Prediction Accuracy Measures for a Nonlinear Model and for Right-Censored Time-to-Event Data

ABSTRACT

This article develops a pair of new prediction summary measures for a nonlinear prediction function with right-censored time-to-event data. The first measure, defined as the proportion of explained variance by a linearly corrected prediction function, quantifies the potential predictive power of the nonlinear prediction function. The second measure, defined as the proportion of explained prediction error by its corrected prediction function, gauges the closeness of the prediction function to its corrected version and serves as a supplementary measure to indicate (by a value less than 1) whether the correction is needed to fulfill its potential predictive power and quantify how much prediction error reduction can be realized with the correction. The two measures together provide a complete summary of the predictive accuracy of the nonlinear prediction function. We motivate these measures by first establishing a variance decomposition and a prediction error decomposition at the population level and then deriving uncensored and censored sample versions of these decompositions. We note that for the least square prediction function under the linear model with no censoring, the first measure reduces to the classical coefficient of determination and the second measure degenerates to 1. We show that the sample measures are consistent estimators of their population counterparts and conduct extensive simulations to investigate their finite sample properties. A real data illustration is provided using the PBC data. Supplementary materials for this article are available online. An R package PAmeasures has been developed and made available via the CRAN R library. Supplementary materials for this article are available online.

KEYWORDS:

• Censoring
• Coefficient of determination
• Cox’s proportional hazards model
• Explained prediction error
• Explained variance

Acknowledgments

The authors thank the associate editor and a referee for their insightful and constructive comments that have led to significant improvements of the article.

Additional information

Funding

The research of Gang Li was partly supported by National Institute of Health Grants P30 CA16042, UL1TR000124-02, and P50 CA211015.