I am building a ML ordered logistic regression. First of all, I really don't know if this is the best way to fit a model to my data, as I am not too confident in ML ordered logit regressions, compared to ML linear regressions. The relevant variables are as follows:

$Y_{ij}$ = support for the European Union, where 0 = no support, 1 = some support and 2 = much support.

$X_1$ = constitutional patriotism as a sumscore of three variables. Possible outcomes are $0,1,...,9$

$X_2$ = cultural patriotism as a sumscore of two variables. Possible outcomes are: $0,1,...,6$

The ML model consists of individuals nested with seven countries. Apart from the variables above, I have added a lot of control variables. First off, I want to create an interaction term including constitutional patriotism (CON) and cultural patriotism (CUL), where both variables are treated as continuous. The problem though, is that the two scales are not comparable. Hence, I thought about scaling the variables like this:

$X_{1 scaled} = X_1/9$ and $X_{2 scaled} = X_2/6$

By doing it this way, I secure that both variables ranges from 0 to 1. Hence, a one unit increase in $X_1$ is comparable to a one unit increase in $X_2$ as they both have minimum = 0 and maximum = 1.

  • Do you think it would be more appropriate to standardize the variables instead, so they have mean 0 and variance 1?

Next I will have to choose between ML linear regression and ML ordered logistic regression. As said before, I am not too confident in the practical use of ordered logit. I wonder how much of a difference it makes, taking into account that ML ordered logit adds substantially more complexity to the model. I will, of course, choose a design that both makes sense, and delivers the best fit to the data. If I run a ML linear regression with my ordinal DV, the stadardized residuals are distributed as follows:

Standardized residuals

This seems to be a problem, as the residuals are clearly not normal distributed.

  • Does this call for a ordered logit, or could I work around the problem in some other way?


  • Do you know of any methods to check how well my model fits the data in a ordered logit regression?

I have thought about calculating the predicted distribution of individuals choosing $Y_{ij}=0$, $Y_{ij}=1$ and $Y_{ij}=2$ compared to the actual distribution found in the data. Also, I want to investigate if the ordered logit violates the proportional odds assumption. However, I am not sure how to do this in a ML framework.

Any suggestions are much appreciated!


There is some room for disagreement here, but I'd expect most statistical people to consider that

  1. As your response variable is just an ordered scale, linear regression would be at best highly dubious and at worst absurd. For example, it might be that substantively a more appropriate scaling is 0, 3, 5 or 0, 4, 6 and your regression results would then be quite different. Ordered logit is more nearly the default method in your situation, although there are other methods. My own take is that the argument that measurement scale implies appropriate model is often overplayed within statistics, but I can't take seriously a linear model for predicting a 0-1-2 ordinal scale. At a minimum, if you choose a linear regression, expect serious flak from some of your peers.

  2. Linear rescalings of your predictors are not needed, or alternatively which you choose is a matter of taste or convenience. The coefficients will adjust. In practice, importance or significance is assessed by looking at sign and magnitude of coefficients relative to their standard errors.

  3. The residuals from your model will have a spiky distribution given categorical responses and predictors, so a kernel density estimate does not seem especially helpful. That said, the distribution of the residuals does not look especially bad here, but the bigger argument by far against a regression model is that it does not match the measurement scale of the response.

You are evidently using Stata; postestimation commands are well documented and there are literature references in the Stata manual, so the tail of your question is asking for standard material.

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  • $\begingroup$ Thanks for your insights! I agree with you. It is probably impossible to justify the use of linear regression with an ordinal DV. However, I don't quite follow comment #2. Yes, the coefficients will adjust, but would it be possible to compare the two scales if they both were in [0;1]? $\endgroup$ – thesixmax May 7 '14 at 14:46
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    $\begingroup$ In linear regression coefficients can be interpreted as derivatives with units of measurement in numerator and denominator and that can be helpful. With ordinal logit, people don't usually even try, but just look at $z$ and $P$-values. Hence rescalings seem a waste of effort. $\endgroup$ – Nick Cox May 7 '14 at 15:42

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