Predictive Comparisons for Non-additive Models... or, Gelman on the Practical Virtues of Statistical Humility

See this paper from Sociological Methodology 2007, titled "Average predictive comparisons for models with nonlinearity, interactions, and variance components".

This paper addresses the issue of making "predictions" based on a model - but where causality is not established. A bonus, in my view, is that right up front, Gelman introduces a way of referring to the "predictions" that is at once descriptive and, hopefully, understandable by non-statisticians as well - without being misleading! The term "predictive comparison" is defined in his Eq. 1, and is said to "correspond to an expected causal effect under a counterfactual assumption(Neyman, 1923, Rubin, 1974, 1990), if it makes sense to consider the inputs causal". How much nicer and more clear it is to refer to this estimate as a "predictive comparison", in cases the inputs are not known to be causal. See his recent post for more on the choice of words to communicate results: "Describing descriptive studies using descriptive language, or the practical virtues of statistical humility".

His statistical contribution in the article is the carefully presented demonstration that in cases where the model is not linear and additive, model-based estimates of the average difference in the outcome variable, for two values of the covariate of interest, the "predictor", should be averaged over values of the remaining covariates as in his Eq. 2, (see Figure 2). After studying Figure 2, you will see how using point estimates for these covariates can get you into trouble. Implementing the improved methodology is a matter for another post...