Semiparametric regression analysis of longitudinal data with informative drop-outs

Informative drop-out arises in longitudinal studies when the subject's follow-up time depends on the unobserved values of the response variable. We specify a semiparametric linear regression model for the repeatedly measured response variable and an accelerated failure time model for the time to informative drop-out. The error terms from the two models are assumed to have a common, but completely arbitrary joint distribution. Using a rank-based estimator for the accelerated failure time model and an artificial censoring device, we construct an asymptotically unbiased estimating function for the linear regression model. The resultant estimator is shown to be consistent and asymptotically normal. A resampling scheme is developed to estimate the limiting covariance matrix. Extensive simulation studies demonstrate that the proposed methods are suitable for practical use. Illustrations with data taken from two AIDS clinical trials are provided.     

Model averaging in the analysis of leukemia mortality among Japanese A-bomb survivors

Epidemiological studies often include numerous covariates, with a variety of possible approaches to control for confounding of the association of primary interest, as well as a variety of possible models for the exposure–response association of interest. Walsh and Kaiser (Radiat Environ Biophys 50:21–35, 2011) advocate a weighted averaging of the models, where the weights are a function of overall model goodness of fit and degrees of freedom. They apply this method to analyses of radiation–leukemia mortality associations among Japanese A-bomb survivors. We caution against such an approach, noting that the proposed model averaging approach prioritizes the inclusion of covariates that are strong predictors of the outcome, but which may be irrelevant as confounders of the association of interest, and penalizes adjustment for covariates that are confounders of the association of interest, but may contribute little to overall model goodness of fit. We offer a simple illustration of how this approach can lead to biased results. The proposed model averaging approach may also be suboptimal as way to handle competing model forms for an exposure–response association of interest, given adjustment for the same set of confounders; alternative approaches, such as hierarchical regression, may provide a more useful way to stabilize risk estimates in this setting.     

Simultaneous inference in general parametric models

Simultaneous inference is a common problem in many areas of application. If multiple null hypotheses are tested simultaneously, the probability of rejecting erroneously at least one of them increases beyond the pre-specified significance level. Simultaneous inference procedures have to be used which adjust for multiplicity and thus control the overall type I error rate. In this paper we describe simultaneous inference procedures in general parametric models, where the experimental questions are specified through a linear combination of elemental model parameters. The framework described here is quite general and extends the canonical theory of multiple comparison procedures in ANOVA models to linear regression problems, generalized linear models, linear mixed effects models, the Cox model, robust linear models, etc. Several examples using a variety of different statistical models illustrate the breadth of the results. For the analyses we use the R add-on package multcomp, which provides a convenient interface to the general approach adopted here.     

Three algorithms and SAS macros for estimating power and sample size for logistic models with one or more independent variables of interest in the presence of covariates

Background

Commonly when designing studies, researchers propose to measure several independent variables in a regression model, a subset of which are identified as the main variables of interest while the rest are retained in a model as covariates or confounders. Power for linear regression in this setting can be calculated using SAS PROC POWER. There exists a void in estimating power for the logistic regression models in the same setting.

Methods

Currently, an approach that calculates power for only one variable of interest in the presence of other covariates for logistic regression is in common use and works well for this special case. In this paper we propose three related algorithms along with corresponding SAS macros that extend power estimation for one or more primary variables of interest in the presence of some confounders.

Results

The three proposed empirical algorithms employ likelihood ratio test to provide a user with either a power estimate for a given sample size, a quick sample size estimate for a given power, and an approximate power curve for a range of sample sizes. A user can specify odds ratios for a combination of binary, uniform and standard normal independent variables of interest, and or remaining covariates/confounders in the model, along with a correlation between variables.

Conclusions

These user friendly algorithms and macro tools are a promising solution that can fill the void for estimation of power for logistic regression when multiple independent variables are of interest, in the presence of additional covariates in the model.

     

The Mantel-Haenszel Procedure Revisited: Models and Generalizations

Several statistical methods have been developed for adjusting the Odds Ratio of the relation between two dichotomous variables X and Y for some confounders Z. With the exception of the Mantel-Haenszel method, commonly used methods, notably binary logistic regression, are not symmetrical in X and Y. The classical Mantel-Haenszel method however only works for confounders with a limited number of discrete strata, which limits its utility, and appears to have no basis in statistical models. Here we revisit the Mantel-Haenszel method and propose an extension to continuous and vector valued Z. The idea is to replace the observed cell entries in strata of the Mantel-Haenszel procedure by subject specific classification probabilities for the four possible values of (X,Y) predicted by a suitable statistical model. For situations where X and Y can be treated symmetrically we propose and explore the multinomial logistic model. Under the homogeneity hypothesis, which states that the odds ratio does not depend on Z, the logarithm of the odds ratio estimator can be expressed as a simple linear combination of three parameters of this model. Methods for testing the homogeneity hypothesis are proposed. The relationship between this method and binary logistic regression is explored. A numerical example using survey data is presented.     

