Logit and probit models are widely used in empirical sociological research. However, the common practice of comparing the coefficients of a given variable across differently specified models fitted to the same sample does not warrant the same interpretation in logits and probits as in linear regression. Unlike linear models, the change in the coefficient of the variable of interest cannot be straightforwardly attributed to the inclusion of confounding variables. The reason for this is that the variance of the underlying latent variable is not identified and will differ between models. We refer to this as the problem of rescaling. We propose a solution that allows researchers to assess the influence of confounding relative to the influence of rescaling, and we develop a test to assess the statistical significance of confounding. A further problem in making comparisons is that, in most cases, the error distribution, and not just its variance, will differ across models. Monte Carlo analyses indicate that other methods that have been proposed for dealing with the rescaling problem can lead to mistaken inferences if the error distributions are very different. In contrast, in all scenarios studied, our approach performs as least as well as, and in some cases better than, others when faced with differences in the error distributions. We present an example of our method using data from the National Education Longitudinal Study.

In a series of recent articles, Karlson, Holm, and Breen (Breen, Karlson, and Holm, 2011, http://papers.ssrn.com/sol3/papers.cfm?abstractid=1730065;

This article presents a method for estimating and interpreting total, direct, and indirect effects in logit or probit models. The method extends the decomposition properties of linear models to these models; it closes the muchdiscussed gap between results based on the ''difference in coefficients'' method and the ''product of coefficients'' method in mediation analysis involving nonlinear probability models models; it reports effects measured on both the logit or probit scale and the probability scale; and it identifies causal mediation effects under the sequential ignorability assumption. We also show that while our method is computationally simpler than other methods, it always performs as well as, or better than, these methods. Further derivations suggest a hitherto unrecognized issue in identifying heterogeneous mediation effects in nonlinear probability models. We conclude the article with an application of our method to data from the National Educational Longitudinal Study of 1988.

Methods textbooks in sociology and other social sciences routinely recommend the use of the logit or probit model when an outcome variable is binary, an ordered logit or ordered probit when it is ordinal, and a multinomial logit when it has more than two categories. But these methodological guidelines take little or no account of a body of work that, over the past 30 years, has pointed to problematic aspects of these nonlinear probability models and, particularly, to difficulties in interpreting their parameters. In this review, we draw on that literature to explain the problems, show how they manifest themselves in research, discuss the strengths and weaknesses of alternatives that have been suggested, and point to lines of further analysis.