Bayesian prediction for small areas using sur models

Sample surveys are usually designed and analyzed to produce estimates for larger areas and/or populations. Nevertheless, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To circumvent such difficulties, borrowing strength from related small areas via modeling becomes essential. In line with this, we propose a hierarchical multivariate Bayes prediction method for small area estimation based on the seemingly unrelated regressions (SUR) model. The performance of the proposed method was evaluated through simulation studies.     

On prediction of population and subpopulation characteristics for future periods

The estimation or prediction of population characteristics based on the sample information is the key issue in survey sampling. If the sample sizes in subpopulations (domains) are large enough, similar methods as used for the whole population can be used to estimate or to predict subpopulations characteristics as well. To estimate or to predict characteristics of domains with small or even zero sample sizes, small area estimation methods “borrowing strength” from other subpopulations or time periods are widely used. We extend this problem and study methods of prediction of future population and subpopulations’ characteristics based on the longitudinal data.     

A Bayesian model for local smoothing in kernel density estimation

A new procedure is proposed for deriving variable bandwidths in univariate kernel density estimation, based upon likelihood cross-validation and an analysis of a Bayesian graphical model. The procedure admits bandwidth selection which is flexible in terms of the amount of smoothing required. In addition, the basic model can be extended to incorporate local smoothing of the density estimate. The method is shown to perform well in both theoretical and practical situations, and we compare our method with those of Abramson (The Annals of Statistics 10: 1217–1223) and Sain and Scott (Journal of the American Statistical Association 91: 1525–1534). In particular, we note that in certain cases, the Sain and Scott method performs poorly even with relatively large sample sizes.We compare various bandwidth selection methods using standard mean integrated square error criteria to assess the quality of the density estimates. We study situations where the underlying density is assumed both known and unknown, and note that in practice, our method performs well when sample sizes are small. In addition, we also apply the methods to real data, and again we believe our methods perform at least as well as existing methods.     

Generalized simultaneous confidence regions for regression coefficients and their ratios

This study constructs a simultaneous confidence region for two combinations of coefficients of linear models and their ratios based on the concept of generalized pivotal quantities. Many biological studies, such as those on genetics, assessment of drug effectiveness, and health economics, are interested in a comparison of several dose groups with a placebo group and the group ratios. The Bonferroni correction and the plug-in method based on the multivariate-t distribution have been proposed for the simultaneous region estimation. However, the two methods are asymptotic procedures, and their performance in finite sample sizes has not been thoroughly investigated. Based on the concept of generalized pivotal quantity, we propose a Bonferroni correction procedure and a generalized variable (GV) procedure to construct the simultaneous confidence regions. To address a genetic concern of the dominance ratio, we conduct a simulation study to empirically investigate the probability coverage and expected length of the methods for various combinations of sample sizes and values of the dominance ratio. The simulation results demonstrate that the simultaneous confidence region based on the GV procedure provides sufficient coverage probability and reasonable expected length. Thus, it can be recommended in practice. Numerical examples using published data sets illustrate the proposed methods.     

Monitoring multivariate simple linear profiles using robust estimators

Brief Abstract

This article focuses on estimation of multivariate simple linear profiles. While outliers may hamper the expected performance of the ordinary regression estimators, this study resorts to robust estimators as the remedy of the estimation problem in presence of contaminated observations. More specifically, three robust estimators M, S and MM are employed. Extensive simulation runs show that in the absence of outliers or for small amount of contamination, the robust methods perform as well as the classical least square method, while for medium and large amounts of contamination the proposed estimators perform considerably better than classical method.     

Integrative Analysis of Cancer Diagnosis Studies with Composite Penalization

In cancer diagnosis studies, high‐throughput gene profiling has been extensively conducted, searching for genes whose expressions may serve as markers. Data generated from such studies have the ‘large d, small n’ feature, with the number of genes profiled much larger than the sample size. Penalization has been extensively adopted for simultaneous estimation and marker selection. Because of small sample sizes, markers identified from the analysis of single data sets can be unsatisfactory. A cost‐effective remedy is to conduct integrative analysis of multiple heterogeneous data sets. In this article, we investigate composite penalization methods for estimation and marker selection in integrative analysis. The proposed methods use the minimax concave penalty (MCP) as the outer penalty. Under the homogeneity model, the ridge penalty is adopted as the inner penalty. Under the heterogeneity model, the Lasso penalty and MCP are adopted as the inner penalty. Effective computational algorithms based on coordinate descent are developed. Numerical studies, including simulation and analysis of practical cancer data sets, show satisfactory performance of the proposed methods.     