Smoothed quantile regression analysis of competing risks

Censored quantile regression models, which offer great flexibility in assessing covariate effects on event times, have attracted considerable research interest. In this study, we consider flexible estimation and inference procedures for competing risks quantile regression, which not only provides meaningful interpretations by using cumulative incidence quantiles but also extends the conventional accelerated failure time model by relaxing some of the stringent model assumptions, such as global linearity and unconditional independence. Current method for censored quantile regressions often involves the minimization of the L₁‐type convex function or solving the nonsmoothed estimating equations. This approach could lead to multiple roots in practical settings, particularly with multiple covariates. Moreover, variance estimation involves an unknown error distribution and most methods rely on computationally intensive resampling techniques such as bootstrapping. We consider the induced smoothing procedure for censored quantile regressions to the competing risks setting. The proposed procedure permits the fast and accurate computation of quantile regression parameter estimates and standard variances by using conventional numerical methods such as the Newton–Raphson algorithm. Numerical studies show that the proposed estimators perform well and the resulting inference is reliable in practical settings. The method is finally applied to data from a soft tissue sarcoma study.     

Risk of lung cancer and leukemia from exposure to ionizing radiation and potential confounders among workers at the Portsmouth Naval Shipyard

Significantly elevated lung cancer deaths and statistically significantly positive linear trends between leukemia mortality and radiation exposure were reported in a previous analysis of Portsmouth Naval Shipyard workers. The purpose of this study was to conduct a modeling-based analysis that incorporates previously unanalyzed confounders in exploring the exposure-response relationship between cumulative external ionizing radiation exposure and mortality from these cancers among radiation-monitored workers in this cohort. The main analyses were carried out with Poisson regression fitted with maximum likelihood in linear excess relative risk models. Sensitivity analyses varying model components and using other regression models were conducted. The positive association between lung cancer risk and ionizing radiation observed previously was no longer present after adjusting for socioeconomic status (smoking surrogate) and welding fume and asbestos exposures. Excesses of leukemia were found to be positively, though not significantly, associated with external ionizing radiation, with or without including potential confounders. The estimated excess relative risk was 10.88% (95% CI -0.90%, 38.77%) per 10 mSv of radiation exposure, which was within the ranges of risk estimates in previous epidemiological studies (-4.1 to 19.0%). These results are limited by many factors and are subject to uncertainties of the exposure and confounder estimates.     

Regularized estimation for the accelerated failure time model

Summary .  In the presence of high-dimensional predictors, it is challenging to develop reliable regression models that can be used to accurately predict future outcomes. Further complications arise when the outcome of interest is an event time, which is often not fully observed due to censoring. In this article, we develop robust prediction models for event time outcomes by regularizing the Gehan's estimator for the accelerated failure time (AFT) model ( Tsiatis, 1996 , Annals of Statistics 18, 305–328) with least absolute shrinkage and selection operator (LASSO) penalty. Unlike existing methods based on the inverse probability weighting and the Buckley and James estimator ( Buckley and James, 1979 , Biometrika 66, 429–436), the proposed approach does not require additional assumptions about the censoring and always yields a solution that is convergent. Furthermore, the proposed estimator leads to a stable regression model for prediction even if the AFT model fails to hold. To facilitate the adaptive selection of the tuning parameter, we detail an efficient numerical algorithm for obtaining the entire regularization path. The proposed procedures are applied to a breast cancer dataset to derive a reliable regression model for predicting patient survival based on a set of clinical prognostic factors and gene signatures. Finite sample performances of the procedures are evaluated through a simulation study.     

Analysis of a Generalized Linear Mixed Model: A Case Study and Simulation Results

A class of generalized linear mixed models can be obtained by introducing random effects in the linear predictor of a generalized linear model, e.g. a split plot model for binary data or count data. Maximum likelihood estimation, for normally distributed random effects, involves high-dimensional numerical integration, with severe limitations on the number and structure of the additional random effects. An alternative estimation procedure based on an extension of the iterative re-weighted least squares procedure for generalized linear models will be illustrated on a practical data set involving carcass classification of cattle. The data is analysed as overdispersed binomial proportions with fixed and random effects and associated components of variance on the logit scale. Estimates are obtained with standard software for normal data mixed models. Numerical restrictions pertain to the size of matrices to be inverted. This can be dealt with by absorption techniques familiar from e.g. mixed models in animal breeding. The final model fitted to the classification data includes four components of variance and a multiplicative overdispersion factor. Basically the estimation procedure is a combination of iterated least squares procedures and no full distributional assumptions are needed. A simulation study based on the classification data is presented. This includes a study of procedures for constructing confidence intervals and significance tests for fixed effects and components of variance. The simulation results increase confidence in the usefulness of the estimation procedure.     

