Linear models that allow perfect estimation

The general Gauss–Markov model, Y = Xβ + e, E(e) = 0, Cov(e) = σ ² V, has been intensively studied and widely used. Most studies consider covariance matrices V that are nonsingular but we focus on the most difficult case wherein C(X), the column space of X, is not contained in C(V). This forces V to be singular. Under this condition there exist nontrivial linear functions of Q′Xβ that are known with probability 1 (perfectly) where ${C(Q)=C(V)^\perp}$ . To treat ${C(X) \not \subset C(V)}$ , much of the existing literature obtains estimates and tests by replacing V with a pseudo-covariance matrix T = V + XUX′ for some nonnegative definite U such that ${C(X) \subset C(T)}$ , see Christensen (Plane answers to complex questions: the theory of linear models, 2002, Chap. 10). We find it more intuitive to first eliminate what is known about Xβ and then to adjust X while keeping V unchanged. We show that we can decompose β into the sum of two orthogonal parts, β = β ₀ + β ₁, where β ₀ is known. We also show that the unknown component of X β is ${X\beta_1 \equiv \tilde{X} \gamma}$ , where ${C(\tilde{X})=C(X)\cap C(V)}$ . We replace the original model with ${Y-X\beta_0=\tilde{X}\gamma+e}$ , E(e) = 0, ${Cov(e)=\sigma^2V}$ and perform estimation and tests under this new model for which the simplifying assumption ${C(\tilde{X}) \subset C(V)}$ holds. This allows us to focus on the part of that parameters that are not known perfectly. We show that this method provides the usual estimates and tests.     

Group Variable Selection with Oracle Property by Weight-Fused Adaptive Elastic Net Model for Strongly Correlated Data

This paper is the generalization of weight-fused elastic net (Fu and Xu, 2012 Fu, G., Xu, Q. (2012). Grouping variable selection by weight fused elastic net for multi-collinear data. Communications in Statistics-Simulation and Computation 41(2):205–221.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), which performs group variable selection by combining weight-fused LASSO(wfLasso) and elastic net (Zou and Hastie, 2005 Zou, H., Hastie, T. (2005). Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 67(2):301–320.[Crossref], [Web of Science ®] , [Google Scholar]) penalties. In this study, the elastic net penalty is replaced by adaptive elastic net penalty (AdaEnet) (Zou and Zhang, 2009 Zou, H., Zhang, H. (2009). On the adaptive elastic-net with a diverging number of parameters. Annals of Statistics 37(4):1733–1751.[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]), and a new group variable selection algorithm with oracle property (Fan and Li, 2001 Fan, J., Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96(456):1348–1360.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]; Zou, 2006 Zou, H. (2006). The adaptive lasso and its oracle properties. Journal of the American Statistical Association 101(476):1418–1429.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) is obtained.     

Estimation of Main Effect When Covariates Have Non-Proportional Hazards

The Cox proportional hazards (PH) regression model has been widely used to analyze survival data in clinical trials and observational studies. In addition to estimating the main treatment or exposure group effect, it is common to adjust for additional covariates using the Cox model. It is well known that violation of the PH assumption can lead to estimates that are biased and difficult to interpret, and model checking has become a routine procedure. However, such checking might focus on the primary group comparisons, and the assumption can still be violated when adjusting for many of the potential covariates. We study the effect of violation of the PH assumption of the covariates on the estimation of the main group effect in the Cox model. The results are summarized in terms of the bias and the coverage properties of the confidence intervals. Overall in randomized clinical trials, the bias caused by misspecifying the PH assumption on the covariates is no more than 15% in absolute value regardless of sample size. In observational studies where the covariates are likely correlated with the group variable, however, the bias can be very severe. The coverage properties largely depend on sample size, as expected, as bias becomes dominating with increasing sample size. These findings should serve as cautionary notes when adjusting for potential confounders in observational studies, as the violation of PH assumption on the confounders can lead to erroneous results.     

Notes on Test Homogeneity of the Odds Ratio in Matched Pairs Under Stratified Sampling: A Monte Carlo Evaluation

Using Monte Carlo simulation, we compare the performance of five asymptotic test procedures and a randomized permutation test procedure for testing the homogeneity of odds ratio under the stratified matched-pair design. We note that the weighted-least-square test procedure is liberal, while Pearson's goodness-of-fit (PGF) test procedure with the continuity correction is conservative. We note that PGF without the continuity correction, the conditional likelihood ratio test procedure, and the randomized permutation test procedure can generally perform well with respect to Type I error. We use the data taken from a case–control study regarding the endometrial cancer incidence published elsewhere to illustrate the use of these test procedures.     