A comparison of subgroup identification methods in clinical drug development: Simulation study and regulatory considerations

With advancement of technologies such as genomic sequencing, predictive biomarkers have become a useful tool for the development of personalized medicine. Predictive biomarkers can be used to select subsets of patients, which are most likely to benefit from a treatment. A number of approaches for subgroup identification were proposed over the last years. Although overviews of subgroup identification methods are available, systematic comparisons of their performance in simulation studies are rare. Interaction trees (IT), model‐based recursive partitioning, subgroup identification based on differential effect, simultaneous threshold interaction modeling algorithm (STIMA), and adaptive refinement by directed peeling were proposed for subgroup identification. We compared these methods in a simulation study using a structured approach. In order to identify a target population for subsequent trials, a selection of the identified subgroups is needed. Therefore, we propose a subgroup criterion leading to a target subgroup consisting of the identified subgroups with an estimated treatment difference no less than a pre‐specified threshold. In our simulation study, we evaluated these methods by considering measures for binary classification, like sensitivity and specificity. In settings with large effects or huge sample sizes, most methods perform well. For more realistic settings in drug development involving data from a single trial only, however, none of the methods seems suitable for selecting a target population. Using the subgroup criterion as alternative to the proposed pruning procedures, STIMA and IT can improve their performance in some settings. The methods and the subgroup criterion are illustrated by an application in amyotrophic lateral sclerosis.     

Multilevel modelling of complex survey data

Summary.  Multilevel modelling is sometimes used for data from complex surveys involving multistage sampling, unequal sampling probabilities and stratification. We consider generalized linear mixed models and particularly the case of dichotomous responses. A pseudolikelihood approach for accommodating inverse probability weights in multilevel models with an arbitrary number of levels is implemented by using adaptive quadrature. A sandwich estimator is used to obtain standard errors that account for stratification and clustering. When level 1 weights are used that vary between elementary units in clusters, the scaling of the weights becomes important. We point out that not only variance components but also regression coefficients can be severely biased when the response is dichotomous. The pseudolikelihood methodology is applied to complex survey data on reading proficiency from the American sample of the 'Program for international student assessment' 2000 study, using the Stata program gllamm which can estimate a wide range of multilevel and latent variable models. Performance of pseudo-maximum-likelihood with different methods for handling level 1 weights is investigated in a Monte Carlo experiment. Pseudo-maximum-likelihood estimators of (conditional) regression coefficients perform well for large cluster sizes but are biased for small cluster sizes. In contrast, estimators of marginal effects perform well in both situations. We conclude that caution must be exercised in pseudo-maximum-likelihood estimation for small cluster sizes when level 1 weights are used.     

Simultaneous estimation for non-crossing multiple quantile regression with right censored data

In this paper, we consider the estimation problem of multiple conditional quantile functions with right censored survival data. To account for censoring in estimating a quantile function, weighted quantile regression (WQR) has been developed by using inverse-censoring-probability weights. However, the estimated quantile functions from the WQR often cross each other and consequently violate the basic properties of quantiles. To avoid quantile crossing, we propose non-crossing weighted multiple quantile regression (NWQR), which estimates multiple conditional quantile functions simultaneously. We further propose the adaptive sup-norm regularized NWQR (ANWQR) to perform simultaneous estimation and variable selection. The large sample properties of the NWQR and ANWQR estimators are established under certain regularity conditions. The proposed methods are evaluated through simulation studies and analysis of a real data set.     

