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In the social sciences I have encountered that it is common to treat ordinal variables as continuous, for example variables originating from rating or Likert scales (strongly disagree, disagree, agree, strongly agree).

This topic has been discussed for example in this post from 2010: Under what conditions should Likert scales be used as ordinal or interval data?

I am looking for a more formal comparison/evaluation especially in the context of regression modeling. Rhemtulla et al. (2012) examine the performance of treating ordinal variables as continuous and make recommendations for structural equation models (SEM). I am not very familiar with SEMs, so I'm not sure if their results would also apply to regression problems.

Does anyone know about similar studies/literature in the context of regression?

Edit: Just to answer the question below: I'm mainly interested in the case where the outcome variable is ordinal (with possibly an ordinal covariate).

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    $\begingroup$ It's not clear to me whether your focus is on ordinal variables as (1) responses or outcomes (2) predictors or explanatory variables (3) either. (2) is easiest as you just compare predictions for different representations of the ordinal variables. (1) and (3) are difficult because you have to compare quite different kinds of models with quite different kinds of prediction. However, even if you treat ordinal responses as continuous that makes your life easier until you have to work out what a prediction of 4.5678 means when your categories are 1,...,5. (Or 1,...4!). $\endgroup$ – Nick Cox Apr 14 '14 at 12:51
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    $\begingroup$ I guess it is (1) and (3). But mainly (1). $\endgroup$ – Francis Apr 14 '14 at 14:04
  • $\begingroup$ Re @Frank's suggestion of penalizing predictors towards ordinality: see the reference given here $\endgroup$ – Scortchi Apr 14 '14 at 17:41
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    $\begingroup$ Another aspect is the number of categories. Without reference, I think there is some simulation research that has shown Likert scales with more than 5 categories can often be treated as continuous. If I remember the reference, I will post it below. Nevertheless, can you explain your major reason why you do not want to model the outcome variable by ordinal regression such as proportional odds models? $\endgroup$ – tomka Apr 14 '14 at 21:56
  • $\begingroup$ @Scortchi I did not know about that R package. Wonderful. tomka if it is $Y$ that is ordinal then the proportional odds, proportional hazards, or other ordinal models may indeed be excellent choices. $\endgroup$ – Frank Harrell Apr 15 '14 at 2:16
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It's important to distinguish, as pointed out by Nick Cox, between iV and dV. As far as dV is concerned, why not use a ordinal regression model, as discussed excellently e.g. by Agresti: http://eu.wiley.com/WileyCDA/WileyTitle/productCd-0470082895.html

I am less sure about the iV case. Standard would perhaps use dummy coding. I suppose this is what Frank Harrell means. Maybe Agresti discusses this as well.

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    $\begingroup$ The abbreviations iV and dV are fairly easily decoded, but (minor point) IV usually means "instrumental variable" to many economists and some others (major point) perpetuate the awful, if still popular, terminology of dependent and independent variables. "Independent" variables usually aren't. At an elementary level, many people still get confused about which is which. $\endgroup$ – Nick Cox Apr 14 '14 at 13:19
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With the luxury of time we would use dummy variables as with nominal predictors, then penalize them (penalized MLE) towards ordinality of effects. Something like that was discussed in a paper by Hans van Houwelingen some years ago. Short of that, we often approximate the effect of ordinal variables by fitting a quadratic effect. It would also not be ridiculous to use AIC to select between a regular nominal dummy variables model and a restricted model that assumed the ordinal predictor was continuous (like the quadratic).

I'm not sure that the SEM results would apply, but they might.

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I have one source, Snijders and Bosker`s (2012) multilevel analysis book, page 310, saying:

"if the number of categories is small (3 or 4), or if it is between 5 and 10, and the distribution cannot well be approximated by a normal distribution, then statistical methods for ordered categorical outcomes can be useful"

My understanding is, if you have at least 10 categories and approximately normally distributed dependent variable, it is safe to treat it as a continuous variable. For a more concrete answer, I would run a small scale simulation analysis.

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    $\begingroup$ It's hard to see how one could implement this advice (which you repeat verbatim in an answer to a different kind of question involving counted responses) until the categories have been assigned meaningful numerical values. Providing exactly the same answer to different questions suggests you ought to consider modifying or qualifying at least one of the answers. If you truly believe this answer is correct and appropriate for both questions, then the right response is to flag one of the questions as a duplicate of the other. $\endgroup$ – whuber May 11 at 14:26
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    $\begingroup$ See my attempt and let me know if it satisfies your concern. $\endgroup$ – Burak Aydin May 11 at 14:40
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If an outcome is ordinal, one should want a method of analysis that is invariant to the codes used to label the levels. For example, suppose the outcome has levels : SD, D, N, A and SA. Then one might label the levels with codes 1, 2, 3, 4, 5. If one analyzes this outcome with a t-test, the p-value is invariant only to location or scale changes. For example, -2, -1, 0, 1, 2 or -4, -2, 0, 2, 4. The p-value from the t-test is not invariant to other codings like -10, -1, 0, 1, 100 or any coding that does not preserve 'distance'. The proportional odds model and the multinomial model give p-values that are invariant to the coding selection. [maybe this point has been made earlier?]

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Liddell & Kruschke (2018) is another source which discusses problems associated with treating ordinal data as continuous. The paper illustrates a number of the problems that can occur.

They advocate using ordered-probit models to deal with ordinal data. While they specifically advocate for a Bayesian approach, they note that Frequentist approaches may also work

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