Understanding the predictive value of continuous markers for censored survival data using a likelihood ratio approach

Andrew M Smith,John P Christodouleas,Wei-Ting Hwang

doi:10.1186/s12874-019-0721-0

Andrew M Smith, John P Christodouleas + Show 1 more

Open Access

https://doi.org/10.1186/s12874-019-0721-0

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Abstract

BackgroundThe likelihood ratio function (LR), the ratio of conditional probabilities of obtaining a specific marker value among those with the event of interest over those without, provides an easily interpretable way to quantify the update of the risk prediction due to the knowledge of the marker value. The LR has been explored for both binary and continuous markers for binary events (e.g., diseased or not), however the use of the LR in censored data has not been fully explored.MethodsWe extend the concept of LR to a time-dependent LR (TD-LR) for survival outcomes that are subject to censoring. Estimation for the TD-LR is done using Kaplan-Meier estimation and a univariate Cox proportional hazards (PH) model. A “scale invariant” approach based on marker quantiles is provided to allow comparison of predictive values between markers with different scales. Relationships to time-dependent receiver-operator characteristic (ROC) curves, area under the curve (AUC), and optimal cut-off values are considered.ResultsThe proposed methods were applied to data from a bladder cancer clinical trial to determine whether the neutrophil-to-lymphocyte ratio (NLR) is a valuable biomarker for predicting overall survival following surgery or combined chemotherapy and surgery. The TD-LR method yielded results consistent with the original findings while providing an easily interpretable three-dimensional surface display of how NLR related to the likelihood of event in the trial data.ConclusionsThe TD-LR provides a more nuanced understanding of the relationship between continuous markers and the likelihood of events in censored survival data. This method also allows more straightforward communication with a clinical audience through graphical presentation.

Highlights

The likelihood ratio function (LR), the ratio of conditional probabilities of obtaining a specific marker value among those with the event of interest over those without, provides an interpretable way to quantify the update of the risk prediction due to the knowledge of the marker value
The time-dependent likelihood ratio (TD-LR) can be estimated using scale-invariant techniques, which can satisfy the need to compare predictive value across markers. These methods are illustrated in an application to evaluate the predictive value of the neutrophil-lymphocyte ratio (NLR) as a prognostic marker for overall survival (OS) using data from the Southwest Oncology Group (SWOG) 8710 clinical trial, a randomized phase III trial assessing radical cystectomy (RC) with or without neoadjuvant chemotherapy (NAC) for muscle-invasive bladder cancer [2, 3]
Estimation Methods To estimate the TD-LR function, we propose to use the Kaplan-Meier nonparametric survival estimator [8] and survival estimates dervied from Cox proportional hazards (PH) models [9]

Summary

Introduction

The likelihood ratio function (LR), the ratio of conditional probabilities of obtaining a specific marker value among those with the event of interest over those without, provides an interpretable way to quantify the update of the risk prediction due to the knowledge of the marker value. One method of summarizing predictive value is the likelihood ratio (LR), the ratio of conditional probabilities of obtaining a specific marker value given event status (i.e., with and without event). The LR is a well-accepted and valuable method of evaluating potential markers because it can be shown that the value of the LR summarizes the predictive value of a marker by quantifying the update to the odds of event obtained by incorporating knowledge of the new marker in question.

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