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I would like to produce a regression analysis model. I have ordinal categorical data. I can use SPSS. I do not know what analysis to perform or what assumptions to check.

What are the statistical tests that can be performed? How can I test the accuracy of the model?

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    $\begingroup$ Ordinary least squares regression is not suitable for ordinal catgeorical dependent variables. If you try searching on 'ordinal response' there are some useful questions and answers. Searching on 'ordered categorical response' finds some more, such as this $\endgroup$
    – Glen_b
    Commented Aug 15, 2013 at 9:44
  • $\begingroup$ Also try this. $\endgroup$
    – Glen_b
    Commented Aug 15, 2013 at 9:57
  • $\begingroup$ I should modify my earlier comment to say "often not suitable" - there are circumstances where it's reasonable to do so. $\endgroup$
    – Glen_b
    Commented Sep 18, 2013 at 1:05

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Relying on SPSS is quite limiting.

If both Y and the Xs are ordinal and the Xs contain more than 3 categories there is only one completely satisfactory solution in my opinion. Use a Bayesian ordinal semiparametric regression model (and here) in a way that also takes into account the ordinal nature of the Xs. This is elegantly implemented in the R brms package as described here. See also this. In brms if an R variable is an ordered factor I think the fitting function automatically handles the ordinal nature of the predictor.

The principled approach to handling ordinal Xs makes full use of the X ordering and does not require a degree of freedom to be spent for each X category. It is roughly described as follows: Assign an indicator variable to each level of each ordinal X other than the first level. Use a Bayesian prior using the outside information about ordinality of Xs by penalizing the regression coefficients of each indicator variable to make the effect monotonic (ever increasing or ever decreasing over levels of that X). Examples:

  • a binary predictor (which is also ordinal) will get a whole parameter (effectively one d.f.)
  • an ordinal predictor with 3 levels will get 2 parameters but their standard errors will be smaller because of the ordinal (monotonicity) restriction; the effective d.f. might be 1.4.
  • an ordinal predictor with 20 levels will get 19 parameters but might have an effective d.f. of 3.3.

Predicted values as you increase the values of one X will be monotonic in that X. Bayesian inference will be exact (no approximations to the likelihood function and no approximations related to the penalization to ordinality). This is the only approach I know that fully respects the ordinal nature of X and that requires no arbitrary metric to represent X.

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Yes, it is possible.

When your dependent variable is ordinal, you want to do ordinal logistic regression. This can be done in SPSS. UCLA's excellent statistics help website has a guide to OLR in SPSS here (with more here).

Regarding your independent variables, you have several options:

  • You can represent them with a standard dummy coding scheme (such as reference cell coding, see my answer here for an explanation).
  • Another approach is to use an ordinal dummy coding scheme (such as difference coding, there is an explanation here).
  • Lastly, Agresti has argued that you can simply replace the ordinal rankings with continuous values that represent your best guesses about the true values. There will naturally be some measurement error associated with this approach, but if you have some knowledge on which to base your guesses they won't be too bad, and you won't use as many degrees of freedom to estimate the effect.

If you use OLR for your analysis, you can get tests of each variable with standard output. In SPSS these tests are reported in the "Parameter Estimates" table. The assumption you need to worry about / check is the proportional odds assumption, which is assessed via the "Test of Parallel Lines". SPSS can output this for you as well. UCLA's guide to OLR in SPSS (linked above) covers both of these issues.

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For ordinal categorical response data, the best thing is usually to use a cumulative multinomial logit model. Those are somewhat non-trivial to implement, though I think SAS can do it and there is (I think) some R package that can do it. I would have to check on which one though. The idea is that you're modeling the probability that the response is less than or equal to each category.

For the dependent variables, you can keep those as categorical, or you can set them to be ordinal values and use a linear trend on your selected values. The latter is usually effective in practice even if the interpretation of the coefficients is challenging. For a logistic model, the Mantel-Haenszel test will let you test the linear trend. For a cumulative multinomial logistic model, there might be an analog that works as well.

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  • $\begingroup$ Edited an objective typo, from Hansel to Haenszel. Otherwise I flag that "best" in para. 1 is opinion and that generally categorical here should be glossed as nominal: that is common but far from universal usage. In my reading, it is more common for categorical to include nominal and ordinal, as e.g. in Agresti's books. $\endgroup$
    – Nick Cox
    Commented Sep 2, 2023 at 10:56

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