Case-cohort analysis with semiparametric transformation models

Semi-parametric transformation models provide a general and flexible class of models for regression analysis of failure time data and many methods have been developed for their estimation. In particular, they include the proportional hazards and proportional odds models as special cases. In this paper, we discuss the situation where one observes left-truncated and interval-censored data, for which it does not seem to exist an established method. For the problem, in contrast to the commonly used conditional approach that may not be efficient, a pairwise pseudo-likelihood method is proposed to recover some missing information in the conditional method. The proposed estimators are proved to be consistent and asymptotically efficient and normal. A simulation study is conducted to assess the empirical performance of the method and suggests that it works well in practical situations. This method is illustrated by using a set of real data arising from an HIV/AIDS cohort study.

A semiparametric transformation model comprises a parametric component for covariate effects and a nonparametric component for the baseline hazard/intensity. The Breslow-type estimator has been proposed for estimating the nonparametric component in some inefficient estimation procedures. We show that introducing weights into this estimator leads to nonparametric maximum likelihood estimation, with the weights depending on the martingale residuals. The weighted Breslow-type estimator suggests an iterative reweighting algorithm for nonparametric maximum likelihood estimation, which can be implemented by a weighted variant of the existing algorithms for inefficient estimation, and can be computationally more efficient than an <scp>em</scp>-type algorithm. The weighting idea is further extended to semiparametric transformation models with mismeasured covariates.

Among several semiparametric models, the Cox proportional hazard model is widely used to assess the association between covariates and the time-to-event when the observed time-to-event is interval-censored. Often, covariates are measured with error. To handle this covariate uncertainty in the Cox proportional hazard model with the interval-censored data, flexible approaches have been proposed. To fill a gap and broaden the scope of statistical applications to analyze time-to-event data with different models, in this paper, a general approach is proposed for fitting the semiparametric linear transformation model to interval-censored data when a covariate is measured with error. The semiparametric linear transformation model is a broad class of models that includes the proportional hazard model and the proportional odds model as special cases. The proposed method relies on a set of estimating equations to estimate the regression parameters and the infinite-dimensional parameter. For handling interval censoring and covariate measurement error, a flexible imputation technique is used. Finite sample performance of the proposed method is judged via simulation studies. Finally, the suggested method is applied to analyze a real data set from an AIDS clinical trial.

Survival data are subject to length-biased sampling when the survival times are left-truncated and the underlying truncation time random variable is uniformly distributed. Substantial efficiency gains can be achieved by incorporating the information about the truncation time distribution in the estimation procedure [Wang (1989) Journal of the American Statistical Association 84, 742-748; Wang (1996) Biometrika 83, 343-354]. Under the semiparametric transformation models, the maximum likelihood method is expected to be fully efficient, yet it is difficult to implement because the full likelihood depends on the nonparametric component in a complicated way. Moreover, its asymptotic properties have not been established. In this article, we extend the martingale estimating equation approach [Chen et al. (2002) Biometrika 89, 659-668; Kim et al. (2013) Journal of the American Statistical Association 108, 217-227] and the pseudo-partial likelihood approach [Severini and Wong (1992) The Annals of Statistics 4, 1768-1802; Zucker (2005) Journal of the American Statistical Association 100, 1264-1277] for semiparametric transformation models with right-censored data to handle left-truncated and right-censored data. In the same spirit of the composite likelihood method [Huang and Qin (2012) Journal of the American Statistical Association 107, 946-957], we further construct another set of unbiased estimating equations by exploiting the special probability structure of length-biased sampling. Thus the number of estimating equations exceeds the number of parameters, and efficiency gains can be achieved by solving a simple combination of these estimating equations. The proposed methods are easy to implement as they do not require additional programming efforts. Moreover, they are shown to be consistent and asymptotically normally distributed. A data analysis of a dementia study illustrates the methods.

This paper discusses regression analysis of interval-censored failure time data arising from semiparametric transformation models in the presence of missing covariates. Although some methods have been developed for the problem, they either apply only to limited situations or may have some computational issues. Corresponding to these, we propose a new and unified two-step inference procedure that can be easily implemented using the existing or standard software. The proposed method makes use of a set of working models to extract partial information from incomplete observations and yields a consistent estimator of regression parameters assuming missing at random. An extensive simulation study is conducted and indicates that it performs well in practical situations. Finally, we apply the proposed approach to an Alzheimer's Disease study that motivated this study.

