I review some key ideas and models in survival analysis with emphasis on modeling the effects of covariates on survival times. I focus on the proportional hazards model of Cox (J R Stat Soc B 34:187-220, 1972), its extensions and alternatives, including the accelerated life model. I briefly describe some models for competing risks data, multiple and repeated event-time data and multivariate survival data.

In studies of affective disorder, individuals are often observed to experience recurrent symptomatic exacerbations warranting hospitalization. Interest may lie in modeling the occurrence of such exacerbations over time and identifying associated risk factors. In some patients, recurrent exacerbations are temporally clustered following disease onset, but cease to occur after a period of time. We develop a dynamic Mover-Stayer model in which a canonical binary variable associated with each event indicates whether the underlying disease has resolved. An individual whose disease process has not resolved will experience events following a standard point process model governed by a latent intensity. When the disease process resolves, the complete data intensity becomes zero and no further event will occur. An expectation-maximization algorithm is described for parametric and semiparametric model fitting based on a discrete time dynamic Mover-Stayer model and a latent intensity-based model of the underlying point process.

Errors were made in copying the Fix–Neyman formulas in equations (5) and (6). In Eq. (5), change “λ1 + λ2” to “λ1λ2”. In one line below, change “λ1 and λ2” to “−λ1 and−λ2”. In lines 2 and 3 below Eq. (5), in the square root, change “−4q01q10” to “+4q01q10”. In line 4 below, in α1, change “λ1−λ2” to “λ2−λ1”, and in α2, change the difference “−” to sum “+”. Delete the next sentence inside the parentheses. Insert “(−λ1,b,−λ2,b)” after “eigenvalues”. In Eq. (6), change “λ1,b + λ2,b” to product “λ1,bλ2,b”. One line below Eq. (7), change (s, t) to (s, s). Two lines below Eq. (7), after (i), replace “Pii (t) → 1 where δi j = 1 if i = j , 0 otherwise” by “Pii (s, t) → 1 as t → s”.

We consider survival or duration times associated with spells (sojourns in some state) or events experienced by individuals in a population over a specified time period. Duration distributions can be estimated from data recorded during followup of panel members in longitudinal surveys, but adjustments for the sample design, population structure and losses to followup are typically required. We provided weighted Kaplan-Meier estimates that allow for these features and, in particular, adjust for dependent loss to followup through the use of inverse probability of censoring weights.

The nonparametric maximum likelihood estimation (NPMLE) of the distribution function from the interval censored (IC) data has been extensively studied in the extant literature. The NPMLE was also developed for the subdistribution functions in an IC competing risks model and in an illness-death model under various interval-censoring scenarios. But the important problem of estimation of the cumulative intensities (CIs) in the interval-censored models has not been considered previously. We develop the NPMLE of the CI in a simple alive/dead model and of the CIs in a competing risks model. Assuming that data are generated by a discrete and finite mixed case interval censoring mechanism we provide a discussion and the simulation study of the asymptotic properties of the NPMLEs of the CIs. In particular we show that they are asymptotically unbiased; in contrast the ad hoc estimators presented in extant literature are substantially biased. We illustrate our methods with the data from a prospective cohort study on the longevity of dental veneers.

Nested case-control (NCC) sampling is widely used in large epidemiological cohort studies for its cost effectiveness, but its data analysis primarily relies on the Cox proportional hazards model. In this paper, we consider a family of linear transformation models for analyzing NCC data and propose an inverse selection probability weighted estimating equation method for inference. Consistency and asymptotic normality of our estimators for regression coefficients are established. We show that the asymptotic variance has a closed analytic form and can be easily estimated. Numerical studies are conducted to support the theory and an application to the Wilms' Tumor Study is also given to illustrate the methodology.

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.

Family-based follow-up study designs are important in epidemiology as they enable investigations of disease aggregation within families. Such studies are subject to methodological complications since data may include multiple endpoints as well as intra-family correlation. The methods herein are developed for the analysis of age of onset with multiple disease types for family-based follow-up studies. The proposed model expresses the marginalized frailty model in terms of the subdistribution hazards (SDH). As with Pipper and Martinussen's (Scand J Stat 30:509-521, 2003) model, the proposed multivariate SDH model yields marginal interpretations of the regression coefficients while allowing the correlation structure to be specified by a frailty term. Further, the proposed model allows for a direct investigation of the covariate effects on the cumulative incidence function since the SDH is modeled rather than the cause specific hazard. A simulation study suggests that the proposed model generally offers improved performance in terms of bias and efficiency when a sufficient number of events is observed. The proposed model also offers type I error rates close to nominal. The method is applied to a family-based study of breast cancer when death in absence of breast cancer is considered a competing risk.

The prevalence of interval censored data is increasing in medical studies due to the growing use of biomarkers to define a disease progression endpoint. Interval censoring results from periodic monitoring of the progression status. For example, disease progression is established in the interval between the clinic visit where progression is recorded and the prior clinic visit where there was no evidence of disease progression. A methodology is proposed for estimation and inference on the regression coefficients in the Cox proportional hazards model with interval censored data. The methodology is based on estimating equations and uses an inverse probability weight to select event time pairs where the ordering is unambiguous. Simulations are performed to examine the finite sample properties of the estimate and a colon cancer data set is used to demonstrate its performance relative to the conventional partial likelihood estimate that ignores the interval censoring.

The current status and panel count data frequently arise from cancer and tumorigenicity studies when events currently occur. A common and widely used class of two sample tests, for current status and panel count data, is the permutation class. We manipulate the double saddlepoint method to calculate the exact mid-p-values of the underlying permutation distributions of this class of tests. Permutation simulations are replaced by analytical saddlepoint computations which provide extremely accurate mid-p-values that are exact for most practical purposes and almost always more accurate than normal approximations. The method is illustrated using two real tumorigenicity panel count data. To compare the saddlepoint approximation with the normal asymptotic approximation, a simulation study is conducted. The speed and accuracy of the saddlepoint method facilitate the calculation of the confidence interval for the treatment effect. The inversion of the mid-p-values to calculate the confidence interval for the mean rate of development of the recurrent event is discussed.