On asymptotically optimal tests under loss of identifiability in semiparametric models

We consider tests of hypotheses when the parameters are not identifiable under the null in semiparametric models, where regularity conditions for profile likelihood theory fail. Exponential average tests based on integrated profile likelihood are constructed and shown to be asymptotically optimal under a weighted average power criterion with respect to a prior on the nonidentifiable aspect of the model. These results extend existing results for parametric models, which involve more restrictive assumptions on the form of the alternative than do our results. Moreover, the proposed tests accommodate models with infinite dimensional nuisance parameters which either may not be identifiable or may not be estimable at the usual parametric rate. Examples include tests of the presence of a change-point in the Cox model with current status data and tests of regression parameters in odds-rate models with right censored data. Optimal tests have not previously been studied for these scenarios. We study the asymptotic distribution of the proposed tests under the null, fixed contiguous alternatives and random contiguous alternatives. We also propose a weighted bootstrap procedure for computing the critical values of the test statistics. The optimal tests perform well in simulation studies, where they may exhibit improved power over alternative tests.Comment: Published in at http://dx.doi.org/10.1214/08-AOS643 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust Inference for Univariate Proportional Hazards Frailty Regression Models

We consider a class of semiparametric regression models which are one-parameter extensions of the Cox [J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220] model for right-censored univariate failure times. These models assume that the hazard given the covariates and a random frailty unique to each individual has the proportional hazards form multiplied by the frailty. The frailty is assumed to have mean 1 within a known one-parameter family of distributions. Inference is based on a nonparametric likelihood. The behavior of the likelihood maximizer is studied under general conditions where the fitted model may be misspecified. The joint estimator of the regression and frailty parameters as well as the baseline hazard is shown to be uniformly consistent for the pseudo-value maximizing the asymptotic limit of the likelihood. Appropriately standardized, the estimator converges weakly to a Gaussian process. When the model is correctly specified, the procedure is semiparametric efficient, achieving the semiparametric information bound for all parameter components. It is also proved that the bootstrap gives valid inferences for all parameters, even under misspecification. We demonstrate analytically the importance of the robust inference in several examples. In a randomized clinical trial, a valid test of the treatment effect is possible when other prognostic factors and the frailty distribution are both misspecified. Under certain conditions on the covariates, the ratios of the regression parameters are still identifiable. The practical utility of the procedure is illustrated on a non-Hodgkin's lymphoma dataset.Comment: Published by the Institute of Mathematical Statistics (http://www.imstat.org) in the Annals of Statistics (http://www.imstat.org/aos/) at http://dx.doi.org/10.1214/00905360400000053

Nonparametric Bounds and Sensitivity Analysis of Treatment Effects

This paper considers conducting inference about the effect of a treatment (or exposure) on an outcome of interest. In the ideal setting where treatment is assigned randomly, under certain assumptions the treatment effect is identifiable from the observable data and inference is straightforward. However, in other settings such as observational studies or randomized trials with noncompliance, the treatment effect is no longer identifiable without relying on untestable assumptions. Nonetheless, the observable data often do provide some information about the effect of treatment, that is, the parameter of interest is partially identifiable. Two approaches are often employed in this setting: (i) bounds are derived for the treatment effect under minimal assumptions, or (ii) additional untestable assumptions are invoked that render the treatment effect identifiable and then sensitivity analysis is conducted to assess how inference about the treatment effect changes as the untestable assumptions are varied. Approaches (i) and (ii) are considered in various settings, including assessing principal strata effects, direct and indirect effects and effects of time-varying exposures. Methods for drawing formal inference about partially identified parameters are also discussed.Comment: Published in at http://dx.doi.org/10.1214/14-STS499 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Adaptive estimation with partially overlapping models

In many problems, one has several models of interest that capture key parameters describing the distribution of the data. Partially overlapping models are taken as models in which at least one covariate effect is common to the models. A priori knowledge of such structure enables efficient estimation of all model parameters. However, in practice, this structure may be unknown. We propose adaptive composite M-estimation (ACME) for partially overlapping models using a composite loss function, which is a linear combination of loss functions defining the individual models. Penalization is applied to pairwise differences of parameters across models, resulting in data driven identification of the overlap structure. Further penalization is imposed on the individual parameters, enabling sparse estimation in the regression setting. The recovery of the overlap structure enables more efficient parameter estimation. An oracle result is established. Simulation studies illustrate the advantages of ACME over existing methods that fit individual models separately or make strong a priori assumption about the overlap structure

Accounting for competing risks in randomized controlled trials: a review and recommendations for improvement

In studies with survival or time-to-event outcomes, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Specialized statistical methods must be used to analyze survival data in the presence of competing risks. We conducted a review of randomized controlled trials with survival outcomes that were published in high-impact general medical journals. Of 40 studies that we identified, 31 (77.5%) were potentially susceptible to competing risks. However, in the majority of these studies, the potential presence of competing risks was not accounted for in the statistical analyses that were described. Of the 31 studies potentially susceptible to competing risks, 24 (77.4%) reported the results of a Kaplan-Meier survival analysis, while only five (16.1%) reported using cumulative incidence functions to estimate the incidence of the outcome over time in the presence of competing risks. The former approach will tend to result in an overestimate of the incidence of the outcome over time, while the latter approach will result in unbiased estimation of the incidence of the primary outcome over time. We provide recommendations on the analysis and reporting of randomized controlled trials with survival outcomes in the presence of competing risks. © 2017 The Authors. Statistics in Medicine published by John Wiley & Sons Ltd

Practical recommendations for reporting Fine-Gray model analyses for competing risk data

In survival analysis, a competing risk is an event whose occurrence precludes the occurrence of the primary event of interest. Outcomes in medical research are frequently subject to competing risks. In survival analysis, there are 2 key questions that can be addressed using competing risk regression models: first, which covariates affect the rate at which events occur, and second, which covariates affect the probability of an event occurring over time. The cause‐specific hazard model estimates the effect of covariates on the rate at which events occur in subjects who are currently event‐free. Subdistribution hazard ratios obtained from the Fine‐Gray model describe the relative effect of covariates on the subdistribution hazard function. Hence, the covariates in this model can also be interpreted as having an effect on the cumulative incidence function or on the probability of events occurring over time. We conducted a review of the use and interpretation of the Fine‐Gray subdistribution hazard model in articles published in the medical literature in 2015. We found that many authors provided an unclear or incorrect interpretation of the regression coefficients associated with this model. An incorrect and inconsistent interpretation of regression coefficients may lead to confusion when comparing results across different studies. Furthermore, an incorrect interpretation of estimated regression coefficients can result in an incorrect understanding about the magnitude of the association between exposure and the incidence of the outcome. The objective of this article is to clarify how these regression coefficients should be reported and to propose suggestions for interpreting these coefficients

Semiparametric Methods for Semi-competing Risks Problem with Censoring and Truncation

Studies of chronic life-threatening diseases often involve both mortality and morbidity. In observational studies, the data may also be subject to administrative left truncation and right censoring. Since mortality and morbidity may be correlated and mortality may censor morbidity, the Lynden-Bell estimator for left truncated and right censored data may be biased for estimating the marginal survival function of the non-terminal event. We propose a semiparametric estimator for this survival function based on a joint model for the two time-to-event variables, which utilizes the gamma frailty specification in the region of the observable data. Firstly, we develop a novel estimator for the gamma frailty parameter under left truncation. Using this estimator, we then derive a closed form estimator for the marginal distribution of the non-terminal event. The large sample properties of the estimators are established via asymptotic theory. The methodology performs well with moderate sample sizes, both in simulations and in an analysis of data from a diabetes registry