Two seemingly paradoxical results in linear models: the variance inflation factor and the analysis of covariance

TL;DR

This paper clarifies the apparent paradox between the variance inflation factor and analysis of covariance results in linear models, showing they are based on different assumptions and contexts, and thus are consistent.

Contribution

It explains the conditions under which adding covariates affects variance in linear models, resolving the perceived paradox by comparing assumptions and implications.

Findings

VIF conditions on treatment indicators, ANCOVA averages over them

Adding covariates reduces variance with bias, not just variance

No real paradox exists between the two results

Abstract

A result from a standard linear model course is that the variance of the ordinary least squares (OLS) coefficient of a variable will never decrease when including additional covariates. The variance inflation factor (VIF) measures the increase of the variance. Another result from a standard linear model or experimental design course is that including additional covariates in a linear model of the outcome on the treatment indicator will never increase the variance of the OLS coefficient of the treatment at least asymptotically. This technique is called the analysis of covariance (ANCOVA), which is often used to improve the efficiency of treatment effect estimation. So we have two paradoxical results: adding covariates never decreases the variance in the first result but never increases the variance in the second result. In fact, these two results are derived under different assumptions. More precisely, the VIF result conditions on the treatment indicators but the ANCOVA result averages over them. Comparing the estimators with and without adjusting for additional covariates in a completely randomized experiment, I show that the former has smaller variance averaging over the treatment indicators, and the latter has smaller variance at the cost of a larger bias conditioning on the treatment indicators. Therefore, there is no real paradox.