Maximum likelihood estimation in semiparametric regression models with censored data

Partial-linear single-index transformation models with censored data

In studies with time-to-event outcomes, multiple, inter-correlated, and time-varying covariates are commonly observed. It is of great interest to model their joint effects by allowing a flexible functional form and to delineate their relative contributions to survival risk. A class of semiparametric transformation (ST) models offers flexible specifications of the intensity function and can be a general framework to accommodate nonlinear covariate effects. In this paper, we propose a partial-linear single-index (PLSI) transformation model that reduces the dimensionality of multiple covariates into a single index and provides interpretable estimates of the covariate effects. We develop an iterative algorithm using the regression spline technique to model the nonparametric single-index function for possibly nonlinear joint effects, followed by nonparametric maximum likelihood estimation. We also propose a nonparametric testing procedure to formally examine the linearity of covariate effects. We conduct Monte Carlo simulation studies to compare the PLSI transformation model with the standard ST model and apply it to NYU Langone Health de-identified electronic health record data on COVID-19 hospitalized patients' mortality and a Veteran's Administration lung cancer trial.

Efficiency of the Breslow estimator in semiparametric transformation models

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.

Fitting the Cox proportional hazards model to big data

The semiparametric Cox proportional hazards model, together with the partial likelihood principle, has been widely used to study the effects of potentially time-dependent covariates on a possibly censored event time. We propose a computationally efficient method for fitting the Cox model to big data involving millions of study subjects. Specifically, we perform maximum partial likelihood estimation on a small subset of the whole data and improve the initial estimator by incorporating the remaining data through one-step estimation with estimated efficient score functions. We show that the final estimator has the same asymptotic distribution as the conventional maximum partial likelihood estimator using the whole dataset but requires only a small fraction of computation time. We demonstrate the usefulness of the proposed method through extensive simulation studies and an application to the UK Biobank data.

Model-based hypothesis tests for the causal mediation of semi-competing risks

Analyzing the causal mediation of semi-competing risks has become important in medical research. Semi-competing risks refers to a scenario wherein an intermediate event may be censored by a primary event but not vice versa. Causal mediation analyses decompose the effect of an exposure on the primary outcome into an indirect (mediation) effect: an effect mediated through a mediator, and a direct effect: an effect not through the mediator. Here we proposed a model-based testing procedure to examine the indirect effect of the exposure on the primary event through the intermediate event. Under the counterfactual outcome framework, we defined a causal mediation effect using counting process. To assess statistical evidence for the mediation effect, we proposed two tests: an intersection-union test (IUT) and a weighted log-rank test (WLR). The test statistic was developed from a semi-parametric estimator of the mediation effect using a Cox proportional hazards model for the primary event and a series of logistic regression models for the intermediate event. We built a connection between the IUT and WLR. Asymptotic properties of the two tests were derived, and the IUT was determined to be a size [Formula: see text] test and statistically more powerful than the WLR. In numerical simulations, both the model-based IUT and WLR can properly adjust for confounding covariates, and the Type I error rates of the proposed methods are well protected, with the IUT being more powerful than the WLR. Our methods demonstrate the strongly significant effects of hepatitis B or C on the risk of liver cancer mediated through liver cirrhosis incidence in a prospective cohort study. The proposed method is also applicable to surrogate endpoint analyses in clinical trials.

Efficacy of high-dose-rate brachytherapy with different radiation source activities among cervical cancer patients and risk factors for long-term outcomes: A 6-year retrospective study

PURPOSE

This study aimed to assess the impact of dose rates due to natural decay of Iridium-192 sources and the risk factors of clinical outcomes for cervical cancer patients treated with high-dose-rate (HDR) brachytherapy.

METHODS AND MATERIALS

Four ninety-four patients were divided into relatively-high-radioactive (rHR), relatively-medium-radioactive (rMR), and relatively-low-radioactive (rLR) groups for retrospective treatment response comparison. The short-term outcomes were evaluated using the 1-month /3-month follow-up results based on RECIST 1.1. Local recurrence-free survival (LRFS) and metastatic recurrence-free survival (MRFS) were selected as long-term outcomes. A class of transformation models with adaptive lasso was applied to assess the risk factors of long-term outcomes.

