Nonparametric discrete survival function estimation with uncertain endpoints using an internal validation subsample

Abstract

Summary When a true survival endpoint cannot be assessed for some subjects, an alternative endpoint that measures the true endpoint with error may be collected, which often occurs when obtaining the true endpoint is too invasive or costly. We develop an estimated likelihood function for the situation where we have both uncertain endpoints for all participants and true endpoints for only a subset of participants. We propose a nonparametric maximum estimated likelihood estimator of the discrete survival function of time to the true endpoint. We show that the proposed estimator is consistent and asymptotically normal. We demonstrate through extensive simulations that the proposed estimator has little bias compared to the naïve Kaplan-Meier survival function estimator, which uses only uncertain endpoints, and more efficient with moderate missingness compared to the complete-case Kaplan-Meier survival function estimator, which uses only available true endpoints. Finally, we apply the proposed method to a dataset for estimating the risk of developing Alzheimer's disease from the Alzheimer's Disease Neuroimaging Initiative.

Assessing treatment effects with surrogate survival outcomes using an internal validation subsample

“…The method can easily be extended to consider multiple covariates, which would be useful in order to adjust for confounding variables or to consider categorical variables with more than two levels. Further study on the number of allowable covariates is warranted; however, based on the events per variable (EPV) testing in Zee and Xie, 8 it is expected that a similar EPV of 4 would apply to multivariable models. This would imply that a minimum of four events should be observed for each parameter to be estimated.…”

“…In semiparametric survival analysis, the parameter of interest is often only the log hazard ratio, so the survival functions can be considered nuisance parameters. Using similar arguments as in Pepe 6 and Zee and Xie, 8 the estimated likelihood is given by…”

“…Also, true and surrogate censoring times can be completely different. Zee and Xie 8 adapted the estimated likelihood method 6 to nonparametrically estimate a survival function assuming real-time validation is possible. The method does not assume known mismeasurement rates of the surrogate outcome and allows for flexible censoring mechanisms.…”

“…In this article, we extend the work of Zee and Xie 8 to a semiparametric estimated likelihood method to estimate a parameter representing a binary covariate effect for discrete survival data with surrogate outcomes on all subjects and true outcomes on a subsample. Although we express our approach with a binary variable for ease of notation, the method can be easily modified for categorical variables with more than two levels.…”