Discussion: View Thread

This message has been cross posted to the following eGroups: Teaching Statistics in Health Sciences Section and Statistical Consulting Section .
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I have a graduate student from the areas of education and psychology that is looking at stress (IV), satisfaction (DV), well-being (DV) and passion (covariate).  She wants to do regression analysis, but all of the variables are ordinal.  It appears that in her field of study, most paper analyze the data treating it as if it is quantitative.  I am not really comfortable with this.  What do you do when you have a regression problem with both variable ordinal?  Is there a computer program that will do ordinal regression?  Help 

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Grenith Zimmerman
Loma Linda University
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Hi Grenith,

Many disciplines allow for a OLS regression on seemingly ordinal variables especially if they are Likert scaled.  From the sound of the DV's it may be that these are actually variable constructs, derived by adding or averaging a couple of survey items? If so, then you can use an OLS regression no problem, but you'd have to run one for each of the DV's.

If you truly have ordinal DV's, then SPSS has syntax called PLUM that you can use. STATA also has ordinal regression, and R software has a package or two for ordinal regression...here is a link to information on one:

http://www.ats.ucla.edu/stat/r/dae/ologit.htm

In fact, the UCLA website is an awesome resource, you may be able to find more information and annotated output on ordinal regression in different programs.  And I am sure you will receive much advice from this group, a very helpful bunch.

Best,

Elaine
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Elaine Eisenbeisz
Owner and Principal Statistician
Omega Statistics
-------------------------------------------

Hi Grenith,

Whether it is reasonable to treat the variables as numerical depends on several issues, including (a) the number of levels, particularly in the response (dependent) variable and (b) the adequacy of the scores used for the numerical values, specifically in their ability to capture the nature of any relevant associations.  If the number of levels in Y is too small, then a linear regression is probably not good at all and is quite likely to make predictions that lie beyond the range of scores.  

Two alternatives are (1) loglinear modeling of the counts in combinations of variables, if ALL variables are categorical (nominal or ordinal) and the combinations of variable levels are mostly populated, or (2) multinomial regression modeling.  The former is often easier to conduct and interpret; the latter is more flexible in the sense that it allows continuous variables.  

To make use of ordering, you need to add assumptions.  In the loglinear model, you need to substitute scores for the variable levels in the models for associations (interactions) between DV and and IVs of interest.  In the multinomial regression, you need to make assumptions relating to the ways in which the odds ratios of the DV relate to explanatory variables (typically "proportional odds").  

All of these ideas are detailed in the book referenced in my signature below.  Unfortunately, it's not due out until the end of the month...

Good luck.

-Tom.


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Thomas Loughin
Simon Fraser University
Coauthor, Analysis of Categorical Data with R, CRC Press
http://www.crcpress.com/product/isbn/9781439855676
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Also Grenith, I just remembered that I think I've heard that if you have 8 or more levels on the ordinal variable you can use OLS regression.  I don't  have the source...just something  I've picked up in practice. I think one of my statisticians knows a source, he was looking into this a couple of weeks ago actually.

I will ask him about it when I see him tomorrow and post it if he remembers it.
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Elaine Eisenbeisz
Owner and Principal Statistician
Omega Statistics
-------------------------------------------



Original Message:
Sent: 07-02-2014 20:15
From: Thomas Loughin
Subject: Regression Analysis for Ordinal Data

Hi Grenith,

Whether it is reasonable to treat the variables as numerical depends on several issues, including (a) the number of levels, particularly in the response (dependent) variable and (b) the adequacy of the scores used for the numerical values, specifically in their ability to capture the nature of any relevant associations.  If the number of levels in Y is too small, then a linear regression is probably not good at all and is quite likely to make predictions that lie beyond the range of scores.  

Two alternatives are (1) loglinear modeling of the counts in combinations of variables, if ALL variables are categorical (nominal or ordinal) and the combinations of variable levels are mostly populated, or (2) multinomial regression modeling.  The former is often easier to conduct and interpret; the latter is more flexible in the sense that it allows continuous variables.  

To make use of ordering, you need to add assumptions.  In the loglinear model, you need to substitute scores for the variable levels in the models for associations (interactions) between DV and and IVs of interest.  In the multinomial regression, you need to make assumptions relating to the ways in which the odds ratios of the DV relate to explanatory variables (typically "proportional odds").  

All of these ideas are detailed in the book referenced in my signature below.  Unfortunately, it's not due out until the end of the month...

Good luck.

-Tom.


-------------------------------------------
Thomas Loughin
Simon Fraser University
Coauthor, Analysis of Categorical Data with R, CRC Press
http://www.crcpress.com/product/isbn/9781439855676
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Given how cheap computing is, I suggest doing both, then plotting the results against each other.