Improving parameter estimation using constrained optimization methods

This paper presents a methodology based on transforming estimation methods in optimization problems in order to incorporate in a natural way some constraints that contain extra information not considered by standard estimation methods, with the aim of improving the quality of the parameter estimates. We include here three types of such information: bounds for the cumulative distribution function, bounds for the quantiles, and any restrictions on the parameters such as those imposed by the support of the random variable under consideration. The method is quite general and can be applied to many estimation methods such as the maximum likelihood (ML), the method of moments (MOM), the least squares, the least absolute values, and the minimax methods. The performances of the obtained estimates from several families of distributions are investigated for the ML and the MOM, using simulations. The simulation results show that for small sample sizes important gains can be achieved with respect to the case where the above information is ignored. In addition, we discuss sensitivity analysis methods for assessing the influence of observations on the proposed estimators. The method applies to both univariate and multivariate data.     

A comparison of some robust,adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.

     

A comparison of some robust, adaptive, and partially adaptive estimators of regression models

Numerous estimation techniques for regression models have been proposed. These procedures differ in how sample information is used in the estimation procedure. The efficiency of least squares (OLS) estimators implicity assumes normally distributed residuals and is very sensitive to departures from normality, particularly to "outliers" and thick-tailed distributions. Lead absolute deviation (LAD) estimators are less sensitive to outliers and are optimal for laplace random disturbances, but not for normal errors. This paper reports monte carlo comparisons of OLS,LAD, two robust estimators discussed by huber, three partially adaptiveestimators, newey's generalized method of moments estimator, and an adaptive maximum likelihood estimator based on a normal kernal studied by manski. This paper is the first to compare the relative performance of some adaptive robust estimators (partially adaptive and adaptive procedures) with some common nonadaptive robust estimators. The partially adaptive estimators are based on three flxible parametric distributions for the errors. These include the power exponential (Box-Tiao) and generalized t distributions, as well as a distribution for the errors, which is not necessarily symmetric. The adaptive procedures are "fully iterative" rather than one step estimators. The adaptive estimators have desirable large sample properties, but these properties do not necessarily carry over to the small sample case.

The monte carlo comparisons of the alternative estimators are based on four different specifications for the error distribution: a normal, a mixture of normals (or variance-contaminated normal), a bimodal mixture of normals, and a lognormal. Five hundred samples of 50 are used. The adaptive and partially adaptive estimators perform very well relative to the other estimation procedures considered, and preliminary results suggest that in some important cases they can perform much better than OLS with 50 to 80% reductions in standard errors.     

Some robustness issues for comparing multiple logistic regression slopes to a control for small samples

For comparing several logistic regression slopes to that of a control for small sample sizes, Dasgupta et al. (2001) proposed an "asymptotic" small-sample test and a "pivoted" version of that test statistic. Their results show both methods perform well in terms of Type I error control and marginal power when the response is related to the explanatory variable via a logistic regression model. This study finds, via Monte Carlo simulations, that when the underlying relationship is probit, complementary log-log, linear, or even non-monotonic, the "asymptotic" and the "pivoted" small-sample methods perform fairly well in terms of Type I error control and marginal power. Unlike their large sample competitors, they are generally robust to departures from the logistic regression model.     

Adaptive LASSO for varying-coefficient partially linear measurement error models

This paper extends the adaptive LASSO (ALASSO) for simultaneous parameter estimation and variable selection to a varying-coefficient partially linear model where some of the covariates are subject to measurement errors of an additive form. We draw comparisons with the SCAD, and prove that both the ALASSO and the SCAD attain the oracle property under this setup. We further develop an algorithm in the spirit of LARS for finding the solution path of the ALASSO in practical applications. Finite sample properties of the proposed methods are examined in a simulation study, and a real data example based on the U.S. Department of Agriculture's Continuing Survey of Food Intakes by Individuals (CSFII) is considered.     

Statistical analysis of accelerated temperature cycling test based on Coffin-Manson model

Abstract

This paper investigates the statistical analysis of grouped accelerated temperature cycling test data when the product lifetime follows a Weibull distribution. A log-linear acceleration equation is derived from the Coffin-Manson model. The problem is transformed to a constant-stress accelerated life test with grouped data and multiple acceleration variables. The Jeffreys prior and reference priors are derived. Maximum likelihood estimation and Bayesian estimation with objective priors are obtained by applying the technique of data augmentation. A simulation study shows that both of these two methods perform well when sample size is large, and the Bayesian method gives better performance under small sample sizes.     

