A Semiparametric Joint Model for Longitudinal and Survival Data with Application to Hemodialysis Study

Summary .  In many longitudinal clinical studies, the level and progression rate of repeatedly measured biomarkers on each subject quantify the severity of the disease and that subject's susceptibility to progression of the disease. It is of scientific and clinical interest to relate such quantities to a later time-to-event clinical endpoint such as patient survival. This is usually done with a shared parameter model. In such models, the longitudinal biomarker data and the survival outcome of each subject are assumed to be conditionally independent given subject-level severity or susceptibility (also called frailty in statistical terms). In this article, we study the case where the conditional distribution of longitudinal data is modeled by a linear mixed-effect model, and the conditional distribution of the survival data is given by a Cox proportional hazard model. We allow unknown regression coefficients and time-dependent covariates in both models. The proposed estimators are maximizers of an exact correction to the joint log likelihood with the frailties eliminated as nuisance parameters, an idea that originated from correction of covariate measurement error in measurement error models. The corrected joint log likelihood is shown to be asymptotically concave and leads to consistent and asymptotically normal estimators. Unlike most published methods for joint modeling, the proposed estimation procedure does not rely on distributional assumptions of the frailties. The proposed method was studied in simulations and applied to a data set from the Hemodialysis Study.     

On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial

To estimate an overall treatment difference with data from a randomized comparative clinical study, baseline covariates are often utilized to increase the estimation precision. Using the standard analysis of covariance technique for making inferences about such an average treatment difference may not be appropriate, especially when the fitted model is nonlinear. On the other hand, the novel augmentation procedure recently studied, for example, by Zhang and others (2008. Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics 64, 707-715) is quite flexible. However, in general, it is not clear how to select covariates for augmentation effectively. An overly adjusted estimator may inflate the variance and in some cases be biased. Furthermore, the results from the standard inference procedure by ignoring the sampling variation from the variable selection process may not be valid. In this paper, we first propose an estimation procedure, which augments the simple treatment contrast estimator directly with covariates. The new proposal is asymptotically equivalent to the aforementioned augmentation method. To select covariates, we utilize the standard lasso procedure. Furthermore, to make valid inference from the resulting lasso-type estimator, a cross validation method is used. The validity of the new proposal is justified theoretically and empirically. We illustrate the procedure extensively with a well-known primary biliary cirrhosis clinical trial data set.     

Constructing confidence intervals for selected parameters

In large-scale problems, it is common practice to select important parameters by a procedure such as the Benjamini and Hochberg procedure and construct confidence intervals (CIs) for further investigation while the false coverage-statement rate (FCR) for the CIs is controlled at a desired level. Although the well-known BY CIs control the FCR, they are uniformly inflated. In this paper, we propose two methods to construct shorter selective CIs. The first method produces shorter CIs by allowing a reduced number of selective CIs. The second method produces shorter CIs by allowing a prefixed proportion of CIs containing the values of uninteresting parameters. We theoretically prove that the proposed CIs are uniformly shorter than BY CIs and control the FCR asymptotically for independent data. Numerical results confirm our theoretical results and show that the proposed CIs still work for correlated data. We illustrate the advantage of the proposed procedures by analyzing the microarray data from a HIV study.     

Testing Hypotheses about Occupancy Rates for Clustered Populations

The use of the Pearson chi-square statistic for testing hypotheses on biological populations is not appropriate when the individuals are distributed by clusters. In the case where the clusters are distributed independently of each other, we propose an asymptotically chi-square distributed test statistic taking into account the cluster size distribution. An example provided by European Corn Borer eggs data is used to illustrate the test procedure.     

Simultaneous confidence corridors for mean functions in functional data analysis of imaging data

Motivated by recent work involving the analysis of biomedical imaging data, we present a novel procedure for constructing simultaneous confidence corridors for the mean of imaging data. We propose to use flexible bivariate splines over triangulations to handle an irregular domain of the images that is common in brain imaging studies and in other biomedical imaging applications. The proposed spline estimators of the mean functions are shown to be consistent and asymptotically normal under some regularity conditions. We also provide a computationally efficient estimator of the covariance function and derive its uniform consistency. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions from two populations of imaging data. Through Monte Carlo simulation studies, we examine the finite sample performance of the proposed method. Finally, the proposed method is applied to analyze brain positron emission tomography data in two different studies. One data set used in preparation of this article was obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database.     