RESULTS

No significant difference was identified in short- or long-term outcomes of different radioactive groups. Subgroup analyses demonstrated similar findings. In multivariate factor analysis, advanced stage was significantly associated with higher risk of local recurrence and metastatic recurrence (HR = 1.66, 95%confidence interval [CI] = 1.14-2.43, p = 0.008; HR = 1.57, 95%CI = 1.23-2.00, p < 0.001). Significant associations were observed between local recurrence and pathology, and between metastatic recurrence and pre-treatment serum indices, respectively (HR = 8.62, 95%CI = 2.28-32.60, p = 0.002; HR = 1.98, 95%CI=1.20-2.26, p = 0.008).

CONCLUSIONS

Overall, there was no significant difference in long- or short-term efficacy of the HDR brachytherapy among the groups with different levels of activity of radiation sources. Stage, pathology, and pretreatment serum indices were crucial factors that affected the long-term outcomes.

Asymptotic Properties for Cumulative Probability Models for Continuous Outcomes

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.

A semiparametric Cox-Aalen transformation model with censored data

We propose a broad class of so-called Cox-Aalen transformation models that incorporate both multiplicative and additive covariate effects on the baseline hazard function within a transformation. The proposed models provide a highly flexible and versatile class of semiparametric models that include the transformation models and the Cox-Aalen model as special cases. Specifically, it extends the transformation models by allowing potentially time-dependent covariates to work additively on the baseline hazard and extends the Cox-Aalen model through a predetermined transformation function. We propose an estimating equation approach and devise an expectation-solving (ES) algorithm that involves fast and robust calculations. The resulting estimator is shown to be consistent and asymptotically normal via modern empirical process techniques. The ES algorithm yields a computationally simple method for estimating the variance of both parametric and nonparametric estimators. Finally, we demonstrate the performance of our procedures through extensive simulation studies and applications in two randomized, placebo-controlled human immunodeficiency virus (HIV) prevention efficacy trials. The data example shows the utility of the proposed Cox-Aalen transformation models in enhancing statistical power for discovering covariate effects.

Spearman-like correlation measure adjusting for covariates in bivariate survival data

We propose an extension of Spearman's correlation for censored continuous and discrete data that permits covariate adjustment. Previously proposed nonparametric and semiparametric Spearman's correlation estimators require either nonparametric estimation of the bivariate survival surface or parametric assumptions about the dependence structure. In practice, nonparametric estimation of the bivariate survival surface is difficult, and parametric assumptions about the correlation structure may not be satisfied. Therefore, we propose a method that requires neither and uses only the marginal survival distributions. Our method estimates the correlation of probability-scale residuals, which has been shown to equal Spearman's correlation when there is no censoring. Because this method relies only on marginal distributions, it tends to be less variable than the previously suggested nonparametric estimators, and the confidence intervals are easily constructed. Although under censoring, it is biased for Spearman's correlation as our simulations show, it performs well under moderate censoring, with a smaller mean squared error than nonparametric approaches. We also extend it to partial (adjusted), conditional, and partial-conditional correlation, which makes it particularly relevant for practical applications. We apply our method to estimate the correlation between time to viral failure and time to regimen change in a multisite cohort of persons living with HIV in Latin America.

Survival analysis with a random change-point

Contemporary works in change-point survival models mainly focus on an unknown universal change-point shared by the whole study population. However, in some situations, the change-point is plausibly individual-specific, such as when it corresponds to the telomere length or menopausal age. Also, maximum-likelihood-based inference for the fixed change-point parameter is notoriously complicated. The asymptotic distribution of the maximum-likelihood estimator is non-standard, and computationally intensive bootstrap techniques are commonly used to retrieve its sampling distribution. This article is motivated by a breast cancer study, where the disease-free survival time of the patients is postulated to be regulated by the menopausal age, which is unobserved. As menopausal age varies across patients, a fixed change-point survival model may be inadequate. Therefore, we propose a novel proportional hazards model with a random change-point. We develop a nonparametric maximum-likelihood estimation approach and devise a stable expectation-maximization algorithm to compute the estimators. Because the model is regular, we employ conventional likelihood theory for inference based on the asymptotic normality of the Euclidean parameter estimators, and the variance of the asymptotic distribution can be consistently estimated by a profile-likelihood approach. A simulation study demonstrates the satisfactory finite-sample performance of the proposed methods, which yield small bias and proper coverage probabilities. The methods are applied to the motivating breast cancer study.

