English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 into the regression. 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.

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