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The transformation model with partly interval-censored data offers a highly flexible modeling framework that can simultaneously support multiple common survival models and a wide variety of censored data types. However, the real data may contain unexplained heterogeneity that cannot be entirely explained by covariates and may be brought on by a variety of unmeasured regional characteristics. Due to this, we introduce the conditionally autoregressive prior into the transformation model with partly interval-censored data and take the spatial frailty into account. An efficient Markov chain Monte Carlo method is proposed to handle the posterior sampling and model inference. The approach is simple to use and does not include any challenging Metropolis steps owing to four-stage data augmentation. Through several simulations, the suggested method's empirical performance is assessed and then the method is used in a leukemia study.

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Semiparametric transformation models for failure time data consist of a parametric regression component and an unspecified cumulative baseline hazard. The nonparametric maximum likelihood estimator (NPMLE) of the cumulative baseline hazard can be summarized in terms of weights introduced into a Breslow-type estimator (Weighted Breslow). At any given time point, the weights invoke an integral over the future of the cumulative baseline hazard, which presents theoretical and computational challenges. A simpler non-MLE Breslow-type estimator (Breslow) was derived earlier from a martingale estimating equation (MEE) setting observed and expected counts of failures equal, conditional on the past history. Despite much successful theoretical and computational development, the simpler Breslow estimator continues to be commonly used as a compromise between simplicity and perceived loss of full efficiency. In this paper we derive the relative efficiency of the Breslow estimator and consider the properties of the two estimators using simulations and real data on prostate cancer survival.

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We aim to evaluate the marginal effects of covariates on time-to-disability in the elderly under the semi-competing risks framework, as death dependently censors disability, not vice versa. It becomes particularly challenging when time-to-disability is subject to interval censoring due to intermittent assessments. A left truncation issue arises when the age time scale is applied. We develop a flexible two-parameter copula-based semiparametric transformation model for semi-competing risks data subject to interval censoring and left truncation. The two-parameter copula quantifies both upper and lower tail dependence between two margins. The semiparametric transformation models incorporate proportional hazards and proportional odds models in both margins. We propose a two-step sieve maximum likelihood estimation procedure and study the sieve estimators' asymptotic properties. Simulations show that the proposed method corrects biases in the marginal method. We demonstrate the proposed method in a large-scale Chinese Longitudinal Healthy Longevity Study and provide new insights into preventing disability in the elderly. The proposed method could be applied to the general semi-competing risks data with intermittently assessed disease status.

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Joint analysis of recurrent and nonrecurrent terminal events has attracted substantial attention in literature. However, there lacks formal methodology for such analysis when the event time data are on discrete scales, even though some modeling and inference strategies have been developed for discrete-time survival analysis. We propose a discrete-time joint modeling approach for the analysis of recurrent and terminal events where the two types of events may be correlated with each other. The proposed joint modeling assumes a shared frailty to account for the dependence among recurrent events and between the recurrent and the terminal terminal events. Also, the joint modeling allows for time-dependent covariates and rich families of transformation models for the recurrent and terminal events. A major advantage of our approach is that it does not assume a distribution for the frailty, nor does it assume a Poisson process for the analysis of the recurrent event. The utility of the proposed analysis is illustrated by simulation studies and two real applications, where the application to the biochemists' rank promotion data jointly analyzes the biochemists' citation numbers and times to rank promotion, and the application to the scleroderma lung study data jointly analyzes the adverse events and off-drug time among patients with the symptomatic scleroderma-related interstitial lung disease.

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Leveraging information in aggregate data from external sources to improve estimation efficiency and prediction accuracy with smaller scale studies has drawn a great deal of attention in recent years. Yet, conventional methods often either ignore uncertainty in the external information or fail to account for the heterogeneity between internal and external studies. This article proposes an empirical likelihood-based framework to improve the estimation of the semiparametric transformation models by incorporating information about the t-year subgroup survival probability from external sources. The proposed estimation procedure incorporates an additional likelihood component to account for uncertainty in the external information and employs a density ratio model to characterize population heterogeneity. We establish the consistency and asymptotic normality of the proposed estimator and show that it is more efficient than the conventional pseudopartial likelihood estimator without combining information. Simulation studies show that the proposed estimator yields little bias and outperforms the conventional approach even in the presence of information uncertainty and heterogeneity. The proposed methodologies are illustrated with an analysis of a pancreatic cancer study.

