Random effects models for count data

Random effects models are considered for count data obtained in a cross or nested classification. The main feature of the proposed models is the use of the additive effects on the original scale in contrast to the commonly used log scale. The rationale behind this approach is given. The estimation of variance components is based on the usual mean square approach. Directly analogous results to those from the analysis of variance models for continuous data are obtained. The usual Poisson dispersion test procedure can be used not only to test for no overall random effects but also to assess the adequacy of the model. Individual variance component can be tested by using the usual F-test. To get a reliable estimate, a large number of factor levels seem to be required.     

Blinded sample size re‐estimation in superiority and noninferiority trials: bias versus variance in variance estimation

The internal pilot study design allows for modifying the sample size during an ongoing study based on a blinded estimate of the variance thus maintaining the trial integrity. Various blinded sample size re‐estimation procedures have been proposed in the literature. We compare the blinded sample size re‐estimation procedures based on the one‐sample variance of the pooled data with a blinded procedure using the randomization block information with respect to bias and variance of the variance estimators, and the distribution of the resulting sample sizes, power, and actual type I error rate. For reference, sample size re‐estimation based on the unblinded variance is also included in the comparison. It is shown that using an unbiased variance estimator (such as the one using the randomization block information) for sample size re‐estimation does not guarantee that the desired power is achieved. Moreover, in situations that are common in clinical trials, the variance estimator that employs the randomization block length shows a higher variability than the simple one‐sample estimator and in turn the sample size resulting from the related re‐estimation procedure. This higher variability can lead to a lower power as was demonstrated in the setting of noninferiority trials. In summary, the one‐sample estimator obtained from the pooled data is extremely simple to apply, shows good performance, and is therefore recommended for application. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust estimation and design procedures for the random effects model

The authors discuss two robust estimators for estimating variance components in the random effects model, and they obtain finite‐sample breakdown points for the estimators. Based on the finite‐sample breakdown point, they propose a criterion for selecting robust designs. With robust designs, one can get efficient and reliable estimates for variance components regardless of outliers which may happen in the experiment. The authors give examples to show the performance of robust estimators and to compare robust designs with optimal designs based on the traditional analysis of variance estimation method.     

A calibrated imputation method for secondary data analysis of survey data

In practical survey sampling, missing data are unavoidable due to nonresponse, rejected observations by editing, disclosure control, or outlier suppression. We propose a calibrated imputation approach so that valid point and variance estimates of the population (or domain) totals can be computed by the secondary users using simple complete‐sample formulae. This is especially helpful for variance estimation, which generally require additional information and tools that are unavailable to the secondary users. Our approach is natural for continuous variables, where the estimation may be either based on reweighting or imputation, including possibly their outlier‐robust extensions. We also propose a multivariate procedure to accommodate the estimation of the covariance matrix between estimated population totals, which facilitates variance estimation of the ratios or differences among the estimated totals. We illustrate the proposed approach using simulation data in supplementary materials that are available online.     

Properties of the beta regression model for small area estimation of proportions and application to estimation of poverty rates

Abstract

Linear mixed effects models have been popular in small area estimation problems for modeling survey data when the sample size in one or more areas is too small for reliable inference. However, when the data are restricted to a bounded interval, the linear model may be inappropriate, particularly if the data are near the boundary. Nonlinear sampling models are becoming increasingly popular for small area estimation problems when the normal model is inadequate. This paper studies the use of a beta distribution as an alternative to the normal distribution as a sampling model for survey estimates of proportions which take values in (0, 1). Inference for small area proportions based on the posterior distribution of a beta regression model ensures that point estimates and credible intervals take values in (0, 1). Properties of a hierarchical Bayesian small area model with a beta sampling distribution and logistic link function are presented and compared to those of the linear mixed effect model. Propriety of the posterior distribution using certain noninformative priors is shown, and behavior of the posterior mean as a function of the sampling variance and the model variance is described. An example using 2010 Small Area Income and Poverty Estimates (SAIPE) data is given, and a numerical example studying small sample properties of the model is presented.     

Criteria for estimating the variance function used in the asymptotic quasi-likelihood approach

The estimation of the variance function of a linear regression model used in the asymptotic quasi-likelihood approach is considered. It is shown that the variance function used in the determination of the asymptotic quasi-likelihood estimates encompasses the variance functions commonly found in the literature. Selection criteria of the most appropriate estimate of the variance function for given data are established. These criteria are based on a graphical technique and a chi-squared test.     