Marginal proportional hazards models for multivariate interval-censored data

Multivariate interval-censored data arise when there are multiple types of events or clusters of study subjects, such that the event times are potentially correlated and when each event is only known to occur over a particular time interval. We formulate the effects of potentially time-varying covariates on the multivariate event times through marginal proportional hazards models while leaving the dependence structures of the related event times unspecified. We construct the nonparametric pseudolikelihood under the working assumption that all event times are independent, and we provide a simple and stable EM-type algorithm. The resulting nonparametric maximum pseudolikelihood estimators for the regression parameters are shown to be consistent and asymptotically normal, with a limiting covariance matrix that can be consistently estimated by a sandwich estimator under arbitrary dependence structures for the related event times. We evaluate the performance of the proposed methods through extensive simulation studies and present an application to data from the Atherosclerosis Risk in Communities Study.

Semiparametric estimation of the transformation model by leveraging external aggregate data in the presence of population heterogeneity

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.

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.

Marginal semiparametric transformation models for clustered multivariate competing risks data

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.

Semiparametric single-index models for optimal treatment regimens with censored outcomes

There is a growing interest in precision medicine, where a potentially censored survival time is often the most important outcome of interest. To discover optimal treatment regimens for such an outcome, we propose a semiparametric proportional hazards model by incorporating the interaction between treatment and a single index of covariates through an unknown monotone link function. This model is flexible enough to allow non-linear treatment-covariate interactions and yet provides a clinically interpretable linear rule for treatment decision. We propose a sieve maximum likelihood estimation approach, under which the baseline hazard function is estimated nonparametrically and the unknown link function is estimated via monotone quadratic B-splines. We show that the resulting estimators are consistent and asymptotically normal with a covariance matrix that attains the semiparametric efficiency bound. The optimal treatment rule follows naturally as a linear combination of the maximum likelihood estimators of the model parameters. Through extensive simulation studies and an application to an AIDS clinical trial, we demonstrate that the treatment rule derived from the single-index model outperforms the treatment rule under the standard Cox proportional hazards model.

Joint analysis of multivariate failure time data with latent variables

We propose a joint modeling approach to investigate the observed and latent risk factors of the multivariate failure times of interest. The proposed model comprises two parts. The first part is a distribution-free confirmatory factor analysis model that characterizes the latent factors by correlated multiple observed variables. The second part is a multivariate additive hazards model that assesses the observed and latent risk factors of the failure times. A hybrid procedure that combines the borrow-strength estimation approach and the asymptotically distribution-free generalized least square method is developed to estimate the model parameters. The asymptotic properties of the proposed estimators are derived. Simulation studies demonstrate that the proposed method performs well for practical settings. An application to a study concerning the risk factors of multiple diabetic complications is provided.

SEMIPARAMETRIC LATENT-CLASS MODELS FOR MULTIVARIATE LONGITUDINAL AND SURVIVAL DATA

In long-term follow-up studies, data are often collected on repeated measures of multivariate response variables as well as on time to the occurrence of a certain event. To jointly analyze such longitudinal data and survival time, we propose a general class of semiparametric latent-class models that accommodates a heterogeneous study population with flexible dependence structures between the longitudinal and survival outcomes. We combine nonparametric maximum likelihood estimation with sieve estimation and devise an efficient EM algorithm to implement the proposed approach. We establish the asymptotic properties of the proposed estimators through novel use of modern empirical process theory, sieve estimation theory, and semiparametric efficiency theory. Finally, we demonstrate the advantages of the proposed methods through extensive simulation studies and provide an application to the Atherosclerosis Risk in Communities study.