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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.

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Regression models for continuous outcomes frequently require a transformation of the outcome, which is often specified a priori or estimated from a parametric family. Cumulative probability models (CPMs) nonparametrically estimate the transformation by treating the continuous outcome as if it is ordered categorically. They thus represent a flexible analysis approach for continuous outcomes. However, it is difficult to establish asymptotic properties for CPMs due to the potentially unbounded range of the transformation. Here we show asymptotic properties for CPMs when applied to slightly modified data where bounds, one lower and one upper, are chosen and the outcomes outside the bounds are set as two ordinal categories. We prove the uniform consistency of the estimated regression coefficients and of the estimated transformation function between the bounds. We also describe their joint asymptotic distribution, and show that the estimated regression coefficients attain the semiparametric efficiency bound. We show with simulations that results from this approach and those from using the CPM on the original data are very similar when a small fraction of the data are modified. We reanalyze a dataset of HIV-positive patients with CPMs to illustrate and compare the approaches.

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Multivariate survival models are often used in studying multiple outcomes for right-censored data. However, the outcomes of interest often have competing risks, where standard multivariate survival models may lead to invalid inferences. For example, patients who had stem cell transplantation may experience multiple types of infections after transplant while reconstituting their immune system, where death without experiencing infections is a competing risk for infections. Such competing risks data often suffer from cluster effects due to a matched pair design or correlation within study centers. The cumulative incidence function (CIF) is widely used to summarize competing risks outcomes. Thus, it is often of interest to study direct covariate effects on the CIF. Most literature on clustered competing risks data analyses is limited to the univariate proportional subdistribution hazards model with inverse probability censoring weighting which requires correctly specifying the censoring distribution. We propose a marginal semiparametric transformation model for multivariate competing risks outcomes. The proposed model does not require modeling the censoring distribution, accommodates nonproportional subdistribution hazards structure, and provides a platform for joint inference of all causes and outcomes.

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This article discusses regression analysis of clustered interval-censored failure time data in the presence of a cured fraction or subgroup. Such data often occur in many areas, including epidemiological studies, medical studies, and social sciences. For the problem, a class of semiparametric transformation nonmixture cure models is presented and for estimation, the maximum likelihood estimation procedure is derived. For the implementation of the proposed method, we develop a novel EM algorithm based on a Poisson variable-based augmentation. An extensive simulation study is conducted and suggests that the proposed approach works well in practical situations. Finally the method is applied to an example that motivated this study.

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Left-truncation poses extra challenges for the analysis of complex time-to-event data. We propose a general semiparametric regression model for left-truncated and right-censored competing risks data that is based on a novel weighted conditional likelihood function. Targeting the subdistribution hazard, our parameter estimates are directly interpretable with regard to the cumulative incidence function. We compare different weights from recent literature and develop a heuristic interpretation from a cure model perspective that is based on pseudo risk sets. Our approach accommodates external time-dependent covariate effects on the subdistribution hazard. We establish consistency and asymptotic normality of the estimators and propose a sandwich estimator of the variance. In comprehensive simulation studies we demonstrate solid performance of the proposed method. Comparing the sandwich estimator with the inverse Fisher information matrix, we observe a bias for the inverse Fisher information matrix and diminished coverage probabilities in settings with a higher percentage of left-truncation. To illustrate the practical utility of the proposed method, we study its application to a large HIV vaccine efficacy trial dataset.

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Causal mediation analysis aims to investigate the mechanism linking an exposure and an outcome. However, studies regarding mediation effects on survival outcomes are limited, particularly in multi-mediator settings. The existing multi-mediator analyses for survival outcomes are either performed under special model specifications such as probit models or additive hazard models, or they assume a rare outcome. Here, we propose a novel multi-mediation analysis based on the widely used Cox proportional hazards model without the rare outcome assumption. We develop a methodology under a counterfactual framework to identify path-specific effects (PSEs) of the exposure on the outcome through the mediator(s) and derive the closed-form formula for PSEs on a transformed survival time. Moreover, we show that the convolution of an extreme value and Gaussian random variables converges to another Gaussian, provided that the variance of the original Gaussian gets large. Based on that, we further derive closed-form expressions for PSEs on survival probabilities. Asymptotic properties are established for both estimators. Extensive simulation is conducted to evaluate the finite sample performance of our proposed estimators and to compare with existing methods. The utility of the proposed method is illustrated in a hepatitis study of liver cancer risk.

