Logit/Probit: How to compare Coeficients between groups (n and m size)

[Juan L. Sobreira]

I have a sample divided into two groups:
large (2000) and Small Companies (4000).

I have specified my model using probit for one of each groups becasue I need to know diferences between groups:

a) significative dependents variables in one and other model.(This is not a problem)
b) diference between groups (using coeficients or marginal effects).

It is possible to do this when sub.sample size are unequals?

Could anybody help me?
(Note.- I am working with Stata 14)

[Stephen Jenkins]

Whether you can compare probit/logit coefficients across groups in any meaningful way is a controversial issue. See inter alia the following (taken from Rich Williams' webpages):
Allison, Paul. 1999. Comparing Logit and Probit Coefficients Across Groups. Sociological Methods and Research 28(2): 186-208.

Karlson, Kristian B., Anders Holm and Richard Breen. 2011. Comparing Regression Coefficients between Same-Sample Nested Models using Logit and Probit: A New Method. Sociological Methodology.

Kohler, Ulrich, Kristian B. Carlson and Anders Holm. 2011. Comparing Coefficients of nested nonlinear probability models. The Stata Journal.

Long, J. Scott. 2009. Group comparisons in logit and probit using predicted probabilities. Working Paper, June 25, 2009.(http://www.indiana.edu/~jslsoc/files_research/groupdif/groupwithprobabilities/groups-with-prob-2009-06-25.pdf )

Williams, Richard. 2009. Using Heterogeneous Choice Models to Compare Logit and Probit Coefficients across Groups. Sociological Methods & Research 37(4): 531-559. A pre-publication version is available at http://www.nd.edu/~rwilliam/oglm/RW_Hetero_Choice.pdf.

Williams, Richard. 2010. Fitting Heterogeneous Choice Models with oglm. The Stata Journal 10(4):540-567. A pre-publication version is available at http://www.nd.edu/~rwilliam/oglm/oglm_Stata.pdf.

[Carlo Lazzaro]

Juan:
can't you simply plug a dummy (i.large, coded 0 for small and 1 for large companies) in the right-hand side of your equation and look for its simple effect or (conditional one, if you also include interaction between i.large and some other predictor)?

[Stephen Jenkins]

Fitting a pooled regression and using interactions assumes that the error variance is the same for large companies and small companies. So, yes, it's one way of solving the identification problem ... but is also problematic for the reasons discussed in the literature cited in post #2

[Carlo Lazzaro]

Stephen:
yes, the "quick and dirty" approach that I suggested may be problematic as well...just a way to lift oneself out of the pond if nothing better can be done.

[Juan L. Sobreira]

Dear Carlo and Stephen,

I will read carefully the Stephen's references.
It is clear I was absollutly wrong!!!,....So(I am Sorry!!!) it is not possible to create a new a variable with two posibles values: coeficients value for probit regression (SMES) when SMES=1 and coeficientes value for probit regression (large) and compare mean by SME (0 or 1).

[Maarten Buis]

Stephen (and Rich) accurately describe the state of the literature when they say that it is problematic to compare logit coefficients, odds ratios or probit coefficients across groups. However, as some people on this list know, I have for a long time been somewhat critical of that conclusion. I have finaly taken some time to collect my arguments and write them up. A first draft of that paper can be found here: http://maartenbuis.nl/wp/oddsratio.html

[Frank Popham]

There is a related literature in epidemiology with recent papers emphasising the difference between marginal and conditional odds ratio and that both can be estimated using the same data and confounding (or not) variables ( using inverse probability weighting and a marginal structural model and logistic regression respectively). See for example http://www.academia.edu/6961654/Esti...collapsibility and http://smm.sagepub.com/content/early...05804.abstract

[Richard Williams]

Just to place things in perspective: it is true that heteroskedastic errors can bias comparisons across groups. But, lots of things can produce biased coefficients, e.g. Omitted variables, not including squared terms, failure to make proper transformations of variables, e.g. Logging them. You should be aware of the potential threats and realize how your results or interpretations qmight be wrong but at the same time you don't want to become so paralyzed that you don't do anything.

[Juan L. Sobreira]

Dear Mr. Williams,

Great advice. Thank you

I have just realized that It is important to define limits around hipothesis in order to know how exacts conclusions could be and, at the same time, to define next research challanges and steps (under several point of views such as statistical, survey and sample definitions, ...)

[Richard Williams]

I need to expand this into a real paper sometime but here is a summary of some of the problems with group comparisons and the proposals for dealing with them. (This was done before Maarten posted his most recent ideas.)

http://www3.nd.edu/~rwilliam/stats3/RW_ESRA2013.pdf