Solved – Ordered Logit Versus OLS

least squaresordered-logitordinal-data

I have an ordinal dependent variable. (I am looking at state bond ratings.) It ranges from 1 to 9. The 1st quantile value is 7– so it is highly skewed with very few low values.

I understand some prefer OLS over ordered logit because the coefficients are easier to interpret and OLS is fairly robust. However, it assumes each category has an equal distance apart.

In order to estimate my coefficients with an ordered logit, I have to collapse a few categories in my DV (and lose information). Specifically, my scale is now 1 to 4. The OLS results (with the collapsed DV or the 1 to 4 scale) are similar to the ordered logit's, however, when I estimate the parameters for my model using OLS with out collapsing the DV (or the 1 to 9 scale), the results are substantially different.

In your opinion, between the OLS or ordered logit models, which would you prefer? With the ordered logit I am losing information by combining the lower categories, but my dependent variable is ordinal.

Best Answer

Long posts have appeared on this site about this subject.

You do not have to collapse any categories when using ordinal regression, and you usually shouldn't. The only issue you'll run into is computer time if you have more than a few hundred distinct $Y$ values. However in the R rms package orm function you can efficiently fit thousands of intercepts, just as you can in JMP.

I don't feel that OLS is very robust. Ordinal (semiparametric) regression is much more robust, and makes no distributional assumption for any one set of predictor values.

Related Solutions

Solved – Ordinal regression: proportional odds assumption

First off, are your two independent variables being adjusted as factors or numerically coded responses and is there an interaction term for the two? The reason I ask is because the test of proportional odds grows very sensitive with small cell counts. For this reason, I often find it justifiable to adjust input variables as their ordinally coded values (1: poor, 2: fair-to-poor, etc.). Doing so allows information to be borrowed across groups, proportionality is assessed so that an associated difference in the odds of a more favorable response comparing units differing by 1 in the predictor are consistent with odds of an even more favorable response (the rough and contrived interpretation of the test of proportional odds).

If your numeric coding still fails to give valid proportionality, it is possible to get consistent cumulative odds ratios estimates by collapsing adjacent categories like the two bottom box responses.

Thirdly, another powered test of association between an ordinal response and two ordinal factors is a plain old linear regression model. Using robust standard errors, you get valid confidence intervals despite the distribution of the errors. This tends to be less powerful that categorical methods, but with fewer pitfalls due to zero cell counts.

Lastly, as a comment, robust standard errors allow consistent estimation of the mean model in most circumstances. I'm not sure if these are implemented in SPSS, but R and SAS use these frequently. As with the proportional hazards assumption in the Cox model, when this "model based assumption check" fails, it does not mean the model results are entirely invalid, it's just that the effect estimates are "averaged" over their inconsistent proportionality. For instance, if proportional odds model has excessive numbers of respondents giving top box responses, and a predictor shows a large association for the top box response but smaller association for other cumulative measures, then you'll find that the cumulative odds ratio is a weighted combination of the several thresholded odds ratios, with a higher weight placed upon the top box OR.

Solved – Ordered Probit and categorical variables

Results from an ordered logit/probit regression are always unintuitive, but categorical explanatory variables are as meaningful as continuous ones. I'd even say that they are easier to interpret.

For a concrete example, you could look at Dobson, An Introduction to Generalizer Linear Models, 2002, 2nd ed., Chapter 8. In her "car preferences" example, the dependent variable is the importance of air conditioning and power steering (three levels: "no or little importance", "important", "very important") and the two explanatory variables are gender (male or female, coded as 1 and 0) and age (18-23, 24-40, >40, coded as age2440 = 1 or 0, and agegt40 = 1 or 0).

Fitting an ordered probit model you get (I've used R, MASS library, polr() function):

Coefficients:
   male age2440 agegt40 
-0.3467  0.6817  1.3288 

Intercepts:
  NoImp|Imp Imp|VeryImp 
    0.01844     0.97594

Then you can compute the probabilities for women (male = 0) over 40 (age2440 = 0, agegt40 = 1):

NoImp     Imp VeryImp 
0.095   0.267   0.638

and for men over 40 (male = 1):

NoImp     Imp VeryImp 
0.168   0.330   0.502

Their difference is the gender partial effect:

 NoImp     Imp VeryImp 
-0.073  -0.063   0.136

I think that it's meaningful ;-)

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Related Solutions

Solved – Ordinal regression: proportional odds assumption

Solved – Ordered Probit and categorical variables

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