Clustered restricted mean survival time regression

For multicenter randomized trials or multilevel observational studies, the Cox regression model has long been the primary approach to study the effects of covariates on time-to-event outcomes. A critical assumption of the Cox model is the proportionality of the hazard functions for modeled covariates, violations of which can result in ambiguous interpretations of the hazard ratio estimates. To address this issue, the restricted mean survival time (RMST), defined as the mean survival time up to a fixed time in a target population, has been recommended as a model-free target parameter. In this article, we generalize the RMST regression model to clustered data by directly modeling the RMST as a continuous function of restriction times with covariates while properly accounting for within-cluster correlations to achieve valid inference. The proposed method estimates regression coefficients via weighted generalized estimating equations, coupled with a cluster-robust sandwich variance estimator to achieve asymptotically valid inference with a sufficient number of clusters. In small-sample scenarios where a limited number of clusters are available, however, the proposed sandwich variance estimator can exhibit negative bias in capturing the variability of regression coefficient estimates. To overcome this limitation, we further propose and examine bias-corrected sandwich variance estimators to reduce the negative bias of the cluster-robust sandwich variance estimator. We study the finite-sample operating characteristics of proposed methods through simulations and reanalyze two multicenter randomized trials.     

A tunable multivariable nonlinear robust observer for biological systems

This paper presents a robust nonlinear asymptotic observer with adjustable convergence rate with a great potential of applicability for biological systems in which the main state variables are difficult and expensive to measure or such measurements do not exist. This observer scheme is based on the classical asymptotic observer, which is modified to allow the tuning of the convergence rate. It is shown that the proposed observer provides fast and satisfactory estimates when facing load disturbances, system failures and parameter uncertainty while maintaining the excellent robustness and stability properties of the classical asymptotic observer. The implementation of the tunable observer is carried out by numerical simulations of a mathematical model of an anaerobic digestion process used for wastewater treatment. The key results are examined and further developed.     

Regression analysis of doubly censored failure time data with applications to AIDS studies

In many epidemiological studies, the survival time of interest is the elapsed time between two related events, the originating event and the failure event, and the times of the occurrences of both events are right or interval censored. We discuss the regression analysis of such studies and a simple estimating equation approach is proposed under the proportional hazards model. The method can easily be implemented and does not involve any iteration among unknown parameters, as full likelihood approaches proposed in the literature do. The asymptotic properties of the proposed regression coefficient estimates are derived and an AIDS cohort study is analyzed to illustrate the proposed approach.     

Exact Maximum Likelihood Estimates of Cox's Regression Parameters Based on Categorized Factorial Data

This paper describes how Cox's Proportional Hazards model may be used to analyze dichotomized factorial data obtained from a right-censored epidemiological study where time to response is of interest. Exact maximum likelihood estimates of the relative mortality rates are obtained for any number of prognostic factors, as well as their joint asymptotic sampling distribution. These rates represent excess mortality due to the various levels of the prognostic factors. The results are used to discuss the effect of the factors on the survival probability distribution of a cohort of industrial workers who have been exposed to a carcinogen. Kaplan-Meier estimates of the survival function of the internal control population are used to determine the expected number of deaths in the study population. This method differs from the usual lite-table procedure. Asymptotic tests are proposed for some simultaneous and conditional statistical hypotheses.     

Adaptive design: estimation and inference with censored data in a semiparametric model

In this article, we provide a method of estimation for the treatment effect in the adaptive design for censored survival data with or without adjusting for risk factors other than the treatment indicator. Within the semiparametric Cox proportional hazards model, we propose a bias-adjusted parameter estimator for the treatment coefficient and its asymptotic confidence interval at the end of the trial. The method for obtaining an asymptotic confidence interval and point estimator is based on a general distribution property of the final test statistic from the weighted linear rank statistics at the interims with or without considering the nuisance covariates. The computation of the estimates is straightforward. Extensive simulation studies show that the asymptotic confidence intervals have reasonable nominal probability of coverage, and the proposed point estimators are nearly unbiased with practical sample sizes.     