The use of power transformations in small area estimation

Sample surveys are usually designed and analysed to produce estimates for larger areas. Nevertheless, sample sizes are often not large enough to give adequate precision for small area estimates of interest. To overcome such difficulties, borrowing strength from related small areas via modelling becomes essential. In line with this, we propose components of variance models with power transformations for small area estimation. This paper reports the results of a study aimed at incorporating the power transformation in small area estimation for improving the quality of small area predictions. The proposed methods are demonstrated on satellite data in conjunction with survey data to estimate mean acreage under a specified crop for counties in Iowa.     

Inference on the Kumaraswamy distribution

Many lifetime distribution models have successfully served as population models for risk analysis and reliability mechanisms. The Kumaraswamy distribution is one of these distributions which is particularly useful to many natural phenomena whose outcomes have lower and upper bounds or bounded outcomes in the biomedical and epidemiological research. This article studies point estimation and interval estimation for the Kumaraswamy distribution. The inverse estimators (IEs) for the parameters of the Kumaraswamy distribution are derived. Numerical comparisons with maximum likelihood estimation and biased-corrected methods clearly indicate the proposed IEs are promising. Confidence intervals for the parameters and reliability characteristics of interest are constructed using pivotal or generalized pivotal quantities. Then, the results are extended to the stress–strength model involving two Kumaraswamy populations with different parameter values. Construction of confidence intervals for the stress–strength reliability is derived. Extensive simulations are used to demonstrate the performance of confidence intervals constructed using generalized pivotal quantities.     

On simultaneous confidence intervals based on rank-estimates with application to analysis of gene expression data

Abstract

Inferential methods based on ranks present robust and powerful alternative methodology for testing and estimation. In this article, two objectives are followed. First, develop a general method of simultaneous confidence intervals based on the rank estimates of the parameters of a general linear model and derive the asymptotic distribution of the pivotal quantity. Second, extend the method to high dimensional data such as gene expression data for which the usual large sample approximation does not apply. It is common in practice to use the asymptotic distribution to make inference for small samples. The empirical investigation in this article shows that for methods based on the rank-estimates, this approach does not produce a viable inference and should be avoided. A method based on the bootstrap is outlined and it is shown to provide a reliable and accurate method of constructing simultaneous confidence intervals based on rank estimates. In particular it is shown that commonly applied methods of normal or t-approximation are not satisfactory, particularly for large-scale inferences. Methods based on ranks are uniquely suitable for analysis of microarray gene expression data since they often involve large scale inferences based on small samples containing a large number of outliers and violate the assumption of normality. A real microarray data is analyzed using the rank-estimate simultaneous confidence intervals. Viability of the proposed method is assessed through a Monte Carlo simulation study under varied assumptions.     

Simultaneous inference based on rank statistics in linear models

A class of simultaneous tests based on the aligned rank transform (ART) statistics is proposed for linear functions of parameters in linear models. The asymptotic distributions are derived. The stability of the finite sample behaviour of the sampling distribution of the ART technique is studied by comparing the simulated upper quantiles of its sampling distribution with those of the multivariate t-distribution. Simulation also shows that the tests based on ART have excellent small sample properties and because of their robustness perform better than the methods based on the least-squares estimates.     

Phase-I monitoring of general linear profiles in multistage processes

In this article, the general linear profile-monitoring problem in multistage processes is addressed. An approach based on the U statistic is first proposed to remove the effect of the cascade property in multistage processes. Then, the T² chart and a likelihood ratio test (LRT)-based scheme on the adjusted parameters are constructed for Phase-I monitoring of the parameters of general linear profiles in each stage. Using simulation experiments, the performance of the proposed methods is evaluated and compared in terms of the signal probability for both weak and strong autocorrelations, for processes with two and three stages, as well as for two sample sizes. According to the results, the effect of the cascade property is effectively removed and hence each stage can be monitored independently. In addition, the result shows that the LRT approach provides significantly better results than the T² method and outperforms it under different shift and autocorrelation scenarios. Moreover, the proposed methods perform better when larger sample sizes are used in the process. Two illustrative examples, including a real case and a simulated example, are used to show the applicability of the proposed methods.