Also, carefully check whether the assumptions of OLS re the residuals are met.

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Peter Flom
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Catherine Durso
Lecturer
University of Denver
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Specifically, compare the results of treating the variables as continuous and as categorical, right?


Original Message:
Sent: 07-02-2014 20:43
From: Peter Flom
Subject: Regression Analysis for Ordinal Data

Given how cheap computing is, I suggest doing both, then plotting the results against each other.

Also, carefully check whether the assumptions of OLS re the residuals are met.

-------------------------------------------
Peter Flom
-------------------------------------------

I think you can just do multinomial logistic regression (also called polytomous regression) using SAS - either PROC LOGISTIC or PROC CATMOD does it, if my memory is correct.

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[Daniel] [Jeske]
[Professor and Chair]
[Department of Statistics]
[University of California - Riverside]
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Original Message:
Sent: 07-02-2014 23:53
From: Catherine Durso
Subject: Regression Analysis for Ordinal Data



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Catherine Durso
Lecturer
University of Denver
-------------------------------------------
Specifically, compare the results of treating the variables as continuous and as categorical, right?

You certainly can do both multinomial and ordinal logistic regression in SAS and also in R

In SAS, if the DV is categorical, PROC LOGISTIC assumes you want a multinomial model unless you specify otherwise.

See my paper http://www.nesug.org/proceedings/nesug05/an/an2.pdf for more

The problem with multinomial is that you get a lot of parameters to estimate and interpret, especially when the DV has many levels.

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Peter Flom
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Original Message:
Sent: 07-03-2014 01:02
From: Daniel Jeske
Subject: Regression Analysis for Ordinal Data

I think you can just do multinomial logistic regression (also called polytomous regression) using SAS - either PROC LOGISTIC or PROC CATMOD does it, if my memory is correct.

-------------------------------------------
[Daniel] [Jeske]
[Professor and Chair]
[Department of Statistics]
[University of California - Riverside]
-------------------------------------------



Original Message:
Sent: 07-02-2014 23:53
From: Catherine Durso
Subject: Regression Analysis for Ordinal Data



-------------------------------------------
Catherine Durso
Lecturer
University of Denver
-------------------------------------------
Specifically, compare the results of treating the variables as continuous and as categorical, right?

In SPSS CATREG (categorical regression) can fit the regression with different asumptions about the level of measuerement.
If you do not find a meaningful difference in fir and tests between assuming the level of measurement is only ordinal and assuming interval level of measurement you may want to consider the more well known ordinary regression and just mention that ordinal regression gave very comparable results.

SPSS has several procedures that will fit ordinal regression is you do not want to test the diffeernce in fit between ordial and interval level of regression.

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Arthur Kendall
Social Research Consultants
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Original Message:
Sent: 07-03-2014 05:41
From: Peter Flom
Subject: Regression Analysis for Ordinal Data

You certainly can do both multinomial and ordinal logistic regression in SAS and also in R

In SAS, if the DV is categorical, PROC LOGISTIC assumes you want a multinomial model unless you specify otherwise.

See my paper http://www.nesug.org/proceedings/nesug05/an/an2.pdf for more

The problem with multinomial is that you get a lot of parameters to estimate and interpret, especially when the DV has many levels.

-------------------------------------------
Peter Flom
-------------------------------------------



Original Message:
Sent: 07-03-2014 01:02
From: Daniel Jeske
Subject: Regression Analysis for Ordinal Data

I think you can just do multinomial logistic regression (also called polytomous regression) using SAS - either PROC LOGISTIC or PROC CATMOD does it, if my memory is correct.

-------------------------------------------
[Daniel] [Jeske]
[Professor and Chair]
[Department of Statistics]
[University of California - Riverside]
-------------------------------------------

see my previous response about using CATREG.  Deonding on the actual response scale used, it is not not unusual for psychological to be be not very discrepant from interval level of measure. This especially true when summative scales are used. i.e., when several items are used as repeated measures of a construct.

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------



Original Message:
Sent: 07-02-2014 20:43
From: Peter Flom
Subject: Regression Analysis for Ordinal Data

Given how cheap computing is, I suggest doing both, then plotting the results against each other.

Also, carefully check whether the assumptions of OLS re the residuals are met.

-------------------------------------------
Peter Flom
-------------------------------------------

Here's another approach.  Ordinal variables are bounded (e.g. 1-5, 1-10), a situation similar to that seen in a logistic regresion.  Therefore, first change the scale to probabilities (i.e. 10 = .95, 9 = .85, etc.).  Then convert these to logistics and your results wil;l be bounded.  This way your dependent variable is logistic and you can use it in a regression analysis. 