The robustness of the likelihood ratio test,the nonparametric sum rank test,and f-ratio tests when the populations are from the negative binomial family

In this study, the robustness of power and significance level of several statistical testing methods was evaluated under the assumption that the test populations were from Poisson, negative binomial, or geometric distributions. The F-ratio test, with or without appropriate transformations, was shown to be both safe and robust for all distributions examined.     

A structured framework for assessing sensitivity to missing data assumptions in longitudinal clinical trials

The objective of this research was to demonstrate a framework for drawing inference from sensitivity analyses of incomplete longitudinal clinical trial data via a re‐analysis of data from a confirmatory clinical trial in depression. A likelihood‐based approach that assumed missing at random (MAR) was the primary analysis. Robustness to departure from MAR was assessed by comparing the primary result to those from a series of analyses that employed varying missing not at random (MNAR) assumptions (selection models, pattern mixture models and shared parameter models) and to MAR methods that used inclusive models. The key sensitivity analysis used multiple imputation assuming that after dropout the trajectory of drug‐treated patients was that of placebo treated patients with a similar outcome history (placebo multiple imputation). This result was used as the worst reasonable case to define the lower limit of plausible values for the treatment contrast. The endpoint contrast from the primary analysis was ? 2.79 (p = .013). In placebo multiple imputation, the result was ? 2.17. Results from the other sensitivity analyses ranged from ? 2.21 to ? 3.87 and were symmetrically distributed around the primary result. Hence, no clear evidence of bias from missing not at random data was found. In the worst reasonable case scenario, the treatment effect was 80% of the magnitude of the primary result. Therefore, it was concluded that a treatment effect existed. The structured sensitivity framework of using a worst reasonable case result based on a controlled imputation approach with transparent and debatable assumptions supplemented a series of plausible alternative models under varying assumptions was useful in this specific situation and holds promise as a generally useful framework. Copyright © 2012 John Wiley & Sons, Ltd.     

A likelihood ratio test for goodness‐of‐fit of recessive and dominant models for case–control studies

Testing goodness‐of‐fit of commonly used genetic models is of critical importance in many applications including association studies and testing for departure from Hardy–Weinberg equilibrium. Case–control design has become widely used in population genetics and genetic epidemiology, thus it is of interest to develop powerful goodness‐of‐fit tests for genetic models using case–control data. This paper develops a likelihood ratio test (LRT) for testing recessive and dominant models for case–control studies. The LRT statistic has a closed‐form formula with a simple $\chi^{2}(1)$ null asymptotic distribution, thus its implementation is easy even for genome‐wide association studies. Moreover, it has the same power and optimality as when the disease prevalence is known in the population. The Canadian Journal of Statistics 41: 341–352; 2013 © 2013 Statistical Society of Canada     

人皆可以为尧舜:论颜元的"偏圣"观 ——兼谈明清圣人普遍化思潮

颜元承孟子性善之旨,认为"人无有不善",但他亦不得不承认人的气禀有偏正之分.在"偏亦为善"的基础上,颜元认为,气质偏驳之人能够通过后天的学习有所作为,最终成为伯夷、伊尹、柳下惠等这样的"偏圣".颜元继承了孟子、朱熹关于"偏圣"的说法,却一改孟、朱对其"虽为圣但吾不取"的态度,对"偏圣"给予正面的认可.对于气质偏驳之人如何成圣,颜元亦提供了"就其质性之近而学,近夷学夷,近惠学惠"的方案.颜元的"圣人观"受到明代以来圣人普遍化思潮的影响,他吸收了王阳明"以才力论偏圣"之思想,进而发展为"以气质论偏圣",极大地扩充了"圣人"的范畴,提供了更为多样化的成圣道路,较诸王阳明,走得更远.     

区块链技术与企业绩效:公司治理结构的调节作用

当前,我国正在全面迈进数字经济新时代.区块链技术作为数字经济最为重要的基础技术设施之一,正在加速推动新一轮产业变革和数字创新,被越来越多的企业接受和应用.区块链技术对企业绩效的作用机制是什么?针对以上问题,本研究通过倾向得分匹配方法(PSM)选取2014-2019年中国沪深A股上市公司为研究样本,以当年是否引入区块链技术作为公司治理手段为哑变量,通过计量分析研究了区块链技术应用与企业绩效之间的关系,并进一步探索了公司治理结构在区块链技术应用与企业绩效关系间的调节作用.研究发现,区块链技术应用对企业绩效有促进作用;当公司治理结构呈现较高独董比例和较高股权集中度时,区块链技术应用对企业绩效的促进作用得到加强.