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Left-truncated data often arise in epidemiology and individual follow-up studies due to a biased sampling plan since subjects with shorter survival times tend to be excluded from the sample. Moreover, the survival time of recruited subjects are often subject to right censoring. In this article, a general class of semiparametric transformation models that include proportional hazards model and proportional odds model as special cases is studied for the analysis of left-truncated and right-censored data. We propose a conditional likelihood approach and develop the conditional maximum likelihood estimators (cMLE) for the regression parameters and cumulative hazard function of these models. The derived score equations for regression parameter and infinite-dimensional function suggest an iterative algorithm for cMLE. The cMLE is shown to be consistent and asymptotically normal. The limiting variances for the estimators can be consistently estimated using the inverse of negative Hessian matrix. Intensive simulation studies are conducted to investigate the performance of the cMLE. An application to the Channing House data is given to illustrate the methodology.

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It is desirable to adjust Spearman's rank correlation for covariates, yet existing approaches have limitations. For example, the traditionally defined partial Spearman's correlation does not have a sensible population parameter, and the conditional Spearman's correlation defined with copulas cannot be easily generalized to discrete variables. We define population parameters for both partial and conditional Spearman's correlation through concordance-discordance probabilities. The definitions are natural extensions of Spearman's rank correlation in the presence of covariates and are general for any orderable random variables. We show that they can be neatly expressed using probability-scale residuals (PSRs). This connection allows us to derive simple estimators. Our partial estimator for Spearman's correlation between X and Y adjusted for Z is the correlation of PSRs from models of X on Z and of Y on Z, which is analogous to the partial Pearson's correlation derived as the correlation of observed-minus-expected residuals. Our conditional estimator is the conditional correlation of PSRs. We describe estimation and inference, and highlight the use of semiparametric cumulative probability models, which allow preservation of the rank-based nature of Spearman's correlation. We conduct simulations to evaluate the performance of our estimators and compare them with other popular measures of association, demonstrating their robustness and efficiency. We illustrate our method in two applications, a biomarker study and a large survey.

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We study the application of a widely used ordinal regression model, the cumulative probability model (CPM), for continuous outcomes. Such models are attractive for the analysis of continuous response variables because they are invariant to any monotonic transformation of the outcome and because they directly model the cumulative distribution function from which summaries such as expectations and quantiles can easily be derived. Such models can also readily handle mixed type distributions. We describe the motivation, estimation, inference, model assumptions, and diagnostics. We demonstrate that CPMs applied to continuous outcomes are semiparametric transformation models. Extensive simulations are performed to investigate the finite sample performance of these models. We find that properly specified CPMs generally have good finite sample performance with moderate sample sizes, but that bias may occur when the sample size is small. Cumulative probability models are fairly robust to minor or moderate link function misspecification in our simulations. For certain purposes, the CPMs are more efficient than other models. We illustrate their application, with model diagnostics, in a study of the treatment of HIV. CD4 cell count and viral load 6 months after the initiation of antiretroviral therapy are modeled using CPMs; both variables typically require transformations, and viral load has a large proportion of measurements below a detection limit.

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Receiver operating characteristic curves and the area under the curves (AUC) are often used to compare the discriminatory ability of potentially correlated biomarkers. Many biomarkers are subject to limit of detection due to the instrumental limitation in measurements and may not be normally distributed. Standard parametric methods assuming normality can lead to biased results when the normality assumption is violated. We propose new estimation and inference procedures for the AUCs of biomarkers subject to limit of detection by using the semiparametric transformation model allowing for heteroscedasticity. We obtain the nonparametric maximum likelihood estimators by maximizing the likelihood for the observed data with limit of detection. The proposed estimators are shown to be consistent, asymptotically normal, and asymptotically efficient. Additionally, we propose a Wald type test statistic to compare the AUCs of 2 potentially correlated biomarkers with limit of detection. Extensive simulation studies demonstrate that the proposed method is robust to nonnormality while performing as well as its parametric counterpart when the normality assumption is true. An application to an autism study is provided.

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