Basic reproduction ratio for a fishery model in a patchy environment

We present a dynamical model of a multi-site fishery. The fish stock is located on a discrete set of fish habitats where it is catched by the fishing fleet. We assume that fishes remain on fishing habitats while the fishing vessels can move at a fast time scale to visit the different fishing sites. We use the existence of two time scales to reduce the dimension of the model : we build an aggregated model considering the habitat fish densities and the total fishing effort. We explore a regulation procedure, which imposes an average residence time in patches. Several equilibria exist, a Fishery Free Equilibria (FFEs) as well as a Sustainable Fishery Equilibria (SFEs). We show that the dynamics depends on a threshold which is similar to a basic reproduction ratio for the fishery. When the basic reproduction ratio is less or equal to 1, one of the FFEs is globally asymptotically stable (GAS), otherwise one of the SFEs is GAS.     

Distribution-free R sample tests for the hypothesis of parallelism of response profiles

In this paper two nonparametric tests are given for testing the nullhypothesis of parallelity of response curves in r samples. The first procedure is done by a permutation test, whose practical applicability is ensured by a FORTRAN-subroutine available from the author. As the computational work and time grows rapidly with sample sizes, a Monte-Carlo-solution is optionally given by the same procedure. Alternatively an asymptotically distribution-free test based on quadratic forms is constructed for large sample situations.     

A Positive Stable Frailty Model for Clustered Failure Time Data with Covariate‐Dependent Frailty

Summary In this article, we propose a positive stable shared frailty Cox model for clustered failure time data where the frailty distribution varies with cluster‐level covariates. The proposed model accounts for covariate‐dependent intracluster correlation and permits both conditional and marginal inferences. We obtain marginal inference directly from a marginal model, then use a stratified Cox‐type pseudo‐partial likelihood approach to estimate the regression coefficient for the frailty parameter. The proposed estimators are consistent and asymptotically normal and a consistent estimator of the covariance matrix is provided. Simulation studies show that the proposed estimation procedure is appropriate for practical use with a realistic number of clusters. Finally, we present an application of the proposed method to kidney transplantation data from the Scientific Registry of Transplant Recipients.     

A resolution of the ascertainment sampling problem. III. Pedigrees.

When nuclear families are sampled by an ascertainment procedure whose properties are not known, biased estimates of genetic parameters will arise if an incorrect specification of the ascertainment procedure is made. Elsewhere we have put forward a resolution of this problem by introducing an ascertainment-assumption-free (AAF) method, for nuclear family data, which gives asymptotically unbiased estimators no matter what the true nature of the ascertainment process. In the present paper we extend this method to cover pedigree data. Problems that arise with pedigrees but not with families--for example, the question of which families in a pedigree are "ascertainable"--are also considered. Comparisons of numerical results for pedigrees and nuclear families are also made.     

Non-linear ordination in several dimensions

A method is described for fitting a model in which site characteristics are represented by a set of orthogonal axes, while the probability that a particular species will be present, and its quantity if it is present, are each related to the axes of the system by symmetrical hypersurfaces with an optimum, decreasing asymptotically to zero as conditions depart from the optimum. The method needs initial estimates of the positions of the sample sites within the axis system. Given these starting points, they can be progressively improved by an iterative procedure. Results are reported of an extensive series of tests using artificial data, and of an analysis of field data from brigalow woodland in Queensland.     

Monotone spline‐based least squares estimation for panel count data with informative observation times

This article discusses the statistical analysis of panel count data when the underlying recurrent event process and observation process may be correlated. For the recurrent event process, we propose a new class of semiparametric mean models that allows for the interaction between the observation history and covariates. For inference on the model parameters, a monotone spline‐based least squares estimation approach is developed, and the resulting estimators are consistent and asymptotically normal. In particular, our new approach does not rely on the model specification of the observation process. The proposed inference procedure performs well through simulation studies, and it is illustrated by the analysis of bladder tumor data.     