Sample size re‐estimation incorporating prior information on a nuisance parameter

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter–based sample size re‐estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta‐analytic‐predictive approach. To incorporate external information into the sample size re‐estimation, we propose to update the meta‐analytic‐predictive prior based on the results of the internal pilot study and to re‐estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re‐estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior‐data conflict is present, incorporating external information into the sample size re‐estimation improves the operating characteristics compared to the traditional approach. In the case of a prior‐data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re‐estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re‐estimation, the potential gains should be balanced against the risks.     

Corrected empirical Bayes confidence intervals in nested error regression models

In the small area estimation, the empirical best linear unbiased predictor (EBLUP) or the empirical Bayes estimator (EB) in the linear mixed model is recognized to be useful because it gives a stable and reliable estimate for a mean of a small area. In practical situations where EBLUP is applied to real data, it is important to evaluate how much EBLUP is reliable. One method for the purpose is to construct a confidence interval based on EBLUP. In this paper, we obtain an asymptotically corrected empirical Bayes confidence interval in a nested error regression model with unbalanced sample sizes and unknown components of variance. The coverage probability is shown to satisfy the confidence level in the second-order asymptotics. It is numerically revealed that the corrected confidence interval is superior to the conventional confidence interval based on the sample mean in terms of the coverage probability and the expected width of the interval. Finally, it is applied to the posted land price data in Tokyo and the neighboring prefecture.     

Estimation using penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models

We consider two estimation schemes based on penalized quasilikelihood and quasi-pseudo-likelihood in Poisson mixed models. The asymptotic bias in regression coefficients and variance components estimated by penalized quasilikelihood (PQL) is studied for small values of the variance components. We show the PQL estimators of both regression coefficients and variance components in Poisson mixed models have a smaller order of bias compared to those for binomial data. Unbiased estimating equations based on quasi-pseudo-likelihood are proposed and are shown to yield consistent estimators under some regularity conditions. The finite sample performance of these two methods is compared through a simulation study.     

On a variance stabilizing model and its application to genomic data

In this paper, we propose a model based on a class of symmetric distributions, which avoids the transformation of data, stabilizes the variance of the observations, and provides robust estimation of parameters and high flexibility for modeling different types of data. Probabilistic and statistical aspects of this new model are developed throughout the article, which include mathematical properties, estimation of parameters and inference. The obtained results are illustrated by means of real genomic data.     

Accounting for Uncertainty in Heteroscedasticity in Nonlinear Regression

Toxicologists and pharmacologists often describe toxicity of a chemical using parameters of a nonlinear regression model. Thus estimation of parameters of a nonlinear regression model is an important problem. The estimates of the parameters and their uncertainty estimates depend upon the underlying error variance structure in the model. Typically, a priori the researcher would not know if the error variances are homoscedastic (i.e., constant across dose) or if they are heteroscedastic (i.e., the variance is a function of dose). Motivated by this concern, in this paper we introduce an estimation procedure based on preliminary test which selects an appropriate estimation procedure accounting for the underlying error variance structure. Since outliers and influential observations are common in toxicological data, the proposed methodology uses M-estimators. The asymptotic properties of the preliminary test estimator are investigated; in particular its asymptotic covariance matrix is derived. The performance of the proposed estimator is compared with several standard estimators using simulation studies. The proposed methodology is also illustrated using a data set obtained from the National Toxicology Program.     

Graphical technique for comparing designs for random models

Methods for comparing designs for a random (or mixed) linear model have focused primarily on criteria based on single-valued functions. In general, these functions are difficult to use, because of their complex forms, in addition to their dependence on the model's unknown variance components. In this paper, a graphical approach is presented for comparing designs for random models. The one-way model is used for illustration. The proposed approach is based on using quantiles of an estimator of a function of the variance components. The dependence of these quantiles on the true values of the variance components is depicted by plotting the so-called quantile dispersion graphs (QDGs), which provide a comprehensive picture of the quality of estimation obtained with a given design. The QDGs can therefore be used to compare several candidate designs. Two methods of estimation of variance components are considered, namely analysis of variance and maximum-likelihood estimation.     

On Quasi Likelihood Equations with Non-parametric Weights

To apply the quasi likelihood method one needs both the mean and the variance functions to determine its optimal weights. If the variance function is unknown, then the weights should be acquired from the data. One way to do so is by adaptive estimation, which involves non-parametric estimation of the variance function. Adaptation, however, also brings in noise that hampers its improvement for moderate samples. In this paper we introduce an alternative method based not on the estimation of the variance function, but on the penalized minimization of the asymptotic variance of the estimator. By doing so we are able to retain a restricted optimality under the smoothness condition, however strong that condition may be. This is important because for moderate sample sizes we need to impose a strong smoothness constraint to damp the noise—often stronger than would be adequate for the adaptive method. We will give a rigorous development of the related asymptotic theory, and provide the simulation evidence for the advantage of this method.     