Estimating asymptotic size using the largest individuals per sample

Summary Estimates of asymptotic size are especially useful for comparative studies of taxonomic groups in which animals mature at small sizes relative to their final asymptotic sizes. The largest individuals per sample can provide reasonable estimates of asymptotic size if three conditions are met: 1) at least some adults in a population are near their final asymptotic size, 2) samples of a reasonable size are likely to contain a largest individual that is near the average asymptotic size for the members of its sex, and 3) the coefficient of variation in asymptotic size is small for the members of each sex. In the current study, we show that all three of these conditions are met for one species of Anolis lizards (A. limifrons). For a series of samples from the genus Anolis, the largest individual per sample produces estimates of asymptotic size that are virtually identical to those produced by fitting field data on growth rates to nonlinear growth equations. These results suggest that the largest individual method can provide reasonable estimates of asymptotic size for the members of this genus, and imply that this method may also be useful for estimating asymptotic sizes in other taxa that satisfy the criteria listed above.     

An asymptotic maximum principle for essentially linear evolution models

Recent work on mutation-selection models has revealed that, under specific assumptions on the fitness function and the mutation rates, asymptotic estimates for the leading eigenvalue of the mutation-reproduction matrix may be obtained through a low-dimensional maximum principle in the limit N-->infinity (where N, or N(d) with d> or =1, is proportional to the number of types). In order to extend this variational principle to a larger class of models, we consider here a family of reversible matrices of asymptotic dimension N(d) and identify conditions under which the high-dimensional Rayleigh-Ritz variational problem may be reduced to a low-dimensional one that yields the leading eigenvalue up to an error term of order 1/N. For a large class of mutation-selection models, this implies estimates for the mean fitness, as well as a concentration result for the ancestral distribution of types.     

Bootstrap and Asymptotic Prediction Criterion Estimates for Binomial Proportions in Insemination Data

Model choice techniques are proposed for logistic regression, based on prediction criterion estimation similar to Akaike's information criterion. For artificial insemination data of cattle, we wish to study a factor influence on success proportion; tests standard methods don't always seem suitable for prediction objective. Two prediction criterion estimate methods are applied to these data: simulated bootstrap and asymptotic estimates. Some empirical properties of this estimate are studied.     

Under-smoothed kernel confidence intervals for the hazard ratio based on censored data

In cancer clinical trials, it is often of interest in estimating the ratios of hazard rates at some specific time points during the study from two independent populations. In this paper, we consider nonparametric confidence interval procedures for the hazard ratio based on kernel estimates for the hazard rates with under-smoothing bandwidths. Two methods are used to derive the confidence intervals: one based on the asymptotic normality of the ratio of the kernel estimates for the hazard rates in two populations and another through Fieller's Theorem. The performances of the proposed confidence intervals are evaluated through Monte-Carlo simulations and applied to the analysis of data from a clinical trial on early breast cancer.     

Estimation of linkage in the presence of multiplicative viability effects

Log-linear analysis of contingency tables is applied to trihybrid backcross data to estimate linkage and viability. Whereas nonadditive viability differences perturb recombination estimates in the classical analysis, this statistical procedure yields maximum likelihood crossover frequency estimates in the presence of multiplicative viability effects. Other advantages of this method include: (1) estimation of viability effects of gene substitution at each locus, (2) estimation of asymptotic confidence intervals on recombination frequencies and viabilities, and (3) it tests the null hypothesis of no interference and no viability interactions. Extensions to cover more loci and to allow certain kinds of epistasis are easily made. Relative merits of the proposed and classical methods are discussed.     