Bayesian transformation models with partly interval-censored data

In many scientific fields, partly interval-censored data, which consist of exactly observed and interval-censored observations on the failure time of interest, appear frequently. However, methodological developments in the analysis of partly interval-censored data are relatively limited and have mainly focused on additive or proportional hazards models. The general linear transformation model provides a highly flexible modeling framework that includes several familiar survival models as special cases. Despite such nice features, the inference procedure for this class of models has not been developed for partly interval-censored data. We propose a fully Bayesian approach coped with efficient Markov chain Monte Carlo methods to fill this gap. A four-stage data augmentation procedure is introduced to tackle the challenges presented by the complex model and data structure. The proposed method is easy to implement and computationally attractive. The empirical performance of the proposed method is evaluated through two simulation studies, and the model is then applied to a dental health study.

A semiparametric mixture model approach for regression analysis of partly interval-censored data with a cured subgroup

Failure time data with a cured subgroup are frequently confronted in various scientific fields and many methods have been proposed for their analysis under right or interval censoring. However, a cure model approach does not seem to exist in the analysis of partly interval-censored data, which consist of both exactly observed and interval-censored observations on the failure time of interest. In this article, we propose a two-component mixture cure model approach for analyzing such type of data. We employ a logistic model to describe the cured probability and a proportional hazards model to model the latent failure time distribution for uncured subjects. We consider maximum likelihood estimation and develop a new expectation-maximization algorithm for its implementation. The asymptotic properties of the resulting estimators are established and the finite sample performance of the proposed method is examined through simulation studies. An application to a set of real data on childhood mortality in Nigeria is provided.

The ROC of Cox proportional hazards cure models with application in cancer studies

With recent advancement in cancer screening and treatment, many patients with cancers are identified at early stage and clinically cured. Importantly, uncured patients should be treated timely before the cancer progresses to advanced stages for which therapeutic options are rather limited. It is also crucial to identify uncured subjects among patients with early-stage cancers for clinical trials to develop effective adjuvant therapies. Thus, it is of interest to develop statistical predictive models with as high accuracy as possible in predicting the latent cure status. The receiver operating characteristic curve (ROC) and the area under the ROC curve (AUC) are among the most widely used statistical metrics for assessing predictive accuracy or discriminatory power for a dichotomous outcome (cured/uncured). Yet the conventional AUC cannot be directly used due to incompletely observed cure status. In this article, we proposed new estimates of the ROC curve and its AUC for predicting latent cure status in Cox proportional hazards (PH) cure models and transformation cure models. We developed explicit formulas to estimate sensitivity, specificity, the ROC and its AUC without requiring to know the patient cure status. We also developed EM type estimates to approximate sensitivity, specificity, ROC and AUC conditional on observed data. Numerical studies were used to assess their finite-sample performance of the proposed methods. Both methods are consistent and have similar efficiency as shown in our numerical studies. A melanoma dataset was used to demonstrate the utility of the proposed estimates of the ROC curve for the latent cure status. We also have developed an [Formula: see text] package called [Formula: see text] to efficiently compute the proposed estimates.

Semiparametric linear transformation models for indirectly observed outcomes

We propose a regression framework to analyze outcomes that are indirectly observed via one or multiple proxies. Semiparametric transformation models, including Cox proportional hazards regression, turn out to be well suited to model the association between the covariates and the unobserved outcome. By coupling this regression model to a semiparametric measurement model, we can estimate these associations without requiring calibration data and without imposing strong functional assumptions on the relationship between the unobserved outcome and its proxy. When multiple proxies are available, we propose a data-driven aggregation resulting in an improved proxy. We empirically validate the proposed methodology in a simulation study, revealing good finite sample properties, especially when multiple proxies are aggregated. The methods are demonstrated on two case studies.

