Assessing the predictive value of a binary surrogate for a binary true endpoint based on the minimum probability of a prediction error

Abstract

The individual causal association (ICA) has recently been introduced as a metric of surrogacy in a causal-inference framework. The ICA is defined on the unit interval and quantifies the association between the individual causal effect on the surrogate (Delta S) and true (Delta T) endpoint. In addition, the ICA offers a general assessment of the surrogate predictive value, taking value 1 when there is a deterministic relationship between Delta T and Delta S, and value 0 when both causal effects are independent. However, when one moves away from the previous two extreme scenarios, the interpretation of the ICA becomes challenging. In the present work, a new metric of surrogacy, the minimum probability of a prediction error (PPE), is introduced when both endpoints are binary, ie, the probability of erroneously predicting the value of Delta T using Delta S. Although the PPE has a more straightforward interpretation than the ICA, its magnitude is bounded above by a quantity that depends on the true endpoint. For this reason, the reduction in prediction error (RPE) attributed to the surrogate is defined. The RPE always lies in the unit interval, taking value 1 if prediction is perfect and 0 if Delta S conveys no information on Delta T. The methodology is illustrated using data from two clinical trials and a user-friendly R package Surrogate is provided to carry out the validation exercise.