Maximum likelihood and pseudo score approaches for parametric time-to-event analysis with informative entry times

We develop a maximum likelihood estimating approach for time-to-event Weibull regression models with outcome-dependent sampling, where sampling of subjects is dependent on the residual fraction of the time left to developing the event of interest. Additionally, we propose a two-stage approach which proceeds by iteratively estimating, through a pseudo score, the Weibull parameters of interest (i.e., the regression parameters) conditional on the inverse probability of sampling weights; and then re-estimating these weights (given the updated Weibull parameter estimates) through the profiled full likelihood. With these two new methods, both the estimated sampling mechanism parameters and the Weibull parameters are consistently estimated under correct specification of the conditional referral distribution. Standard errors for the regression parameters are obtained directly from inverting the observed information matrix in the full likelihood specification and by either calculating bootstrap or robust standard errors for the hybrid pseudo score/profiled likelihood approach. Loss of efficiency with the latter approach is considered. Robustness of the proposed methods to misspecification of the referral mechanism and the time-to-event distribution is also briefly examined. Further, we show how to extend our methods to the family of parametric time-to-event distributions characterized by the generalized gamma distribution. The motivation for these two approaches came from data on time to cirrhosis from hepatitis C viral infection in patients referred to the Edinburgh liver clinic. We analyze these data here.Comment: Published in at http://dx.doi.org/10.1214/14-AOAS725 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

The versatility of multi-state models for the analysis of longitudinal data with unobservable features.

Multi-state models provide a convenient statistical framework for a wide variety of medical applications characterized by multiple events and longitudinal data. We illustrate this through four examples. The potential value of the incorporation of unobserved or partially observed states is highlighted. In addition, joint modelling of multiple processes is illustrated with application to potentially informative loss to follow-up, mis-measured or missclassified data and causal inference

Multiple Imputation of Missing Composite Outcomes in Longitudinal Data.

In longitudinal randomised trials and observational studies within a medical context, a composite outcome-which is a function of several individual patient-specific outcomes-may be felt to best represent the outcome of interest. As in other contexts, missing data on patient outcome, due to patient drop-out or for other reasons, may pose a problem. Multiple imputation is a widely used method for handling missing data, but its use for composite outcomes has been seldom discussed. Whilst standard multiple imputation methodology can be used directly for the composite outcome, the distribution of a composite outcome may be of a complicated form and perhaps not amenable to statistical modelling. We compare direct multiple imputation of a composite outcome with separate imputation of the components of a composite outcome. We consider two imputation approaches. One approach involves modelling each component of a composite outcome using standard likelihood-based models. The other approach is to use linear increments methods. A linear increments approach can provide an appealing alternative as assumptions concerning both the missingness structure within the data and the imputation models are different from the standard likelihood-based approach. We compare both approaches using simulation studies and data from a randomised trial on early rheumatoid arthritis patients. Results suggest that both approaches are comparable and that for each, separate imputation offers some improvement on the direct imputation of a composite outcome

Exploring the existence of a stayer population with mover-stayer counting process models: application to joint damage in psoriatic arthritis.

Many psoriatic arthritis patients do not progress to permanent joint damage in any of the 28 hand joints, even under prolonged follow-up. This has led several researchers to fit models that estimate the proportion of stayers (those who do not have the propensity to experience the event of interest) and to characterize the rate of developing damaged joints in the movers (those who have the propensity to experience the event of interest). However, when fitted to the same data, the paper demonstrates that the choice of model for the movers can lead to widely varying conclusions on a stayer population, thus implying that, if interest lies in a stayer population, a single analysis should not generally be adopted. The aim of the paper is to provide greater understanding regarding estimation of a stayer population by comparing the inferences, performance and features of multiple fitted models to real and simulated data sets. The models for the movers are based on Poisson processes with patient level random effects and/or dynamic covariates, which are used to induce within-patient correlation, and observation level random effects are used to account for time varying unobserved heterogeneity. The gamma, inverse Gaussian and compound Poisson distributions are considered for the random effects

Trivariate mover-stayer counting process models for investigating joint damage in psoriatic arthritis.

In psoriatic arthritis, many patients do not develop permanent joint damage even after a prolonged follow-up. This has led several authors to consider the possibility of a subpopulation of stayers (those who do not have the propensity to experience the event of interest), as opposed to assuming the entire population consist of movers (those who have the propensity to experience the event of interest). In addition, it is recognised that the damaged joints process may act very differently across different joint areas, particularly the hands, feet and large joints. From a clinical perspective, interest lies in identifying possible relationships between the damaged joints processes in these joint areas for the movers and estimating the proportion of stayers in these joint areas, if they exist. For this purpose, this paper proposes a novel trivariate mover-stayer model consisting of mover-stayer truncated negative binomial margins, and patient-level dynamic covariates and random effects in the models for the movers and stayers, respectively. The model is then extended to have a two-level mover-stayer structure for its margins so that the nature of the stayer property can be investigated. A particularly attractive feature of the proposed models is that only an optimisation routine is required in their model fitting procedures. © 2016 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd

A multistate model for events defined by prolonged observation.

Time-to-event and similar analyses can be problematic if the event of interest is operationally defined by some condition being true for a prolonged period of time. A particular example of this, remission in psoriatic arthritis, is considered in detail for illustration. A 3-state model is proposed for characterizing the transition rates into and out of remission. Remission is linked to an initial and subsequent state for the purpose of introducing the condition that remission must be of some duration to be clinically meaningful. The model is compared with alternative approaches that have been used in such situations. These involve 2-state models where the duration of remission is allowed for through different definitions for the time of entry into remission. Both definitions are linked to prolonged observation of a particular clinical state