Probabilistic noninvasive prediction of wall properties of abdominal aortic aneurysms using Bayesian regression

Multiple patient-specific parameters, such as wall thickness, wall strength, and constitutive properties, are required for the computational assessment of abdominal aortic aneurysm (AAA) rupture risk. Unfortunately, many of these quantities are not easily accessible and could only be determined by invasive procedures, rendering a computational rupture risk assessment obsolete. This study investigates two different approaches to predict these quantities using regression models in combination with a multitude of noninvasively accessible, explanatory variables. We have gathered a large dataset comprising tensile tests performed with AAA specimens and supplementary patient information based on blood analysis, the patients medical history, and geometric features of the AAAs. Using this unique database, we harness the capability of state-of-the-art Bayesian regression techniques to infer probabilistic models for multiple quantities of interest. After a brief presentation of our experimental results, we show that we can effectively reduce the predictive uncertainty in the assessment of several patient-specific parameters, most importantly in thickness and failure strength of the AAA wall. Thereby, the more elaborate Bayesian regression approach based on Gaussian processes consistently outperforms standard linear regression. Moreover, our study contains a comparison to a previously proposed model for the wall strength.     

Conditional generalized estimating equations for the analysis of clustered and longitudinal data

Summary .   A common and important problem in clustered sampling designs is that the effect of within-cluster exposures (i.e., exposures that vary within clusters) on outcome may be confounded by both measured and unmeasured cluster-level factors (i.e., measurements that do not vary within clusters). When some of these are ill/not accounted for, estimation of this effect through population-averaged models or random-effects models may introduce bias. We accommodate this by developing a general theory for the analysis of clustered data, which enables consistent and asymptotically normal estimation of the effects of within-cluster exposures in the presence of cluster-level confounders. Semiparametric efficient estimators are obtained by solving so-called conditional generalized estimating equations. We compare this approach with a popular proposal by Neuhaus and Kalbfleisch (1998, Biometrics 54, 638–645) who separate the exposure effect into a within- and a between-cluster component within a random intercept model. We find that the latter approach yields consistent and efficient estimators when the model is linear, but is less flexible in terms of model specification. Under nonlinear models, this approach may yield inconsistent and inefficient estimators, though with little bias in most practical settings.     

Joint semiparametric models for case-cohort designs

Two-phase studies such as case-cohort and nested case-control studies are widely used cost-effective sampling strategies. In the first phase, the observed failure/censoring time and inexpensive exposures are collected. In the second phase, a subgroup of subjects is selected for measurements of expensive exposures based on the information from the first phase. One challenging issue is how to utilize all the available information to conduct efficient regression analyses of the two-phase study data. This paper proposes a joint semiparametric modeling of the survival outcome and the expensive exposures. Specifically, we assume a class of semiparametric transformation models and a semiparametric density ratio model for the survival outcome and the expensive exposures, respectively. The class of semiparametric transformation models includes the proportional hazards model and the proportional odds model as special cases. The density ratio model is flexible in modeling multivariate mixed-type data. We develop efficient likelihood-based estimation and inference procedures and establish the large sample properties of the nonparametric maximum likelihood estimators. Extensive numerical studies reveal that the proposed methods perform well under practical settings. The proposed methods also appear to be reasonably robust under various model mis-specifications. An application to the National Wilms Tumor Study is provided.     

Frailty‐Based Competing Risks Model for Multivariate Survival Data

Summary In this work, we provide a new class of frailty‐based competing risks models for clustered failure times data. This class is based on expanding the competing risks model of Prentice et al. (1978, Biometrics 34 , 541–554) to incorporate frailty variates, with the use of cause‐specific proportional hazards frailty models for all the causes. Parametric and nonparametric maximum likelihood estimators are proposed. The main advantages of the proposed class of models, in contrast to the existing models, are: (1) the inclusion of covariates; (2) the flexible structure of the dependency among the various types of failure times within a cluster; and (3) the unspecified within‐subject dependency structure. The proposed estimation procedures produce the most efficient parametric and semiparametric estimators and are easy to implement. Simulation studies show that the proposed methods perform very well in practical situations.     

