Gologit vs series of binary logistic regressions

[Santiago Valdivieso]

I have dependent variable with four categories, I am running a gologit regression and I have to explain the resulting coefficients in an intuitive manner. I understand that the latter are *almost* the same as the ones that I would get if I ran many binary logistic regressions. However, there the differences arise "because the gologit model estimates all the parameters simultaneously whereas the separate logistic regressions estimate them one cumulative logit at a time", (Williams; 2016)¹. I think that I would have no problem explaining the coefficients just as the result of binary regressions, but I don't get how to explain the intuition of estimating "all parameters simultaneously". Can someone help me?


¹https://www3.nd.edu/~rwilliam/rwpubs/UnderstandingGologit2016.pdf

[Richard Williams]

Much the same thing happens with mlogit -- the results are almost the same as what you get with a series of binary logits. The explanation for why mlogit is not exactly identical to a series of binary logits was presented in

https://www.stata.com/statalist/arch.../msg00952.html

I suspect a similar explanation applies for gologit, but I've never formally worked it out.

A few other points:

If you are running gologit2 without proportionality constraints on any variables, you don't gain any parsimony. In that case you might as well use the better known mlogit. As is noted on p. 18 of the article you mention above,

There are several other issues to be aware of with the gologit/partial proportional odds model. First, it probably works best when relatively few of the variables in the model violate the proportional odds assumption. If several variables violate the assumption, then the gologit model offers little in the way of parsimony and more widely known techniques such as multinomial logit may be superior.

If you want help with interpretation, this handout may be helpful:

https://www3.nd.edu/~rwilliam/xsoc73994/Margins05.pdf

[Santiago Valdivieso]

Dear Richard Williams ,

In your article you mention that gologit gives "results that are similar to what we get with the series of binary logistic regressions/ cumulative logit models (...) and can be interpreted the same way". From that, do I understand correctly that while the coefficients differ in a small amount from binary logistic regressions, I can still interpret the coefficient B_ij as "the expected change in the Log Odds of Y_j = 1 given a marginal increase in X_i, holding everything else constant"? Where Y_j=1 is the same as saying Y*>j for j = 1, 2, 3, ..., J-1? Or should I include something else for that interpretation to be correct?

With respect to your second point, with a set of binary logistic regressions, the results I get are pretty similar in relation to the gologit. But when I run the mlogit, the results change meaningfully. I suspect that this is due to how my data is distributed and my categories are defined. In any case, gologit seems to be more robust and that's why it was chosen.

Finally, related to the margins command, I am a little bit confused on whether the default output when someone writes

reg y i.x
margins x

makes assumptions about the values of the other covariates (e.g., taking their average value), or if it just calculates:

(or anything else).

Thank you in advance!

[Richard Williams]

For help on margins, and on AMEs versus MEMs, see

https://www3.nd.edu/~rwilliam/xsoc73994/Margins01.pdf

If you want more on marginal effects with ordinal DVs, see

https://www3.nd.edu/~rwilliam/xsoc73994/Margins05.pdf

gologit2 and mlogit are parameterized differently. With gologit2, the panels correspond to 1 versus 2,3,4, then 1,2 versus 3,4, then 1,2,3 versus 4, i.e. the dependent variable is collapsed differently in each panel.

With mlogit, it is 2 vs 1, then 3 vs 1, then 4 vs 1, i.e. each category is contrasted with the reference category.

Therefore the gologit2 and mlogit coefficients should look different from each other.

[Santiago Valdivieso]

Thank you, Richard! But related to my first point, can you expand more on this:

In your article you mention that gologit gives "results that are similar to what we get with the series of binary logistic regressions/ cumulative logit models (...) and can be interpreted the same way". From that, do I understand correctly that while the coefficients differ in a small amount from binary logistic regressions, I can still interpret the coefficient B_ij as "the expected change in the Log Odds of Y_j = 1 given a marginal increase in X_i, holding everything else constant"? Where Y_j=1 is the same as saying Y*>j for j = 1, 2, 3, ..., J-1? Or should I include something else for that interpretation to be correct?

?

[Santiago Valdivieso]

Thank you, Richard Williams ! But related to my first point, can you expand more on this:

In your article you mention that gologit gives "results that are similar to what we get with the series of binary logistic regressions/ cumulative logit models (...) and can be interpreted the same way". From that, do I understand correctly that while the coefficients differ in a small amount from binary logistic regressions, I can still interpret the coefficient B_ij as "the expected change in the Log Odds of Y_j = 1 given a marginal increase in X_i, holding everything else constant"? Where Y_j=1 is the same as saying Y*>j for j = 1, 2, 3, ..., J-1? Or should I include something else for that interpretation to be correct?
?

[Richard Williams]

I think what you say is fine. I’m not sure why you would want to stress that point. I think marginal effects and adjusted predictions are the best way to make results more interpretable.