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Ihor Kowalysko
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Original Message:
Sent: 07-03-2014 08:06
From: Arthur Kendall
Subject: Regression Analysis for Ordinal Data

see my previous response about using CATREG.  Deonding on the actual response scale used, it is not not unusual for psychological to be be not very discrepant from interval level of measure. This especially true when summative scales are used. i.e., when several items are used as repeated measures of a construct.

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------

There are some good ideas in this stream. However, there is a lot of confusion. If the ordinal variable is a dependent variable, that is one thing. If the ordinal variable is an independent, that is entirely different. You don't use multinomial methods for prediction of a continuous dependent with an ordinal IV. So, when people make suggestions, it would be good to consider if your suggestions are for the analysis of an ORDINAL DV or with an ORDINAL IV for a CONTINUOUS DV.

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Paul Thompson
Director, Methodology and Data Analysis Center
Sanford Research/USD
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:29
From: Ihor Kowalysko
Subject: Regression Analysis for Ordinal Data

Here's another approach.  Ordinal variables are bounded (e.g. 1-5, 1-10), a situation similar to that seen in a logistic regresion.  Therefore, first change the scale to probabilities (i.e. 10 = .95, 9 = .85, etc.).  Then convert these to logistics and your results wil;l be bounded.  This way your dependent variable is logistic and you can use it in a regression analysis. 

-------------------------------------------
Ihor Kowalysko
-------------------------------------------



Original Message:
Sent: 07-03-2014 08:06
From: Arthur Kendall
Subject: Regression Analysis for Ordinal Data

see my previous response about using CATREG.  Deonding on the actual response scale used, it is not not unusual for psychological to be be not very discrepant from interval level of measure. This especially true when summative scales are used. i.e., when several items are used as repeated measures of a construct.

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------

Hello Grenith,

Here is the reference I mentioned yesterday. It might be useful?  It mentions that an ordinal variable with 7 levels or more can be treated as continuous if you can assume the underlying construct would be continuous:

http://www.ncbi.nlm.nih.gov/pubmed/11725929

Have a great holiday everyone!

Elaine

-------------------------------------------
Elaine Eisenbeisz
Owner and Principal Statistician
Omega Statistics
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:48
From: Paul Thompson
Subject: Regression Analysis for Ordinal Data

There are some good ideas in this stream. However, there is a lot of confusion. If the ordinal variable is a dependent variable, that is one thing. If the ordinal variable is an independent, that is entirely different. You don't use multinomial methods for prediction of a continuous dependent with an ordinal IV. So, when people make suggestions, it would be good to consider if your suggestions are for the analysis of an ORDINAL DV or with an ORDINAL IV for a CONTINUOUS DV.

-------------------------------------------
Paul Thompson
Director, Methodology and Data Analysis Center
Sanford Research/USD
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:29
From: Ihor Kowalysko
Subject: Regression Analysis for Ordinal Data

Here's another approach.  Ordinal variables are bounded (e.g. 1-5, 1-10), a situation similar to that seen in a logistic regresion.  Therefore, first change the scale to probabilities (i.e. 10 = .95, 9 = .85, etc.).  Then convert these to logistics and your results wil;l be bounded.  This way your dependent variable is logistic and you can use it in a regression analysis. 

-------------------------------------------
Ihor Kowalysko
-------------------------------------------

Is this timely or what? I am just answering a reviewer who didn't like my use of the APGAR (10 level ordinal) score in a continuous manner. Your reference is just what the neonatologist ordered!

-------------------------------------------
Paul Thompson
Director, Methodology and Data Analysis Center
Sanford Research/USD
-------------------------------------------



Original Message:
Sent: 07-03-2014 11:08
From: Elaine Eisenbeisz
Subject: Regression Analysis for Ordinal Data

Hello Grenith,

Here is the reference I mentioned yesterday. It might be useful?  It mentions that an ordinal variable with 7 levels or more can be treated as continuous if you can assume the underlying construct would be continuous:

http://www.ncbi.nlm.nih.gov/pubmed/11725929

Have a great holiday everyone!

Elaine

-------------------------------------------
Elaine Eisenbeisz
Owner and Principal Statistician
Omega Statistics
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:48
From: Paul Thompson
Subject: Regression Analysis for Ordinal Data

There are some good ideas in this stream. However, there is a lot of confusion. If the ordinal variable is a dependent variable, that is one thing. If the ordinal variable is an independent, that is entirely different. You don't use multinomial methods for prediction of a continuous dependent with an ordinal IV. So, when people make suggestions, it would be good to consider if your suggestions are for the analysis of an ORDINAL DV or with an ORDINAL IV for a CONTINUOUS DV.

