and standard deviation are two essential summary measures. However, many of the clinical reports tend not to directly record the sample mean and standard deviation. Instead, the sample size, median, minimum and maximum values and/or the first and third quartiles are reported. As a result, researchers have to transform the reported information to the sample mean and standard deviation for further compute the effect size. Since most of the popular transformation methods were developed upon the normality assumption of the underlying data, it is necessary to perform a pre-test before transforming the summary statistics. In this article, we had introduced test statistics for three popular scenarios in meta-analysis. We suggests medical researchers to perform a normality test of the selected studies before using them to conduct further analysis. Moreover, we applied three different case studies to demonstrate the usage of the newly proposed test statistics. The real data case studies indicate that the new test statistics are easy to apply in practice and by following the recommended path to conduct the meta-analysis, researchers can obtain more reliable conclusions., Comment: 48 pages, 11 figures and 10 tables

For testing a group of hypotheses, tremendous $p$-value combination methods have been developed and widely applied since 1930's. Some methods (e.g., the minimal $p$-value) are optimal for sparse signals, and some others (e.g., Fisher's combination) are optimal for dense signals. To address a wide spectrum of signal patterns, this paper proposes a unifying family of statistics, called TFisher, with general $p$-value truncation and weighting schemes. Analytical calculations for the $p$-value and the statistical power of TFisher under general hypotheses are given. Optimal truncation and weighting parameters are studied based on Bahadur Efficiency (BE) and the proposed Asymptotic Power Efficiency (APE), which is superior to BE for studying the signal detection problem. A soft-thresholding scheme is shown to be optimal for signal detection in a large space of signal patterns. When prior information of signal pattern is unavailable, an omnibus test, oTFisher, can adapt to the given data. Simulations evidenced the accuracy of calculations and validated the theoretical properties. The TFisher tests were applied to analyzing a whole exome sequencing data of amyotrophic lateral sclerosis. Relevant tests and calculations have been implemented into an R package $TFisher$ and published on the CRAN., Comment: 46 pages, 15 figures

In some clinical studies, researchers may report the five number summary (including the sample median, the first and third quartiles, and the minimum and maximum values) rather than the sample mean and standard deviation. To conduct meta-analysis for pooling studies, one needs to first estimate the sample mean and standard deviation from the five number summary. A number of studies have been proposed in the recent literature to solve this problem. However, none of the existing estimators for the standard deviation is satisfactory for practical use. After a brief review of the existing literature, we point out that Wan et al.'s method (BMC Med Res Methodol 14:135, 2014) has a serious limitation in estimating the standard deviation from the five number summary. To improve it, we propose a smoothly weighted estimator by incorporating the sample size information and derive the optimal weight for the new estimator. For ease of implementation, we also provide an approximation formula of the optimal weight and a shortcut formula for estimating the standard deviation from the five number summary. The performance of the proposed estimator is evaluated through two simulation studies. In comparison with Wan et al.'s estimator, our new estimator provides a more accurate estimate for normal data and performs favorably for non-normal data. In real data analysis, our new method is also able to provide a more accurate estimate of the true sample standard deviation than the existing method. In this paper, we propose an optimal estimator of the standard deviation from the five number summary. Together with the optimal mean estimator in Luo et al. (Stat Methods Med Res, in press, 2017), our new methods have improved the existing literature and will make a solid contribution to meta-analysis and evidence-based medicine., Comment: 18 pages, 5 figures, 2 tables

We propose a likelihood ratio test framework for testing normal mean vectors in high-dimensional data under two common scenarios: the one-sample test and the two-sample test with equal covariance matrices. We derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure. In comparison with the diagonal Hotelling's tests, our proposed test statistics display some interesting characteristics. In particular, they are a summation of the log-transformed squared $t$-statistics rather than a direct summation of those components. More importantly, to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses, we do not need the requirement that the covariance matrices follow a diagonal matrix structure. As a consequence, our proposed test methods are very flexible and readily applicable in practice. Simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods., Comment: 38 pages, 5 figures and 6 tables

In this paper, we consider the single-index measurement error model with mismeasured covariates in the nonparametric part. To solve the problem, we develop a simulation-extrapolation (SIMEX) algorithm based on the local linear smoother and the estimating equation. For the proposed SIMEX estimation, it is not needed to assume the distribution of the unobserved covariate. We transform the boundary of a unit ball in $\mathbb{R}^p$ to the interior of a unit ball in $\mathbb{R}^{p-1}$ by using the constraint $\|\beta\|=1$. The proposed SIMEX estimator of the index parameter is shown to be asymptotically normal under some regularity conditions. We also derive the asymptotic bias and variance of the estimator of the unknown link function. Finally, the performance of the proposed method is examined by simulation studies and is illustrated by a real data example., Comment: 25 pages, 7 figures