Screening methods for linear errors-in-variables models in high dimensions

Microarray studies, in order to identify genes associated with an outcome of interest, usually produce noisy measurements for a large number of gene expression features from a small number of subjects. One common approach to analyzing such high-dimensional data is to use linear errors-in-variables (EIV) models; however, current methods for fitting such models are computationally expensive. In this paper, we present two efficient screening procedures, namely, corrected penalized marginal screening (PMSc) and corrected sure independence screening (SISc), to reduce the number of variables for final model building. Both screening procedures are based on fitting corrected marginal regression models relating the outcome to each contaminated covariate separately, which can be computed efficiently even with a large number of features. Under mild conditions, we show that these procedures achieve screening consistency and reduce the number of features substantially, even when the number of covariates grows exponentially with sample size. In addition, if the true covariates are weakly correlated, we show that PMSc can achieve full variable selection consistency. Through a simulation study and an analysis of gene expression data for bone mineral density of Norwegian women, we demonstrate that the two new screening procedures make estimation of linear EIV models computationally scalable in high-dimensional settings, and improve finite sample estimation and selection performance compared with estimators that do not employ a screening stage.     

Cole–Cole, linear and multivariate modeling of capacitance data for on-line monitoring of biomass

This work evaluates three techniques of calibrating capacitance (dielectric) spectrometers used for on-line monitoring of biomass: modeling of cell properties using the theoretical Cole–Cole equation, linear regression of dual-frequency capacitance measurements on biomass concentration, and multivariate (PLS) modeling of scanning dielectric spectra. The performance and robustness of each technique is assessed during a sequence of validation batches in two experimental settings of differing signal noise. In more noisy conditions, the Cole–Cole model had significantly higher biomass concentration prediction errors than the linear and multivariate models. The PLS model was the most robust in handling signal noise. In less noisy conditions, the three models performed similarly. Estimates of the mean cell size were done additionally using the Cole–Cole and PLS models, the latter technique giving more satisfactory results.     

Prediction of maize single-cross hybrid performance: support vector machine regression versus best linear prediction

Accurate prediction of the phenotypic performance of a hybrid plant based on the molecular fingerprints of its parents should lead to a more cost-effective breeding programme as it allows to reduce the number of expensive field evaluations. The construction of a reliable prediction model requires a representative sample of hybrids for which both molecular and phenotypic information are accessible. This phenotypic information is usually readily available as typical breeding programmes test numerous new hybrids in multi-location field trials on a yearly basis. Earlier studies indicated that a linear mixed model analysis of this typically unbalanced phenotypic data allows to construct ɛ-insensitive support vector machine regression and best linear prediction models for predicting the performance of single-cross maize hybrids. We compare these prediction methods using different subsets of the phenotypic and marker data of a commercial maize breeding programme and evaluate the resulting prediction accuracies by means of a specifically designed field experiment. This balanced field trial allows to assess the reliability of the cross-validation prediction accuracies reported here and in earlier studies. The limits of the predictive capabilities of both prediction methods are further examined by reducing the number of training hybrids and the size of the molecular fingerprints. The results indicate a considerable discrepancy between prediction accuracies obtained by cross-validation procedures and those obtained by correlating the predictions with the results of a validation field trial. The prediction accuracy of best linear prediction was less sensitive to a reduction of the number of training examples compared with that of support vector machine regression. The latter was, however, better at predicting hybrid performance when the size of the molecular fingerprints was reduced, especially if the initial set of markers had a low information content.     

Functional mixture regression

In functional linear models (FLMs), the relationship between the scalar response and the functional predictor process is often assumed to be identical for all subjects. Motivated by both practical and methodological considerations, we relax this assumption and propose a new class of functional regression models that allow the regression structure to vary for different groups of subjects. By projecting the predictor process onto its eigenspace, the new functional regression model is simplified to a framework that is similar to classical mixture regression models. This leads to the proposed approach named as functional mixture regression (FMR). The estimation of FMR can be readily carried out using existing software implemented for functional principal component analysis and mixture regression. The practical necessity and performance of FMR are illustrated through applications to a longevity analysis of female medflies and a human growth study. Theoretical investigations concerning the consistent estimation and prediction properties of FMR along with simulation experiments illustrating its empirical properties are presented in the supplementary material available at Biostatistics online. Corresponding results demonstrate that the proposed approach could potentially achieve substantial gains over traditional FLMs.     

Model checking for ROC regression analysis

Summary .   The receiver operating characteristic (ROC) curve is a prominent tool for characterizing the accuracy of a continuous diagnostic test. To account for factors that might influence the test accuracy, various ROC regression methods have been proposed. However, as in any regression analysis, when the assumed models do not fit the data well, these methods may render invalid and misleading results. To date, practical model-checking techniques suitable for validating existing ROC regression models are not yet available. In this article, we develop cumulative residual-based procedures to graphically and numerically assess the goodness of fit for some commonly used ROC regression models, and show how specific components of these models can be examined within this framework. We derive asymptotic null distributions for the residual processes and discuss resampling procedures to approximate these distributions in practice. We illustrate our methods with a dataset from the cystic fibrosis registry.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.