-------------------------------------------
Paul Thompson
Director, Methodology and Data Analysis Center
Sanford Research/USD
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:29
From: Ihor Kowalysko
Subject: Regression Analysis for Ordinal Data

Here's another approach.  Ordinal variables are bounded (e.g. 1-5, 1-10), a situation similar to that seen in a logistic regresion.  Therefore, first change the scale to probabilities (i.e. 10 = .95, 9 = .85, etc.).  Then convert these to logistics and your results wil;l be bounded.  This way your dependent variable is logistic and you can use it in a regression analysis. 

-------------------------------------------
Ihor Kowalysko
-------------------------------------------

Serendipity Rocks! :)

-------------------------------------------
Elaine Eisenbeisz
Owner and Principal Statistician
Omega Statistics
-------------------------------------------



Original Message:
Sent: 07-03-2014 11:28
From: Paul Thompson
Subject: Regression Analysis for Ordinal Data

Is this timely or what? I am just answering a reviewer who didn't like my use of the APGAR (10 level ordinal) score in a continuous manner. Your reference is just what the neonatologist ordered!

-------------------------------------------
Paul Thompson
Director, Methodology and Data Analysis Center
Sanford Research/USD
-------------------------------------------



Original Message:
Sent: 07-03-2014 11:08
From: Elaine Eisenbeisz
Subject: Regression Analysis for Ordinal Data

Hello Grenith,

Here is the reference I mentioned yesterday. It might be useful?  It mentions that an ordinal variable with 7 levels or more can be treated as continuous if you can assume the underlying construct would be continuous:

http://www.ncbi.nlm.nih.gov/pubmed/11725929

Have a great holiday everyone!

Elaine

-------------------------------------------
Elaine Eisenbeisz
Owner and Principal Statistician
Omega Statistics
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:48
From: Paul Thompson
Subject: Regression Analysis for Ordinal Data

There are some good ideas in this stream. However, there is a lot of confusion. If the ordinal variable is a dependent variable, that is one thing. If the ordinal variable is an independent, that is entirely different. You don't use multinomial methods for prediction of a continuous dependent with an ordinal IV. So, when people make suggestions, it would be good to consider if your suggestions are for the analysis of an ORDINAL DV or with an ORDINAL IV for a CONTINUOUS DV.

-------------------------------------------
Paul Thompson
Director, Methodology and Data Analysis Center
Sanford Research/USD
-------------------------------------------

I think Ihor's idea to re-scale the ordinal variables as quantities ranging from >0 to <1 is a great idea, especially when done the way he proposes using "probability"=(score-0.5)/max(score). Quantities that lie in this range are wonderfully amenable to something called beta regression, in which the underlying assumption is that the error terms follow the Beta distribution.  The parameter estimates from beta regression are in logits, and have much the same interpretation they have in logistic regression.

I am not an expert in beta regression, so I do not know if it would be a novel application to apply beta regression to ordinal variables. But I suspect it is close enough to being novel to get Dr. Zimmerman's student's future article some citations, should she choose to give it a whirl. SAS and Stata both do beta regression. I assume that R does, too. I don't know about SPSS.

Also, to illustrate how flexible beta regression can be, Swearingen et al (2011) published a well-executed application of beta regression to a biomedical problem where one wouldn't normally expect to see it: analyzing the brain volume affected by ischemic stroke. The free article can be found here: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3186717/. It is well worth reading.

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:29
From: Ihor Kowalysko
Subject: Regression Analysis for Ordinal Data

Here's another approach.  Ordinal variables are bounded (e.g. 1-5, 1-10), a situation similar to that seen in a logistic regresion.  Therefore, first change the scale to probabilities (i.e. 10 = .95, 9 = .85, etc.).  Then convert these to logistics and your results wil;l be bounded.  This way your dependent variable is logistic and you can use it in a regression analysis. 

-------------------------------------------
Ihor Kowalysko
-------------------------------------------



Original Message:
Sent: 07-03-2014 08:06
From: Arthur Kendall
Subject: Regression Analysis for Ordinal Data

see my previous response about using CATREG.  Deonding on the actual response scale used, it is not not unusual for psychological to be be not very discrepant from interval level of measure. This especially true when summative scales are used. i.e., when several items are used as repeated measures of a construct.

-------------------------------------------
Arthur Kendall
Social Research Consultants
-------------------------------------------

I find this a very interesting discussion and wanted to chime in...

