  With a growing interest in using non-representative samples to train
prediction models for numerous outcomes it is necessary to account for the
sampling design that gives rise to the data in order to assess the generalized
predictive utility of a proposed prediction rule. After learning a prediction
rule based on a non-uniform sample, it is of interest to estimate the rule's
error rate when applied to unobserved members of the population. Efron (1986)
proposed a general class of covariance penalty inflated prediction error
estimators that assume the available training data are representative of the
target population for which the prediction rule is to be applied. We extend
Efron's estimator to the complex sample context by incorporating
Horvitz-Thompson sampling weights and show that it is consistent for the true
generalization error rate when applied to the underlying superpopulation. The
resulting Horvitz-Thompson-Efron (HTE) estimator is equivalent to dAIC, a
recent extension of AIC to survey sampling data, but is more widely applicable.
The proposed methodology is assessed with simulations and is applied to models
predicting renal function obtained from the large-scale NHANES survey.
