Semiparametric Models Versus Faith-based Inference

We appreciate the thoughtful comments by Subramanian and O’Malley to our paper on comparing mixed models and population average models, and the opportunity this response affords us to make a stronger and more general case regarding prevalent misconceptions surrounding statistical estimation. There are several technical points made in the paper that can be debated, but we will focus on what we believe is the crux of their critique—an issue that is widely shared (either explicitly or implicitly) by analyses of a majority of researchers using statistical inference from data to support scientific hypotheses. We start with what we hope is an accurate summary of their argument: nonparametric identifiability of a parameter of interest from the observed data, considering knowledge available on the data-generating distribution, should not be a major concern in deciding on the choice of parameter of interest within a chosen data-generating model. Instead, the scientific question should guide the types of models used to make inferences from data. Thus, the proposed model for the data-generating distribution and the resulting target parameter should not be restricted by what is actually known (and knowable) about the data-generating process. There are times when the parameters of interest are defined only within the context of a mixed model (latent variable model), and thus giving up on one’s parameter of interest for the sake of semiparametric inference, they argue, is counterproductive or even illogical. The authors’ assertion that the generalized-estimating-equation (GEE) approach requires parametric assumptions for interpretable parameters suggests that they failed to understand one of our basic points. This demonstrates a need to reiterate what we meant by defining the parameter of the data-generating distribution nonparametrically, or, in general, in the context of a realistic semiparametric statistical model. Their assertion that, for instance, modeling ranks as a nonparametric solution shows that there has been an unfortunate disconnect between what is meant by a semi-parametric approach and traditional notions of nonparametric statistics (ie, estimating the distribution of ranks). Though persuasively presented, the comment serves to underscore the need for vigorous debate on how we “learn” from data and what has been a consistent failure in our discipline to distinguish, both in estimation and inference, what can be learned from data and what aspects of some models are simply not deducible from data. We propose a new “golden rule” in statistical estimation, namely that one should be able to define what an estimation procedure estimates purely as a function of the data-generating distribution. If, in the words of Box, all models are wrong, then this parameter of the observed data-generating distribution is the only quantity one knows for sure is actually being estimated. Only under further nonidentifiable assumptions can this be interpreted as a parameter of a hypothesized model. It is also known (and we have