Another Look at the Method of Y-Standardization in Logit and Probit Models

Abstract

This article takes another look at the derivation of the method of Y-standardization used in sociological analyses involving comparisons of coefficients across nested logit or probit models. It shows that the method can be derived under less restrictive assumptions than hitherto suggested. Rather than assuming that the logit or probit fixes the variance of the latent error at a known constant, it suffices to assume that the variance of the error is unknown. A further result suggests that using Y-standardization for cross-model comparisons is likely to be biased by model differences in the fit of the latent error to the assumed logistic or normal distribution. Under correct specification Y-standardization recovers an effect size metric similar to Cohen's d.

Keywords:

• attenuation
• comparison
• identification
• logistic
• logit
• probit
• rescaling
• standardization
• Y-standardization
• Y-standardized

Notes

¹For similar derivations, see Amemiya (1981), Maddala (1983), and Powers and Xie (2000, pp. 56–58).

²The intercept is a composite of the intercept in (1), the threshold parameter in (4), and the scale parameter. The intercept is irrelevant for the derivations reported here, and I ignore it throughout.

³In fact, according to the Eq. (7b), the variances of Y* in Panels A and B roughly differ by a factor of 4, being the ratio of the respective squared scale parameters in Models (A) and (B).

⁴In the probit this condition might be met whenever Z and the error in the model including Z are normal (Yatchew & Griliches, 1985). However, the condition is unlikely to be met in applied work.