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

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 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.

We 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.

The 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.

The 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.