Valid Inference Corrected for Outlier Removal

TL;DR

This paper addresses the problem of invalid statistical inference after outlier removal in linear regression and proposes a method using selective inference to correct for this, supported by simulations and real data applications.

Contribution

It introduces a new inferential method that accounts for outlier removal, ensuring valid confidence intervals and p-values in linear regression analysis.

Findings

Standard outlier removal can lead to invalid inference.

The proposed method provides corrected inference post outlier removal.

Simulations and real data demonstrate the effectiveness of the approach.

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 (1) identify and remove outliers by looking at the data and (2) to fit OLS and form confidence intervals and p-values on the remaining data as if this were the original data collected. This standard "detect-and-forget" approach has been shown to be problematic, and in this paper we highlight the fact that it can lead to invalid inference and show how recently developed tools in selective inference can be used to properly account for outlier detection and removal. Our inferential procedures apply to a general class of outlier removal procedures that includes several of the most commonly used approaches. We conduct simulations to corroborate the theoretical results, and we apply our method to three real data sets 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.