On semiparametric modelling, estimation and inference for survival data subject to dependent censoring

When modelling survival data, it is common to assume that the survival time $T$ is conditionally independent of the censoring time $C$ given a set of covariates. However, there are numerous situations in which this assumption is not realistic. The goal of this paper is therefore to develop a semiparametric normal transformation model which assumes that, after a proper nonparametric monotone transformation, the vector $(T, C)$ follows a linear model, and the vector of errors in this bivariate linear model follows a standard bivariate normal distribution with a possibly nondiagonal covariance matrix. We show that this semiparametric model is identifiable, and propose estimators of the nonparametric transformation, the regression coefficients and the correlation between the error terms. It is shown that the estimators of the model parameters and the transformation are consistent and asymptotically normal. We also assess the finite-sample performance of the proposed method by comparing it with an estimation method under a fully parametric model. Finally, our method is illustrated using data from the AIDS Clinical Trial Group 175 study.

Lung function over the life course of paediatric and adult patients with cystic fibrosis from a large multi-centre registry

A key measure of lung function in people with Cystic Fibrosis (CF) is Forced Expiratory Volume in the first second FEV₁ percent predicted (FEV₁pp). This study aimed to address challenges in identifying predictors of FEV₁pp, specifically dealing with non-linearity and the censoring effect of death. Data was obtained from a large multi-centre Australian Cystic Fibrosis Data Registry (ACFDR). A linear mixed model was used to study FEV₁pp as the endpoint. There were 3655 patients (52.4% male) included in our study. Restricted cubic splines were used to fit the non-linear relationship between age of visit and FEV₁pp. The following predictors were found to be significant in the multivariate model: age of patient at visit, BMI z-score, age interaction with lung transplantation, insulin dependent diabetes, cirrhosis/portal hypertension, pancreatic insufficiency, Pseudomonas aeruginosa infection and baseline variability in FEV₁pp. Those with P. aeruginosa infection had a lower mean difference in FEV₁pp of 4.7 units, p < 0.001 compared to those who did not have the infection. Joint modelling with mortality outcome did not materially affect our findings. These models will prove useful for to study the impact of CFTR modulator therapies on rate of change of lung function among patients with CF.

Semiparametric copula-based analysis for treatment effects in the presence of treatment switching

In controlled trials, "treatment switching" occurs when patients in one treatment group switch to alternative treatments during the trial, and poses challenges to treatment effect evaluation owing to crossover of the treatments groups. In this work, we assume that treatment switching can occur after some disease progression event and view the progression and death events as two semicompeting risks. The proposed model consists of a copula model for the joint distribution of time-to-progression (TTP) and overall survival (OS) up to the earlier of the two events, as well as a conditional hazard model for OS subsequent to progression. The copula model facilitates assessing the marginal distributions of TTP and OS separately from the association between the two events, and, in particular, the treatment effect on OS in the absence of treatment switching. The proposed conditional hazard model for death subsequent to progression allows us to assess the treatment switching (crossover) effect on OS given occurrence of progression and covariates. Semiparametric proportional hazards models are employed in the marginal models for TTP and OS. A nonparametric maximum likelihood procedure is developed for model inference, which is verified through asymptotic theory and simulation studies. The proposed analysis is applied to a lung cancer dataset to illustrate its real utility.

Ensemble estimation and variable selection with semiparametric regression models

We consider scenarios in which the likelihood function for a semiparametric regression model factors into separate components, with an efficient estimator of the regression parameter available for each component. An optimal weighted combination of the component estimators, named an ensemble estimator, may be employed as an overall estimate of the regression parameter, and may be fully efficient under uncorrelatedness conditions. This approach is useful when the full likelihood function may be difficult to maximize, but the components are easy to maximize. It covers settings where the nuisance parameter may be estimated at different rates in the component likelihoods. As a motivating example we consider proportional hazards regression with prospective doubly censored data, in which the likelihood factors into a current status data likelihood and a left-truncated right-censored data likelihood. Variable selection is important in such regression modelling, but the applicability of existing techniques is unclear in the ensemble approach. We propose ensemble variable selection using the least squares approximation technique on the unpenalized ensemble estimator, followed by ensemble re-estimation under the selected model. The resulting estimator has the oracle property such that the set of nonzero parameters is successfully recovered and the semiparametric efficiency bound is achieved for this parameter set. Simulations show that the proposed method performs well relative to alternative approaches. Analysis of an AIDS cohort study illustrates the practical utility of the method.