On Measuring Uncertainty of Benchmarked Predictors with Application to Disease Risk Estimate

Empirical Bayes (EB) estimates in general linear mixed models are useful for the small area estimation in the sense of increasing precision of estimation of small area means. However, one potential difficulty of EB is that the overall estimate for a larger geographical area based on a (weighted) sum of EB estimates is not necessarily identical to the corresponding direct estimate such as the overall sample mean. Another difficulty is that EB estimates yield over‐shrinking, which results in the sampling variance smaller than the posterior variance. One way to fix these problems is the benchmarking approach based on the constrained empirical Bayes (CEB) estimators, which satisfy the constraints that the aggregated mean and variance are identical to the requested values of mean and variance. In this paper, we treat the general mixed models, derive asymptotic approximations of the mean squared error (MSE) of CEB and provide second‐order unbiased estimators of MSE based on the parametric bootstrap method. These results are applied to natural exponential families with quadratic variance functions. As a specific example, the Poisson‐gamma model is dealt with, and it is illustrated that the CEB estimates and their MSE estimates work well through real mortality data.     

Multivariate normal mean–variance mixture distribution based on Birnbaum–Saunders distribution

A multivariate normal mean–variance mixture based on a Birnbaum–Saunders (NMVMBS) distribution is introduced and several properties of this new distribution are discussed. A new robust non-Gaussian ARCH-type model is proposed in which there exists a relation between the variance of the observations, and the marginal distributions are NMVMBS. A simple EM-based maximum likelihood estimation procedure to estimate the parameters of this normal mean–variance mixture distribution is given. A simulation study and some real data are used to demonstrate the modelling strength of this new model.     

Adaptive Inference for Multi-Stage Survey Data

Multi-level models can be used to account for clustering in data from multi-stage surveys. In some cases, the intraclass correlation may be close to zero, so that it may seem reasonable to ignore clustering and fit a single-level model. This article proposes several adaptive strategies for allowing for clustering in regression analysis of multi-stage survey data. The approach is based on testing whether the PSU-level variance component is zero. If this hypothesis is retained, then variance estimates are calculated ignoring clustering; otherwise, clustering is reflected in variance estimation. A simple simulation study is used to evaluate the various procedures.     

Using data augmentation to correct for non-ignorable non-response when surrogate data are available: an application to the distribution of hourly pay

Summary.  The paper develops a data augmentation method to estimate the distribution function of a variable, which is partially observed, under a non-ignorable missing data mechanism, and where surrogate data are available. An application to the estimation of hourly pay distributions using UK Labour Force Survey data provides the main motivation. In addition to considering a standard parametric data augmentation method, we consider the use of hot deck imputation methods as part of the data augmentation procedure to improve the robustness of the method. The method proposed is compared with standard methods that are based on an ignorable missing data mechanism, both in a simulation study and in the Labour Force Survey application. The focus is on reducing bias in point estimation, but variance estimation using multiple imputation is also considered briefly.     

Semiparametric analysis of recurrent events: artificial censoring,truncation, pairwise estimation and inference

The analysis of recurrent failure time data from longitudinal studies can be complicated by the presence of dependent censoring. There has been a substantive literature that has developed based on an artificial censoring device. We explore in this article the connection between this class of methods with truncated data structures. In addition, a new procedure is developed for estimation and inference in a joint model for recurrent events and dependent censoring. Estimation proceeds using a mixed U-statistic based estimating function approach. New resampling-based methods for variance estimation and model checking are also described. The methods are illustrated by application to data from an HIV clinical trial as with a limited simulation study.     

Linear models and convex geometry: aspects of non-negative variance estimation 1

Geometric aspects of linear model theory are surveyed as they bear on mean estimation, or variance covariance component estimation. It is outlined that notions associated with linear subspaces suffice for those of the customary procedures which are solely based on linear, or multilinear algebra. While conceptually simple, these methods do not always respect convexity constraints which naturally arise in variance component estimation.

Previous work on negative estimates of variance is reviewed, followed by a more detailed study of the non-negative definite analogue of the MINQUE procedure. Some characterizations are proposed which are based on convex duality theory. Optimal estimators now correspond to (non-linear) projections onto closed convex cones, they are easy to visualise, but hard to compute. No ultimate solution can be recommended, instead the paper concludes with a list of open problems.