Collected Notes on Aldrich-Mckelevey Scaling

TL;DR

This paper reviews Aldrich-McKelvey scaling, proposing improvements with QR decomposition, clarifying its identification constraints, comparing its robustness to heteroskedasticity with mean aggregation, and distinguishing Bayesian from traditional methods.

Contribution

It introduces practical enhancements, clarifies theoretical constraints, and compares robustness and parameterization of Aldrich-McKelvey scaling methods.

Findings

QR decomposition improves estimation accuracy and respondent retention.

A minimum of three external stimuli is necessary for identification.

Aldrich-McKelvey scaling is not more robust to heteroskedasticity than mean aggregation.

Abstract

Aldrich-McKelvey scaling is a method for correcting differential item functioning in ordered rating scales of perceived ideological positions in surveys. In this collection of notes, I present four findings. First, I show that similarly to ordinary least squares, Aldrich-McKelvey scaling can be improved with the use of QR decomposition during the estimation stage. While in theory this might improve accuracy, in practice the main advantage is to retain respondents otherwise lost in the estimation stage. Second, I show that this method leads to a proof of an identification constraint of Aldrich-McKelvey scaling: a minimum of three external stimuli. Third, I show that the common motivation for Aldrich-McKelvey scaling that it is robust to heteroskedasticity as compared to taking the means does not hold up. A review of the literature of prediction aggregation shows taking the mean is equally as robust. However, Aldrich-McKelvey scaling remains robust to rationalization bias and is transparent in its assumptions. Finally, I show that the setup of Bayesian Aldrich-McKelvey Scaling and Aldrich-McKelvey scaling differ from each other in their parameterisation. This is not commonly acknowledged in the literature, and new users of these methods should be aware.