An empirical comparison of two novel transformation models

Continuous response variables are often transformed to meet modeling assumptions, but the choice of the transformation can be challenging. Two transformation models have recently been proposed: semiparametric cumulative probability models (CPMs) and parametric most likely transformation models (MLTs). Both approaches model the cumulative distribution function and require specifying a link function, which implicitly assumes that the responses follow a known distribution after some monotonic transformation. However, the two approaches estimate the transformation differently. With CPMs, an ordinal regression model is fit, which essentially treats each continuous response as a unique category and therefore nonparametrically estimates the transformation; CPMs are semiparametric linear transformation models. In contrast, with MLTs, the transformation is parameterized using flexible basis functions. Conditional expectations and quantiles are readily derived from both methods on the response variable's original scale. We compare the two methods with extensive simulations. We find that both methods generally have good performance with moderate and large sample sizes. MLTs slightly outperformed CPMs in small sample sizes under correct models. CPMs tended to be somewhat more robust to model misspecification and outcome rounding. Except in the simplest situations, both methods outperform basic transformation approaches commonly used in practice. We apply both methods to an HIV biomarker study.

Statistical Generalized Derivative Applied to the Profile Likelihood Estimation in a Mixture of Semiparametric Models

There is a difficulty in finding an estimate of the standard error (SE) of the profile likelihood estimator in the joint model of longitudinal and survival data. The difficulty is on the differentiation of an implicit function that appear in the profile likelihood estimation. We solve the difficulty by introducing the "statistical generalized derivative". The derivative is used to show the asymptotic normality of the estimator with the SE expressed in terms of the profile likelihood score function.

Testing and Estimation of Social Network Dependence With Time to Event Data

Nowadays, events are spread rapidly along social networks. We are interested in whether people's responses to an event are affected by their friends' characteristics. For example, how soon will a person start playing a game given that his/her friends like it? Studying social network dependence is an emerging research area. In this work, we propose a novel latent spatial autocorrelation Cox model to study social network dependence with time-to-event data. The proposed model introduces a latent indicator to characterize whether a person's survival time might be affected by his or her friends' features. We first propose a score-type test for detecting the existence of social network dependence. If it exists, we further develop an EM-type algorithm to estimate the model parameters. The performance of the proposed test and estimators are illustrated by simulation studies and an application to a time-to-event dataset about playing a popular mobile game from one of the largest online social network platforms. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.

Dynamic landmark prediction for mixture data

In kin-cohort studies, clinicians want to provide their patients with the most current cumulative risk of death arising from a rare deleterious mutation. Estimating the cumulative risk is difficult when the genetic mutation status is unknown and only estimated probabilities of a patient having the mutation are available. We estimate the cumulative risk for this scenario using a novel nonparametric estimator that incorporates covariate information and dynamic landmark prediction. Our estimator has improved prediction accuracy over existing estimators that ignore covariate information. It is built within a dynamic landmark prediction framework whereby we can obtain personalized dynamic predictions over time. Compared to current standards, a simple transformation of our estimator provides more efficient estimates of marginal distribution functions in settings where patient-specific predictions are not the main goal. We show our estimator is unbiased and has more predictive accuracy compared to methods that ignore covariate information and landmarking. Applying our method to a Huntington disease study of mortality, we develop dynamic survival prediction curves incorporating gender and familial genetic information.

Penalized estimation of semiparametric transformation models with interval-censored data and application to Alzheimer's disease

Variable selection or feature extraction is fundamental to identify important risk factors from a large number of covariates and has applications in many fields. In particular, its applications in failure time data analysis have been recognized and many methods have been proposed for right-censored data. However, developing relevant methods for variable selection becomes more challenging when one confronts interval censoring that often occurs in practice. In this article, motivated by an Alzheimer's disease study, we develop a variable selection method for interval-censored data with a general class of semiparametric transformation models. Specifically, a novel penalized expectation-maximization algorithm is developed to maximize the complex penalized likelihood function, which is shown to perform well in the finite-sample situation through a simulation study. The proposed methodology is then applied to the interval-censored data arising from the Alzheimer's disease study mentioned above.