Model Evaluation Based on the Distribution of Estimated Absolute Prediction Error

The construction of a reliable, practically useful prediction rule for future response is heavily dependent on the "adequacy" of the fitted regression model. In this article, we consider the absolute prediction error, the expected value of the absolute difference between the future and predicted responses, as the model evaluation criterion. This prediction error is easier to interpret than the average squared error and is equivalent to the mis-classification error for the binary outcome. We show that the distributions of the apparent error and its cross-validation counterparts are approximately normal even under a misspecified fitted model. When the prediction rule is "unsmooth", the variance of the above normal distribution can be estimated well via a perturbation-resampling method. We also show how to approximate the distribution of the difference of the estimated prediction errors from two competing models. With two real examples, we demonstrate that the resulting interval estimates for prediction errors provide much more information about model adequacy than the point estimates alone.

ON THE MERITS OF VOXEL-BASED MORPHOMETRIC PATH-ANALYSIS FOR INVESTIGATING VOLUMETRIC MEDIATION OF A TOXICANT'S INFLUENCE ON COGNITIVE FUNCTION

We previously showed that lifetime cumulative lead dose, measured as lead concentration in the tibia bone by X-ray fluorescence, was associated with persistent and progressive declines in cognitive function and with decreases in MRI-based brain volumes in former lead workers. Moreover, larger region-specific brain volumes were associated with better cognitive function. These findings motivated us to explore a novel application of path analysis to evaluate effect mediation. Voxel-wise path analysis, at face value, represents the natural evolution of voxel-based morphometry methods to answer questions of mediation. Application of these methods to the former lead worker data demonstrated potential limitations in this approach where there was a tendency for results to be strongly biased towards the null hypothesis (lack of mediation). Moreover, a complimentary analysis using anatomically-derived regions of interest volumes yielded opposing results, suggesting evidence of mediation. Specifically, in the ROI-based approach, there was evidence that the association of tibia lead with function in three cognitive domains was mediated through the volumes of total brain, frontal gray matter, and/or possibly cingulate. A simulation study was conducted to investigate whether the voxel-wise results arose from an absence of localized mediation, or more subtle defects in the methodology. The simulation results showed the same null bias evidenced as seen in the lead workers data. Both the lead worker data results and the simulation study suggest that a null-bias in voxel-wise path analysis limits its inferential utility for producing confirmatory results.