Misunderstanding Ordinal Regression clog log output

[Jean Torgigial]

Hi everyone,

I ran an ordinal regression (with clog log link function) and I have my output.

My question is the following: How I understand things is that we put values for independent variables, then add cutoff for each equation. Then we take the inverse link function of these values (here, this inverse is equal to 1-exp(exp(-x)) and then we have cumulative probabilities. By taking the difference for each categories, we have the probability by category, then the category with the highest probability is the category predicted by the model.

However, I also know that for that kind of regression, Stata estimates a latent continuous variable, then cut it (regarding the cutoff values) and then use it to predict the dependent variable.

How these two explanations can be reconciled? I mean, if I use my first explanation, I have everything I need, I don't see where using this latent help me in the process. I hope I am clear enough, because I thought I understood it well but in fact I am confused.

thanks in advance ,

Jean

[Richard Williams]

First off, what program are you using? The user-written oglm or something else?

you might see if these handouts help you with interpretation:

https://www3.nd.edu/~rwilliam/xsoc73994/Ologit01.pdf

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

[Weiwen Ng]

However, I also know that for that kind of regression, Stata estimates a latent continuous variable, then cut it (regarding the cutoff values) and then use it to predict the dependent variable.

How these two explanations can be reconciled? I mean, if I use my first explanation, I have everything I need, I don't see where using this latent help me in the process. I hope I am clear enough, because I thought I understood it well but in fact I am confused.
...

Jean, Stata is not literally estimating a latent continuous variable here (as far as I know). It is only estimating the probability of a higher response.

The latent continuous variable explanation comes from econometrics. It is useful as a way to think about the problem, and econometricians also use that framework to parameterize some problems. You do not need to think about it in the context of this problem. Also, I think your explanation is a bit incorrect. In that framework, the latent continuous variable is the real dependent variable, if you will, but I think that it is not directly estimated.

There are other statistical frameworks that do relate (multiple) ordinal items to a continuous latent variable, and that will go and directly estimate the latent variable. Structural equation modeling and item response theory do this. But, again, this is not what you are doing.

Also, while you didn't ask this, the -margins- post-estimation command can really help you understand your results as well. It's not immediately obvious what those coefficients mean, especially given the unusual link (which I am totally unfamiliar with). Here's an example using a stock Stata dataset:

webuse fullauto
ologit rep77 foreign length mpg
margins, over(foreign)
marginsplot, by(foreign)

Maybe not the best margins plot, but it will do. Side note: the FAQ recommends posting your results in code delimiters, using the # button on the formatting toolbar. It's guaranteed to be readable, whereas screenshots can't be accessed by everyone.

Crossed with Richard's post.

[Jean Torgigial]

Dear Weiwen, Richard,

Thanks for your answers;
I read what you posted. Richard, from page 2 of https://www3.nd.edu/~rwilliam/xsoc73994/Ologit01.pdf, I read "The continuous latent variable Y* has various threshold points. (κ is the Greeksmall letter Kappa.) Your value on the observed variable Y depends on whether or not you have crossed a particular threshold. For example, when M = 3", so what we are doing is putting our values for independent variables then comparing in which threshold we are, and then we have our output.


But at te same time, I read it, from a SPSS tool, (I also ran it on SPSS, where the thresholds are the same and same estimates etc, maybe you remember me from previous post about same data ):
"For example, suppose you have an applicant who wants a 48-month loan (duration), is 22 years old (age), has one credit with the bank (numcred), has no other installment debt (othnstal), and owns her home (housng). Inserting these values into the prediction equations, this applicant has predicted values of -2.78, -1.95, 0.63, and 0.97. (Remember that there is one equation for each category except the last.) Taking the inverse of the complementary log-log link function gives the cumulative probabilities of .06, 0.13, 0.85, and 0.93 (and, of course, 1.0 for the last category). Taking differences gives the following individual category probabilities: category 1: .06, category 2: 0.13-0.06=0.07, category 3: 0.85-0.13=0.72, category 4: 0.93-0.85=0.08, and category 5: 1.0-0.93=0.07. Clearly, category 3 (debt payments current) is the most likely category for this case according to the model, with a predicted probability of 0.72. Thus, you would predict that this applicant would keep her payments current and the account would not become critical." So here, we see that the difference between -2.78 and -1.95 and the one between 0.63 and -1.95 etc. corresponds to the difference between the thresholds (sorry for the picture, it is from the SPSS tool i can only screenshot it, I put it at the end of my post), so we add the thresholds values (=cutoff in stata) to estimate the different equation. Then we take inverse of link function and then from cumulative probabilities we find the probabilities for each category and finally, the one with highest probability comes out as predicted category. This is different from what I read in https://www3.nd.edu/~rwilliam/xsoc73994/Ologit01.pdf, And I read theses two versions of the story everywhere on the Internet. I just can't understand how they can be explained simultaneously.
(I am working on stata but I have exactly same results on SPSS)

Kind Regards,

Jean

[Richard Williams]

Sorry, I am not following this. First off, I don’t agree that you predict a specific category. You predict probabilities of being in each category. So, one category might have a predicted probability of .3, and that may be higher than any other category, but there is still a 70% chance of another outcome. You can say that an outcome is the most likely but that doesn’t necessarily even mean it is likely. I rarely if ever see analyses in which one category is declared to be the winner for each individual case.

Further, as far as I know, Spss and ologit yield the same predicted probabilities.

Sno, I am not sure where the conflicts are supposed to be.

[Jean Torgigial]

Dear Richard,

Sorry, but I don't get it. If I take my screenshot, the one from the previous message. How is it predicting the predicted category for you? we take our independent variables, and then look where we are comparing to the thresholds, and then that's it, right?

But from my previous message too, the explanation from the SPSS tool is different, I just copied/pasted it. This is where I am confused.

thanks for you answer,

Kind Regards,

Jean

[Richard Williams]

The ologit handout I linked to before shows how compute the predicted probabilities for each outcome. You could do it by hand, but it is much easier to use the predict command.

[Jean Torgigial]

Thank you,

Kind Regards,

Jean

[Weiwen Ng]

Jean, in the ologit pdf of Richard’s that you linked, the second page explains that Stata doesn’t estimate Y* directly, but it does estimate Z, which is the linear predictor (XB) for each observation. Y* = Z + a random error term. You don’t know the magnitude of this error term. It could be very large. All you can do is estimate each person’s probability of responding in each answer category. Z is not deterministic.

[Richard Williams]

You can think of Z as being like yhat in an OLS regression. Z is like our best guess as to what Y* equals, but there is an error term so the actual value of Y* may be higher or lower. If we somehow knew the real value of Y*, knowing Y* would tell you the value of Y.

incidentally I suspect there is no conflict between the way I explain it and the way Spss explains it. They are probably different but equivalent approaches.