A multistep protein lysate array quantification method and its statistical properties

The protein lysate array is an emerging technology for quantifying the protein concentration ratios in multiple biological samples. Statistical inference for a parametric quantification procedure has been inadequately addressed in the literature, mainly because the appropriate asymptotic theory involves a problem with the number of parameters increasing with the number of observations. In this article, we develop a multistep procedure for the Sigmoidal models, ensuring consistent estimation of the concentration levels with full asymptotic efficiency. The results obtained in the article justify inferential procedures based on large sample approximations. Simulation studies and real data analysis are used in the article to illustrate the performance of the proposed method in finite samples. The multistep procedure is convenient to work with asymptotically, and is recommended for its statistical efficiency in protein concentration estimation and improved numerical stability by focusing on optimization of lower-dimensional objective functions.     

A semiparametric empirical likelihood method for data from an outcome-dependent sampling scheme with a continuous outcome

Outcome-dependent sampling (ODS) schemes can be a cost effective way to enhance study efficiency. The case-control design has been widely used in epidemiologic studies. However, when the outcome is measured on a continuous scale, dichotomizing the outcome could lead to a loss of efficiency. Recent epidemiologic studies have used ODS sampling schemes where, in addition to an overall random sample, there are also a number of supplemental samples that are collected based on a continuous outcome variable. We consider a semiparametric empirical likelihood inference procedure in which the underlying distribution of covariates is treated as a nuisance parameter and is left unspecified. The proposed estimator has asymptotic normality properties. The likelihood ratio statistic using the semiparametric empirical likelihood function has Wilks-type properties in that, under the null, it follows a chi-square distribution asymptotically and is independent of the nuisance parameters. Our simulation results indicate that, for data obtained using an ODS design, the semiparametric empirical likelihood estimator is more efficient than conditional likelihood and probability weighted pseudolikelihood estimators and that ODS designs (along with the proposed estimator) can produce more efficient estimates than simple random sample designs of the same size. We apply the proposed method to analyze a data set from the Collaborative Perinatal Project (CPP), an ongoing environmental epidemiologic study, to assess the relationship between maternal polychlorinated biphenyl (PCB) level and children's IQ test performance.     

Parametric survival models for interval-censored data with time-dependent covariates

We present a parametric family of regression models for interval-censored event-time (survival) data that accomodates both fixed (e.g. baseline) and time-dependent covariates. The model employs a three-parameter family of survival distributions that includes the Weibull, negative binomial, and log-logistic distributions as special cases, and can be applied to data with left, right, interval, or non-censored event times. Standard methods, such as Newton-Raphson, can be employed to estimate the model and the resulting estimates have an asymptotically normal distribution about the true values with a covariance matrix that is consistently estimated by the information function. The deviance function is described to assess model fit and a robust sandwich estimate of the covariance may also be employed to provide asymptotically robust inferences when the model assumptions do not apply. Spline functions may also be employed to allow for non-linear covariates. The model is applied to data from a long-term study of type 1 diabetes to describe the effects of longitudinal measures of glycemia (HbA1c) over time (the time-dependent covariate) on the risk of progression of diabetic retinopathy (eye disease), an interval-censored event-time outcome.     

Plane Wave Diffraction by a Finite Plate with Impedance Boundary Conditions

In this study we have examined a plane wave diffraction problem by a finite plate having different impedance boundaries. The Fourier transforms were used to reduce the governing problem into simultaneous Wiener-Hopf equations which are then solved using the standard Wiener-Hopf procedure. Afterwards the separated and interacted fields were developed asymptotically by using inverse Fourier transform and the modified stationary phase method. Detailed graphical analysis was also made for various physical parameters we were interested in.     

