Inconsistent interpretation for the ologit coeffcicients

[Nader Mehri]

The interpretation of the logit coefficients and odds ratio in ordered logistic regression looks quite different, though they stem from the same statistical concept. For example, the ologit output posted at the UCLA Statistical Consulting Group web page, interprets the logit coefficients in a very intuitive way but when it comes to the odd ratios of the same coefficients, the interpretation becomes less intuitive. I wonder why they differ in intuitiveness; consistent with the interpretation for the logit coefficients, would it be wrong to say that the odd of higher SES increases by 3% per one unit increase in science score.

Here is the interpretation for both the logit coefficients and the odd ratios from the UCLA webpage!


Parameter Estimates

science – This is the ordered log-odds estimate for a one unit increase in science score on the expected ses level given the other variables are held constant in the model. If a subject were to increase his science score by one point, his ordered log-odds of being in a higher ses category would increase by 0.03 while the other variables in the model are held constant.

socst – This is the ordered log-odds estimate for a one unit increase in socst score on the expected ses level given the other variables are held constant in the model. A one unit increase in socst test scores would result in a 0.0532 unit increase in the ordered log-odds of being in a higher ses category while the other variables in the model are held constant.

female – This is the ordered log-odds estimate of comparing females to males on expected ses given the other variables are held constant in the model. The ordered logit for females being in a higher ses category is 0.4824 less than males when the other variables in the model are held constant.

Odds Ratio Interpretation


The following is the interpretation of the ordered logistic regression in terms of proportional odds ratios and can be obtained by specifying the or option. This part of the interpretation applies to the output below.

a. Odds Ratio – These are the proportional odds ratios for the ordered logit model (a.k.a. proportional odds model) shown earlier. They can be obtained by exponentiating the ordered logit coefficients, e^coef., or by specifying the or option. Recall that ordered logit model estimates a single equation (regression coefficients) over the levels of the dependent variable. Now, if we view the change in levels in a cumulative sense and interpret the coefficients in odds, we are comparing the people who are in groups greater than k versus those who are in groups less than or equal to k, where k is the level of the response variable. The interpretation would be that for a one unit change in the predictor variable, the odds for cases in a group that is greater than k versus less than or equal to k are the proportional odds times larger. For a general discussion of OR, we refer to the following Stata FAQ for binary logistic regression: How do I interpret odds ratios in logistic regression?

science – This is the proportional odds ratio for a one unit increase in science score on ses level given that the other variables in the model are held constant. Thus, for a one unit increase in science test score, the odds of high ses versus the combined middle and low ses categories are 1.03 times greater, given the other variables are held constant in the model. Likewise, for a one unit increase in science test score, the odds of the combined high and middle ses versus low ses are 1.03 times greater, given the other variables are held constant.

socst – This is the proportional odds ratio for a one unit increase in socst score on ses level given that the other variables in the model are held constant. Thus, for a one unit increase in socst test score, the odds of high ses versus the combined middle and low ses are 1.05 times greater, given the other variables are held constant in the model. Likewise, for a one unit increase in socst test score, the odds of the combined high and middle ses versus low ses are 1.05 times greater, given the other variables are held constant.

female – This is the proportional odds ratio of comparing females to males on ses given the other variables in the model are held constant. For females, the odds of high ses versus the combined middle and low ses are 0.6173 times lower than for males, given the other variables are held constant. Likewise, the odds of the combined categories of high and middle ses versus low ses is 0.6173 times lower for females compared to males, given the other variables are held constant in the model.

[Maarten Buis]

I don't see an inconsistency in the UCLA webpage. That is how you interpret an ordered logistic regression.

[Nader Mehri]

I don't see an inconsistency in the UCLA webpage. That is how you interpret an ordered logistic regression.

I wonder why the intrepretation for the coeffcients in logit and odd ratio forms differ in intuitiveness; consistent with the interpretation for the logit coefficients, would it be wrong to simplify the interpretation of odds ratio by saying that the odd of higher SES increases by 3% per one unit increase in science score (i.e., ignoring the cumulative nature of the odds ratio from ordered logistic regression model).

[Maarten Buis]

You cannot ignore the cumulative nature because that is the core of that model.

[Nader Mehri]

You cannot ignore the cumulative nature because that is the core of that model.

Thanks for your clarification! I am curious about how the interpretation for the UCLA example would change if the assumption were violated specifically for the science variable. If the parallel odds assumption for science was violated, we'd have two odds ratios instead of a single proportional OR. Given that high SES is the reference category, how would we interpret the results in this situation? Specifically, would the interpretation for the low ses odds ratio compare "high vs. medium and low". LIkewise, would the OR for the medium ses comapre "high and medium vs. low"? Or would we need to present the results differently?