Objective Testing Procedures in Linear Models: Calibration of the p-values

Abstract.  An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on intrinsic model posterior probabilities is considered. It is proven that these posterior probabilities are simple functions of the classical F -statistic, thus the evaluation of the procedure can be carried out analytically through the frequentist analysis of the posterior probability of the null. An asymptotic analysis proves that, under mild conditions on the design matrix, the procedure is consistent. For any testing hypothesis it is also seen that there is a one-to-one mapping – which we call calibration curve – between the posterior probability of the null hypothesis and the classical bi p -value. This curve adds substantial knowledge about the possible discrepancies between the Bayesian and the p -value measures of evidence for testing hypothesis. It permits a better understanding of the serious difficulties that are encountered in linear models for interpreting the p -values. A specific illustration of the variable selection problem is given.     

Empirical likelihood tests for two‐sample problems via nonparametric density estimation

The authors study the problem of testing whether two populations have the same law by comparing kernel estimators of the two density functions. The proposed test statistic is based on a local empirical likelihood approach. They obtain the asymptotic distribution of the test statistic and propose a bootstrap approximation to calibrate the test. A simulation study is carried out in which the proposed method is compared with two competitors, and a procedure to select the bandwidth parameter is studied. The proposed test can be extended to more than two samples and to multivariate distributions.     

ON THE COVERAGE PROBABILITY OF CONFIDENCE INTERVALS IN REGRESSION AFTER VARIABLE SELECTION

This paper considers a linear regression model with regression parameter vector β. The parameter of interest is θ= a^Tβ where a is specified. When, as a first step, a data‐based variable selection (e.g. minimum Akaike information criterion) is used to select a model, it is common statistical practice to then carry out inference about θ, using the same data, based on the (false) assumption that the selected model had been provided a priori. The paper considers a confidence interval for θ with nominal coverage 1 ‐ α constructed on this (false) assumption, and calls this the naive 1 ‐ α confidence interval. The minimum coverage probability of this confidence interval can be calculated for simple variable selection procedures involving only a single variable. However, the kinds of variable selection procedures used in practice are typically much more complicated. For the real‐life data presented in this paper, there are 20 variables each of which is to be either included or not, leading to 2²⁰ different models. The coverage probability at any given value of the parameters provides an upper bound on the minimum coverage probability of the naive confidence interval. This paper derives a new Monte Carlo simulation estimator of the coverage probability, which uses conditioning for variance reduction. For these real‐life data, the gain in efficiency of this Monte Carlo simulation due to conditioning ranged from 2 to 6. The paper also presents a simple one‐dimensional search strategy for parameter values at which the coverage probability is relatively small. For these real‐life data, this search leads to parameter values for which the coverage probability of the naive 0.95 confidence interval is 0.79 for variable selection using the Akaike information criterion and 0.70 for variable selection using Bayes information criterion, showing that these confidence intervals are completely inadequate.     

Regularization and variable selection via the elastic net

Summary.  We propose the elastic net, a new regularization and variable selection method. Real world data and a simulation study show that the elastic net often outperforms the lasso, while enjoying a similar sparsity of representation. In addition, the elastic net encourages a grouping effect, where strongly correlated predictors tend to be in or out of the model together. The elastic net is particularly useful when the number of predictors ( p ) is much bigger than the number of observations ( n ). By contrast, the lasso is not a very satisfactory variable selection method in the p ≫ n case. An algorithm called LARS-EN is proposed for computing elastic net regularization paths efficiently, much like algorithm LARS does for the lasso.     

Body Weight and Matching With a Physically Attractive Romantic Partner

Matching and attribute trade are two perspectives used to explain mate selection. We investigated patterns of matching and trade, focusing on obesity, using Add Health Romantic Pair data (N = 1,405 couples). Obese individuals, relative to healthy weight individuals, were less likely to have physically attractive partners, with this disadvantage greater for women than men, and greater for White women than Black women. Additional education, a more attractive personality, and better grooming increased the probability of having a physically attractive partner and offset the disadvantage of obesity for some individuals. Unexpectedly, we found women, like men, trade education for their partners’ physical attractiveness. Despite evidence of attribute trade, matching with respect to physical characteristics was the dominant mate selection pattern.     

辽朝选任官吏的方式考述

辽朝是中国历史上一个十分重要的王朝,在其统治中国北方的过程中,注重选拔人才,形成了一套比较系统而又完备的官吏管理法律制度。就管吏的选任方式而言,主要以世选制、科举制为基本方式。此外,还有察举制、征召制、入粟补官法、恩荫、任子制、赏赐与赠与。既有成效,也有教训。     

毫米波第三次谐波振荡器的实现

本文讨论了第三次谐波振荡器的最佳工作状态、器件选择原则以及电路设计,并采用8mm GaAs Gunn二极管实际制作了3mm第三次谐皮振荡器,输出功率达10mW量级。     

Survival Analysis For A Sequenced Clinical Trial With Covariates

In this paper we discuss the survival analysis for a clinical trial in which treatment categories and general prognostic data are realised at different stages during a patient's survival time. In the light of possible strategies for the parsimonious modelling of such data, a corresponding sequence of illustrative analyses is presented. Detailed results are given for a weighted least squares analysis and these generally agree with those obtained by maximum likelihood.     

政府直接配置资源的理由--集体选择的逻辑

本文以亚当·斯密对政府行为的经验分析为基础,通过类推科斯对企业产生原因的分析。从制度经济学的角度说明了政府直接配置资源的理由;运用博弈论为分析工具说明了政府直接配置资源的范围。     

Semiparametric log-linear regression for longitudinal measurements subject to outcome-dependent follow-up

A common problem for longitudinal data analyses is that subjects follow-up is irregular, often related to the past outcome or other factors associated with the outcome measure that are not included in the regression model. Analyses unadjusted for outcome-dependent follow-up yield biased estimates. We propose a longitudinal data analysis that can provide consistent estimates in regression models that are subject to outcome-dependent follow-up. We focus on semiparametric marginal log-link regression with arbitrary unspecified baseline function. Based on estimating equations, the proposed class of estimators are root n consistent and asymptotically normal. We present simulation studies that assess the performance of the estimators under finite samples. We illustrate our approach using data from a health services research study.