Title: Valid Inference Corrected for Outlier Removal,
Abstract: Ordinary least square (OLS) estimation of a linear regression model is
well-known to be highly sensitive to outliers. It is common practice to first
identify and remove outliers by looking at the data then to fit OLS and form
confidence intervals and p-values on the remaining data as if this were the
original data collected. We show in this paper that this "detect-and-forget"
approach can lead to invalid inference, and we propose a framework that
properly accounts for outlier detection and removal to provide valid confidence
intervals and hypothesis tests. Our inferential procedures apply to any outlier
removal procedure that can be characterized by a set of quadratic constraints
on the response vector, and we show that several of the most commonly used
outlier detection procedures are of this form. Our methodology is built upon
recent advances in selective inference (Taylor & Tibshirani 2015), which are
focused on inference corrected for variable selection. We conduct simulations
to corroborate the theoretical results, and we apply our method to two classic
data sets considered in the outlier detection literature to illustrate how our
inferential results can differ from the traditional detect-and-forget strategy.
A companion R package, outference, implements these new procedures with an
interface that matches the functions commonly used for inference with lm in R.