Mean residual life cure models for right-censored data with and without length-biased sampling

We propose a semiparametric mean residual life mixture cure model for right-censored survival data with a cured fraction. The model employs the proportional mean residual life model to describe the effects of covariates on the mean residual time of uncured subjects and the logistic regression model to describe the effects of covariates on the cure rate. We develop estimating equations to estimate the proposed cure model for the right-censored data with and without length-biased sampling, the latter is often found in prevalent cohort studies. In particular, we propose two estimating equations to estimate the effects of covariates in the cure rate and a method to combine them to improve the estimation efficiency. The consistency and asymptotic normality of the proposed estimates are established. The finite sample performance of the estimates is confirmed with simulations. The proposed estimation methods are applied to a clinical trial study on melanoma and a prevalent cohort study on early-onset type 2 diabetes mellitus.     

Maximum Likelihood Methods for Fitting the Burr Type XII Distribution to Life Test Data

This paper describes mathematical and computational methodology for estimating the parameters of the Burr Type XII distribution by the method of maximum likelihood. Expressions for the asymptotic variances and covariances of the parameter estimates are given, and the modality of the log-likelihood and conditional log-likelihood functions is analyzed. As a result of this analysis for various a priori known and unknown parameter combinations, conditions are given which guarantee that the parameter estimates obtained will, indeed, be maximum likelihood estimates. An efficient numerical method for maximizing the conditional log-likelihood function is described, and mathematical expressions are given for the various numerical approximations needed to evaluate the expressions given for the asymptotic variances and covariances of the parameter estimates. The methodology discussed is applied in a numerical example to life test data arising in a clinical setting.     

A note on the bias due to omitted confounders

In observational studies some confounders may be unknown andtherefore omitted from the analysis while others are adjustedfor. Approximations to the functions defining the relationshipbetween the parameters in the full and reduced models are proposedleading to asymptotic bias estimates. Numerical calculationsfor logistic and Poisson regression models show good agreementbetween asymptotic and simulation bias. A data set containingthe relationship between low birth weight and smoking (Hosmer& Lemeshow, 1989) is used as an illustration.     

Additive hazard regression with auxiliary covariates

We consider the additive hazard model when some of the truecovariates are measured only on a randomly selected validationset whereas auxiliary covariates are observed for all studysubjects. An updated pseudoscore estimation approach is proposedfor the parameters of the additive hazard model. It allows oneto fit the model with auxiliary covariates, while leaving thebaseline hazard unspecified. Asymptotic properties of the proposedestimators are established, and consistent standard error estimatorsare developed. Simulations demonstrate that the asymptotic approximationsof the proposed estimates are adequate for practical use. Areal example is used to illustrate the performance of the proposedmethod.     

Estimating linkage disequilibrium between a polymorphic marker locus and a trait locus in natural populations.

Positional cloning of gene(s) underlying a complex trait requires a high-resolution linkage map between the trait locus and genetic marker loci. Recent research has shown that this may be achieved through appropriately modeling and screening linkage disequilibrium between the candidate marker locus and the major trait locus. A quantitative genetics model was developed in the present study to estimate the coefficient of linkage disequilibrium between a polymorphic genetic marker locus and a locus underlying a quantitative trait as well as the relevant genetic parameters using the sample from randomly mating populations. Asymptotic covariances of the maximum-likelihood estimates of the parameters were formulated. Convergence of the EM-based statistical algorithm for calculating the maximum-likelihood estimates was confirmed and its utility to analyze practical data was exploited by use of extensive Monte-Carlo simulations. Appropriateness of calculating the asymptotic covariance matrix in the present model was investigated for three different approaches. Numerical analyses based on simulation data indicated that accurate estimation of the genetic parameters may be achieved if a sample size of 500 is used and if segregation at the trait locus explains not less than a quarter of phenotypic variation of the trait, but the study reveals difficulties in predicting the asymptotic variances of these maximum-likelihood estimates. A comparison was made between the statistical powers of the maximum-likelihood analysis and the previously proposed regression analysis for detecting the disequilibrium.     