The cumulative incidence is the probability of failure from the cause of interest over a certain time period in the presence of other risks. A semiparametric regression model proposed by Fine and Gray (1999) has become the method of choice for formulating the effects of covariates on the cumulative incidence. Its estimation, however, requires modeling of the censoring distribution and is not statistically efficient. In this paper, we present a broad class of semiparametric transformation models which extends the Fine and Gray model, and we allow for unknown causes of failure. We derive the nonparametric maximum likelihood estimators (NPMLEs) and develop simple and fast numerical algorithms using the profile likelihood. We establish the consistency, asymptotic normality, and semiparametric efficiency of the NPMLEs. In addition, we construct graphical and numerical procedures to evaluate and select models. Finally, we demonstrate the advantages of the proposed methods over the existing ones through extensive simulation studies and an application to a major study on bone marrow transplantation.

The "semicompeting risks" include a terminal event and a non-terminal event. The terminal event may censor the non-terminal event but not vice versa. Because times to the two events are usually correlated, the non-terminal event is subject to dependent/informative censoring by the terminal event. We seek to conduct marginal regressions and joint association analyses for the two event times under semicompeting risks. The proposed method is based on the modeling setup where the semiparametric transformation models are assumed for marginal regressions, and a copula model is assumed for the joint distribution. We propose a nonparametric maximum likelihood approach for inferences, which provides a martingale representation for the score function and an analytical expression for the information matrix. Direct theoretical developments and computational implementation are allowed for the proposed approach. Simulations and a real data application demonstrate the utility of the proposed methodology.

We recently proposed a representation of the bivariate survivor function as a mapping of the hazard function for truncated failure time variates. The representation led to a class of estimators that includes van der Laan's repaired nonparametric maximum likelihood estimator (NPMLE) as an important special case. We proposed a Greenwood-like variance estimator for the repaired NPMLE but found somewhat poor agreement between the empirical variance estimates and these analytic estimates for the sample sizes and bandwidths considered in our simulation study. The simulation results also confirmed those of others in showing slightly inferior performance for the repaired NPMLE compared to other competing estimators as well as a sensitivity to bandwidth choice in moderate sized samples. Despite its attractive asymptotic properties, the repaired NPMLE has drawbacks that hinder its practical application. This paper presents a modification of the repaired NPMLE that improves its performance in moderate sized samples and renders it less sensitive to the choice of bandwidth. Along with this modified estimator, more extensive simulation studies of the repaired NPMLE and Greenwood-like variance estimates are presented. The methods are then applied to a real data example.

In this paper, we aim to develop a semiparametric transformation model. Nonparametric transformation functions are modeled with Bayesian P-splines. The transformed variables can be fitted to a general nonlinear mixed model, including linear or nonlinear regression models, mixed effect models, factor analysis models, and other latent variable models as special cases. Markov chain Monte Carlo algorithms are implemented to estimate transformation functions and unknown quantities in the model. The performance of the developed methodology is demonstrated with a simulation study. Its application to a real study on polydrug use is presented.

I suggest an extension of the semiparametric transformation model that specifies a time-varying regression structure for the transformation, and thus allows time-varying structure in the data. Special cases include a stratified version of the usual semiparametric transformation model. The model can be thought of as specifying a first order Taylor expansion of a completely flexible baseline. Large sample properties are derived and estimators of the asymptotic variances of the regression coefficients are given. The method is illustrated by a worked example and a small simulation study. A goodness of fit procedure for testing if the regression effects lead to a satisfactory fit is also suggested.

We consider causal inference in randomized survival studies with right censored outcomes and all-or-nothing compliance, using semiparametric transformation models to estimate the distribution of survival times in treatment and control groups, conditional on covariates and latent compliance type. Estimands depending on these distributions, for example, the complier average causal effect (CACE), the complier effect on survival beyond time t, and the complier quantile effect are then considered. Maximum likelihood is used to estimate the parameters of the transformation models, using a specially designed expectation-maximization (EM) algorithm to overcome the computational difficulties created by the mixture structure of the problem and the infinite dimensional parameter in the transformation models. The estimators are shown to be consistent, asymptotically normal, and semiparametrically efficient. Inferential procedures for the causal parameters are developed. A simulation study is conducted to evaluate the finite sample performance of the estimated causal parameters. We also apply our methodology to a randomized study conducted by the Health Insurance Plan of Greater New York to assess the reduction in breast cancer mortality due to screening.

We investigate efficient estimation strategies for partially linear transformation models with time-dependent covariates under interval censoring. The unknown monotone function is approximated using a monotone -spline basis to enable flexible semiparametric modeling, and we develop a computationally efficient nested hybrid EM algorithm that integrates Newton's method with isotonic regression. To support large-sample inference, we propose a straightforward variance-covariance estimation procedure for the regression parameters and introduce a score test to assess the adequacy of the proportional hazards (PH) specification within the broader class of transformation models. The numerical performance of the penalized estimators is examined extensively and compared with both the time-invariant covariate model by Lu et al. and the semiparametric transformation model by Zeng et al. Finally, the proposed methodology is applied to data from the National Alzheimer's Coordinating Center (NACC) to demonstrate its practical utility in a real-world clinical setting.