The era of big data is coming, and evidence-based medicine is attracting increasing attention to improve decision making in medical practice via integrating evidence from well designed and conducted clinical research. Meta-analysis is a statistical technique widely used in evidence-based medicine for analytically combining the findings from independent clinical trials to provide an overall estimation of a treatment effectiveness. The sample mean and standard deviation are two commonly used statistics in meta-analysis but some trials use the median, the minimum and maximum values, or sometimes the first and third quartiles to report the results. Thus, to pool results in a consistent format, researchers need to transform those information back to the sample mean and standard deviation. In this paper, we investigate the optimal estimation of the sample mean for meta-analysis from both theoretical and empirical perspectives. A major drawback in the literature is that the sample size, needless to say its importance, is either ignored or used in a stepwise but somewhat arbitrary manner, e.g., the famous method proposed by Hozo et al. We solve this issue by incorporating the sample size in a smoothly changing weight in the estimators to reach the optimal estimation. Our proposed estimators not only improve the existing ones significantly but also share the same virtue of the simplicity. The real data application indicates that our proposed estimators are capable to serve as "rules of thumb" and will be widely applied in evidence-based medicine., Comment: 21 pages, 7 figures and 7 tables

RNA-sequencing (RNA-Seq) has become a powerful technology to characterize gene expression profiles because it is more accurate and comprehensive than microarrays. Although statistical methods that have been developed for microarray data can be applied to RNA-Seq data, they are not ideal due to the discrete nature of RNA-Seq data. The Poisson distribution and negative binomial distribution are commonly used to model count data. Recently, Witten (2011) proposed a Poisson linear discriminant analysis for RNA-Seq data. The Poisson assumption may not be as appropriate as negative binomial distribution when biological replicates are available and in the presence of overdispersion (i.e., when the variance is larger than the mean). However, it is more complicated to model negative binomial variables because they involve a dispersion parameter that needs to be estimated. In this paper, we propose a negative binomial linear discriminant analysis for RNA-Seq data. By Bayes' rule, we construct the classifier by fitting a negative binomial model, and propose some plug-in rules to estimate the unknown parameters in the classifier. The relationship between the negative binomial classifier and the Poisson classifier is explored, with a numerical investigation of the impact of dispersion on the discriminant score. Simulation results show the superiority of our proposed method. We also analyze four real RNA-Seq data sets to demonstrate the advantage of our method in real-world applications.

In systematic reviews and meta-analysis, researchers often pool the results of the sample mean and standard deviation from a set of similar clinical trials. A number of the trials, however, reported the study using the median, the minimum and maximum values, and/or the first and third quartiles. Hence, in order to combine results, one may have to estimate the sample mean and standard deviation for such trials. In this paper, we propose to improve the existing literature in several directions. First, we show that the sample standard deviation estimation in Hozo et al. (2005) has some serious limitations and is always less satisfactory in practice. Inspired by this, we propose a new estimation method by incorporating the sample size. Second, we systematically study the sample mean and standard deviation estimation problem under more general settings where the first and third quartiles are also available for the trials. Through simulation studies, we demonstrate that the proposed methods greatly improve the existing methods and enrich the literature. We conclude our work with a summary table that serves as a comprehensive guidance for performing meta-analysis in different situations.

We study the least squares estimator in the residual variance estimation context. We show that the mean squared differences of paired observations are asymptotically normally distributed. We further establish that, by regressing the mean squared differences of these paired observations on the squared distances between paired covariates via a simple least squares procedure, the resulting variance estimator is not only asymptotically normal and root-$n$ consistent, but also reaches the optimal bound in terms of estimation variance. We also demonstrate the advantage of the least squares estimator in comparison with existing methods in terms of the second order asymptotic properties., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ432 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

In this paper, a shrinkage estimator for the population mean is proposed under known quadratic loss functions with unknown covariance matrices. The new estimator is non-parametric in the sense that it does not assume a specific parametric distribution for the data and it does not require the prior information on the population covariance matrix. Analytical results on the improvement of the proposed shrinkage estimator are provided and some corresponding asymptotic properties are also derived. Finally, we demonstrate the practical improvement of the proposed method over existing methods through extensive simulation studies and real data analysis. Keywords: High-dimensional data; Shrinkage estimator; Large $p$ small $n$; $U$-statistic., Comment: Some technical parts of Theorem 3.1 and 3.2 were corrected in this version