# Building a ML ordered logit regression model

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:

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?

Finally:

• 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!

• 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. – Nick Cox May 7 '14 at 15:42