Reducing mean squared error in the analysis of pair-matched case-control studies

The standard estimator of the common odds ratio for pair-matched case-control studies, the stratified estimate, is consistent but it ignores all information from the concordant pairs. At the other extreme, the pooled estimator is more efficient as it uses all the data, but is not consistent. In order to trade between bias and precision, Liang and Zeger (1988, Biometrics 44, 1145-1156) proposed an estimator that is a compromise between the stratified and pooled estimates. In the current paper, the possibility of optimizing the trade-off is explored. Specifically, the family of weighted averages of the stratified and pooled estimates is considered, and the weight that minimizes an asymptotic approximation of mean squared error is derived. In practice, the optimal weight must be estimated from the data so that the estimator is only approximately optimal. Small-sample properties are evaluated via simulations.     

CEDAR: communication efficient distributed analysis for regressions

Electronic health records (EHRs) offer great promises for advancing precision medicine and, at the same time, present significant analytical challenges. Particularly, it is often the case that patient-level data in EHRs cannot be shared across institutions (data sources) due to government regulations and/or institutional policies. As a result, there are growing interests about distributed learning over multiple EHRs databases without sharing patient-level data. To tackle such challenges, we propose a novel communication efficient method that aggregates the optimal estimates of external sites, by turning the problem into a missing data problem. In addition, we propose incorporating posterior samples of remote sites, which can provide partial information on the missing quantities and improve efficiency of parameter estimates while having the differential privacy property and thus reducing the risk of information leaking. The proposed approach, without sharing the raw patient level data, allows for proper statistical inference. We provide theoretical investigation for the asymptotic properties of the proposed method for statistical inference as well as differential privacy, and evaluate its performance in simulations and real data analyses in comparison with several recently developed methods.     

Dose Response Analysis Using Robust Covariance Estimation

A dose response analysis is robustified by estimating the asymptotic covariance of the fitted model parameters by the approximate information sandwich (a sandwich statistic) under a heterogeneous variance. The robust method is described by using a nonlinear four‐parameter regression model. The usual, robust, bootstrap, and jackknife estimates of the asymptotic variance are examined for the bioassay data. Under the response of a normal distribution with changing variances over the dose levels, the performance of the usual and robust variances is investigated by Monte Carlo study. It confirms the robustness of the sandwich estimate and shows the non‐accuracy of the usual asymptotic variance estimates of fitted model parameters under the different forms of nonconstant variance structures.     

Risk assessment for quantitative responses using a mixture model

A problem that frequently occurs in biological experiments with laboratory animals is that some subjects are less susceptible to the treatment than others. A mixture model has traditionally been proposed to describe the distribution of responses in treatment groups for such experiments. Using a mixture dose-response model, we derive an upper confidence limit on additional risk, defined as the excess risk over the background risk due to an added dose. Our focus will be on experiments with continuous responses for which risk is the probability of an adverse effect defined as an event that is extremely rare in controls. The asymptotic distribution of the likelihood ratio statistic is used to obtain the upper confidence limit on additional risk. The method can also be used to derive a benchmark dose corresponding to a specified level of increased risk. The EM algorithm is utilized to find the maximum likelihood estimates of model parameters and an extension of the algorithm is proposed to derive the estimates when the model is subject to a specified level of added risk. An example is used to demonstrate the results, and it is shown that by using the mixture model a more accurate measure of added risk is obtained.     

Local estimation of age-dependent variance components from longitudinal twin data

In the study of longitudinal twin and family data, interest is often in the covariance structure of the data and the decomposition of this covariance structure into genetic and environmental components rather than in estimating the mean function. Various parametric models for covariance structures have been proposed but, e.g., in studies of children where growth spurts occur at various ages, it is difficult to a priori determine an appropriate parametric model for the covariance structure. In particular, there is a general lack of the visualization procedures, such as lowess, that are invaluable in the initial stages of constructing a parametric model for a mean function. Here we use kernel smoothing to modify a cross-sectional approach based on the sample covariance matrices to obtain smoothed estimates of the genetic and environmental variances and correlations for longitudinal twin data. The methods are proposed to be exploratory as an aid to parametric modeling rather than inferential, although approximate asymptotic standard errors are derived in the Appendix.