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A new goodness-of-fit measure for probit models: Surrogate R2.

Dungang Liu1, Xiaorui Zhu1,2, Brandon Greenwell1

  • 1Department of Operations, Business Analytics and Information Systems, University of Cincinnati Carl H. Lindner College of Business, Cincinnati, Ohio, USA.

The British Journal of Mathematical and Statistical Psychology
|October 17, 2022
PubMed
Summary

Researchers developed a new goodness-of-fit measure, surrogate R-squared, for probit models. This measure allows for better comparison of models in social science research, addressing a long-standing need for a comparable metric to ordinary least square R-squared.

Keywords:
OLS R2categorical datamodel comparisonprobit analysispseudo R2surrogate residual

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Area of Science:

  • Social Sciences
  • Statistics
  • Econometrics

Background:

  • Discrete data are common in social science research, necessitating robust inferential models like probit models.
  • Existing goodness-of-fit measures for probit models lack comparability with ordinary least square (OLS) R-squared, hindering model evaluation and comparison across studies.
  • A need exists for a goodness-of-fit measure that provides an interpretable 'explained variation' similar to OLS R-squared for probit models.

Purpose of the Study:

  • To propose a novel goodness-of-fit measure for probit models that mimics the interpretation and utility of OLS R-squared.
  • To introduce the 'surrogate R-squared' measure, based on simulating a continuous surrogate of the discrete response variable.
  • To demonstrate the theoretical and numerical advantages of the surrogate R-squared over existing pseudo R-squared measures.

Main Methods:

  • The study proposes a "surrogate R-squared" by simulating a continuous variable as a surrogate for the original discrete response.
  • A linear model is used to determine the proportion of the surrogate response's variance explained by the explanatory variables.
  • Theoretical derivations and numerical simulations are employed to validate the proposed measure.

Main Results:

  • The surrogate R-squared is shown to approximate the OLS R-squared based on the underlying latent continuous variable.
  • The proposed measure preserves the interpretation of explained variation, a key feature of OLS R-squared.
  • The surrogate R-squared maintains monotonicity between nested models, unlike some existing measures.

Conclusions:

  • The surrogate R-squared measure effectively fills the void for a comparable goodness-of-fit metric in probit models.
  • This new measure offers improved comparability across empirical models and samples in social science research.
  • The surrogate R-squared provides a valuable tool for researchers using probit models, enhancing model inference and interpretation.