Time‐Varying Latent Effect Model for Longitudinal Data with Informative Observation Times

Summary In analysis of longitudinal data, it is not uncommon that observation times of repeated measurements are subject‐specific and correlated with underlying longitudinal outcomes. Taking account of the dependence between observation times and longitudinal outcomes is critical under these situations to assure the validity of statistical inference. In this article, we propose a flexible joint model for longitudinal data analysis in the presence of informative observation times. In particular, the new procedure considers the shared random‐effect model and assumes a time‐varying coefficient for the latent variable, allowing a flexible way of modeling longitudinal outcomes while adjusting their association with observation times. Estimating equations are developed for parameter estimation. We show that the resulting estimators are consistent and asymptotically normal, with variance–covariance matrix that has a closed form and can be consistently estimated by the usual plug‐in method. One additional advantage of the procedure is that it provides a unified framework to test whether the effect of the latent variable is zero, constant, or time‐varying. Simulation studies show that the proposed approach is appropriate for practical use. An application to a bladder cancer data is also given to illustrate the methodology.     

A Booststrap Adaptive Test for Two-way Analysis of Variance

We propose a method to construct adaptive tests based on a bootstrap technique. The procedure leads to a nearly exact adaptive test depending on the size of the sample. With the use of the estimated Pitman's relative efficacy as selector statistic, we show that the adaptive test has a power that is asymptotically equal to the power of it's better component. We apply the idea to construct an adaptive test for two-way analysis of variance model. Finally, we use simulations to observe the behaviour of the method for small sample sizes.     

Adjusting O'Brien's test to control type I error for the generalized nonparametric Behrens-Fisher problem

O'Brien (1984, Biometrics 40, 1079-1087) introduced a simple nonparametric test procedure for testing whether multiple outcomes in one treatment group have consistently larger values than outcomes in the other treatment group. We first explore the theoretical properties of O'Brien's test. We then extend it to the general nonparametric Behrens-Fisher hypothesis problem when no assumption is made regarding the shape of the distributions. We provide conditions when O'Brien's test controls its error probability asymptotically and when it fails. We also provide adjusted tests when the conditions do not hold. Throughout this article, we do not assume that all outcomes are continuous. Simulations are performed to compare the adjusted tests to O'Brien's test. The difference is also illustrated using data from a Parkinson's disease clinical trial.     

The effects of conditioning on probands to correct for multiple ascertainment.

Cannings and Thompson suggested conditioning on the phenotypes of the probands to correct for ascertainment in the analysis of pedigree data. The method assumes single ascertainment and can be expected to yield asymptotically biased parameter estimates except in this specific case. However, because the method is easy to apply, we investigated the degree of bias in the more typical situation of multiple ascertainment, in the hope that the bias might be small and that the method could be applied more generally. To explore the utility of conditioning on probands to correct for multiple ascertainment, we calculated the asymptotic value of the segregation ratio for two versions of the simple Mendelian segregation model on sibship data. For both versions, we found that this asymptotic value decreased approximately linearly as the ascertainment probability increased. When ascertainment was complete, the segregation-ratio estimates were zero, not just asymptotically but for finite sample size as well. In some cases, conditioning on probands actually resulted in greater parameter bias than no ascertainment correction at all. These results hold for a variety of sibship-size distributions, several modes of inheritance, and a wide range of population prevalences of affected individuals.     

Semiparametric inference for a 2-stage outcome-auxiliary-dependent sampling design with continuous outcome

Two-stage design has long been recognized to be a cost-effective way for conducting biomedical studies. In many trials, auxiliary covariate information may also be available, and it is of interest to exploit these auxiliary data to improve the efficiency of inferences. In this paper, we propose a 2-stage design with continuous outcome where the second-stage data is sampled with an "outcome-auxiliary-dependent sampling" (OADS) scheme. We propose an estimator which is the maximizer for an estimated likelihood function. We show that the proposed estimator is consistent and asymptotically normally distributed. The simulation study indicates that greater study efficiency gains can be achieved under the proposed 2-stage OADS design by utilizing the auxiliary covariate information when compared with other alternative sampling schemes. We illustrate the proposed method by analyzing a data set from an environmental epidemiologic study.