IntroductionRandomized controlled trial (RCT) data are analyzed by two challengeable adjusted and non-adjusted approaches. Performing appropriate adjusted analysis leads to a more interpretable and efficient estimation of treatment effects. Semiparametric adjustment approach modifies the estimating equations solved by the marginal treatment effect estimator by adding an augmentation function, which makes use of the baseline covariates and estimate the unbiased marginal treatment effect with improved precision. The effect of the intervention obtained using the semi-parametric adjustment method, similar to the unadjusted method and contrary to the adjusted parametric method, is marginal, resulting in better interpretability. Moreover, due to leveraging baseline covariates, it is more efficient compared to the unadjusted models. This study aimed to estimate the effect of beractant (Beraksurf™, Tekzima Company), compared with the Poractant alfa (Curosurf®, Chiesi Pharmaceuticals), as surfactant replacement therapy, on the need for Intermittent Positive Pressure Ventilation (IPPV) in Neonatal Respiratory Distress Syndrome (NRDS) more precisely by fitting a semi-parametric efficient model adjusted for appropriate covariates.MethodThis study is secondary and we re-analyzing data of a published RCT. This RCT was conducted in the NICU of Alzahra Hospital in Tabriz, Iran for eight months, and 200 infants were assigned to two groups receiving either 100 mg/kg BeraksurfTM (n = 99) or 200 mg/kg Curosurf® (n = 101). The effect of the treatments was evaluated regarding the need for IPPV by fitting semi-parametric logistic regression models, adjusted for the best subset of covariates selected by the forward variable selection algorithm and confounders identified by the expert panel. IPPV in our study was administered via an endotracheal tube, as per the protocol followed in the primary trial. The need for IPPV was determined based on the clinical judgment of neonatologists, considering the infants' respiratory distress levels, oxygen saturation, and arterial blood gas measurements. The decision was made within the first 72 h after surfactant administration.ResultsThe proportions of the required IPPV outcome were 29.3% and 59.4% in the BeraksurfTM group and Curosurf®, respectively. While the unadjusted comparison between the two treatments was significant (OR = .283, 95% CI: (.157, .509), P-value < .001), adjusted OR in semi-parametric logistic regression by adjusting for the best subset of covariates selected by forward method including steroids, Apgar score at min1, and initial Fio2 wasn’t significant (OR = .751, 95% CI: (.510, 1.111), P-value = .151). The efficacy of this semi-parametric model over the unadjusted model was 1.81. The results of the expert-based adjusted model, adjusting for the gestational age, birth weight, and initial FiO2, were consistent with those derived from the R2-based selection, supporting the robustness of our findings (OR = .685, 95% CI: (.449, 1.529), P-value = .078).ConclusionFitting a semi-parametric model adjusted for the baseline covariates resulted in a marginal, unbiased, more efficient and interpretable estimation of BeraksurfTM versus Curosurf® effects. Although the crude model showed that BeraksurfTM was more effective than Curosurf®, the results of the efficient semi-parametric model with adjustment for the best subset of covariates revealed no statistically significant difference between the two drugs regarding their effects. We hope that the use of this method and its findings will contribute to a better understanding of covariate adjustment.

The linear transformation model is a semiparametric model which contains the Cox proportional hazards model and the proportional odds model as special cases. Cai et al. (Biometrika 87:867-878, 2000) have proposed an inference procedure for the linear transformation model with correlated censored observations. In this article, we develop formal and graphical model checking techniques for the linear transformation models based on cumulative sums of martingale-type residuals. The proposed method is illustrated with a clinical trial data.

Change point models are widely used in medical and epidemiological studies to capture the threshold effects of continuous covariates on health outcomes. These threshold effects represent critical points at which the relationship between biomarkers or risk factors and disease risk shifts, often reflecting underlying biological mechanisms or clinically relevant intervention points. While most existing methods focus on right-censored data, interval censoring is common in large-scale clinical trials and follow-up studies, where the exact event times are not observed but are known to fall within time intervals. In this paper, we propose a semiparametric transformation model with an unknown change point for interval-censored data. The model allows flexible transformation functions, including the proportional hazards and proportional odds models, and it accommodates both main effects and their interactions with the threshold variable. Model parameters are estimated via the EM algorithm, with the change point identified through a profile likelihood approach using grid search. We establish the asymptotic properties of the proposed estimators and evaluate their finite-sample performance through extensive simulations, showing good accuracy and coverage properties. The method is further illustrated through an application to the Prostate, Lung, Colorectal, and Ovarian (PLCO) Cancer Screening Trial data.