Biases from Non-Simultaneous Regression with Correlated Covariates: A Case Study from Supernova Cosmology

TL;DR

This paper investigates biases introduced by non-simultaneous regression methods in supernova cosmology when covariates are correlated, deriving formulas for these biases and proposing corrections to improve analysis accuracy.

Contribution

It provides a detailed analysis of biases in non-simultaneous regressions with correlated covariates and offers correction methods, emphasizing the advantages of simultaneous regression techniques.

Findings

Biases depend on covariate correlations and can be mathematically characterized.

Simulations demonstrate the effectiveness of proposed bias corrections.

Using simultaneous regression avoids biases entirely.

Abstract

Several Type Ia supernova analyses make use of non-simultaneous regressions between observed supernova and host galaxy properties and supernova luminosity: first the supernova magnitudes are corrected for their light curve shape and color, and then they are separately corrected for their host galaxy masses. This two-step regression methodology does not introduce any biases when there are no correlations between the variables regressed in each correction step. However, correlations between these covariates will bias estimates of the size of the corrections, as well as estimates of the variance of the final residuals. In this work, we analyze the general case of non-simultaneous regression with correlated covariates to derive the functional forms of these biases. We also simulate this effect on data from the literature to provide corrections to remove these biases from the data sets studied. The biases examined here can be entirely avoided by using simultaneous regression techniques.