I have found Agresti's 2010 "analysis of ordinal categorical data"  invaluable for analyses of ordinal data. In the first chapter, he addresses the concern about OLS (section 1.3).

On the side note generated by this thread and the link to ordinal beta regression:

My collaborators and I have published a paper in the Journal of Vegetation Science where we begin investigating the link between beta regression and ordinal data (Irvine, K. M. and Rodhouse, T. J. (2010), Power analysis for trend in ordinal cover classes: implications for long-term vegetation monitoring, Journal of Vegetation Science, 21: 1152-1161).

 Also, I am presenting the sequel at JSM in Boston where we have developed an ordinal regression model using a beta latent variable in a Bayesian framework. Essentially, we explicitly model the category probabilities which allows for a model that doesn't require the assumption of proportional odds and has nice intuitive appeal for my ecological colleagues. We have also extended the model to deal with zeros or an anchor on the ordinal scale as the support for the beta distribution doesn't include 0 or 1. 


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Kathryn Irvine
Statistician
US Geological Survey
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Original Message:
Sent: 07-07-2014 13:12
From: Eric Siegel
Subject: Regression Analysis for Ordinal Data

I think Ihor's idea to re-scale the ordinal variables as quantities ranging from >0 to <1 is a great idea, especially when done the way he proposes using "probability"=(score-0.5)/max(score). Quantities that lie in this range are wonderfully amenable to something called beta regression, in which the underlying assumption is that the error terms follow the Beta distribution.  The parameter estimates from beta regression are in logits, and have much the same interpretation they have in logistic regression.

I am not an expert in beta regression, so I do not know if it would be a novel application to apply beta regression to ordinal variables. But I suspect it is close enough to being novel to get Dr. Zimmerman's student's future article some citations, should she choose to give it a whirl. SAS and Stata both do beta regression. I assume that R does, too. I don't know about SPSS.

Also, to illustrate how flexible beta regression can be, Swearingen et al (2011) published a well-executed application of beta regression to a biomedical problem where one wouldn't normally expect to see it: analyzing the brain volume affected by ischemic stroke. The free article can be found here: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3186717/. It is well worth reading.

-------------------------------------------
Eric Siegel
Biostatistician
Univ of Arkansas for Medical Sciences of Biostatistics
-------------------------------------------



Original Message:
Sent: 07-03-2014 09:29
From: Ihor Kowalysko
Subject: Regression Analysis for Ordinal Data

Here's another approach.  Ordinal variables are bounded (e.g. 1-5, 1-10), a situation similar to that seen in a logistic regresion.  Therefore, first change the scale to probabilities (i.e. 10 = .95, 9 = .85, etc.).  Then convert these to logistics and your results wil;l be bounded.  This way your dependent variable is logistic and you can use it in a regression analysis. 

-------------------------------------------
Ihor Kowalysko
-------------------------------------------

Hi Grenith,

One of my graduate students and I have just had an article come out that may be of help.

Larrabee gs, B., H. M. Scott and N. M. Bello*. 2014. "Ordinary least squares regression of ordered categorical data: Inferential implications". Journal of Agricultural, Biological and Environmental Statistics. DOI: 10.1007/s13253-014-0176-z

Best,
Nora

-------------------------------------------
Nora M. Bello, PhD DVM
Assistant Professor
Dept. of Statistics
Kansas State University
-------------------------------------------

I take it your problem is what to do with ordinal covariates because if it's just about the response you can run a standard ordinal regression. For that, I recommend package VGAM in R and these references:

Agresti, Analysis of ordinal categorical data, 2nd edition, 2010. The supplement can be found online:

http://www.stat.ufl.edu/~aa/ordinal/R_examples.pdf

Now, if there is an ordinal covariate X with K categories, you can convert it to a quantitative one by mapping categories to numbers like 1, 2, ... K. That's the simplest form that will require only one regression parameter. On the other extreme, you may say that it's a purely categorical covariate that  takes (K - 1) parameters (main effects). There can also be "intermidiate" versions where you treat X as quantitative, but also include X^2 or X^3, or both.  Then you can use AIC/AICc to pick the model with the best bias-variance tradeoff.

Regards,
Nik Tuzov




Original Message:
Sent: 07-03-2014 03:39
From: Nora Bello
Subject: Regression Analysis for Ordinal Data


Hi Grenith,

One of my graduate students and I have just had an article come out that may be of help.

Larrabee gs, B., H. M. Scott and N. M. Bello*. 2014. "Ordinary least squares regression of ordered categorical data: Inferential implications". Journal of Agricultural, Biological and Environmental Statistics. DOI: 10.1007/s13253-014-0176-z

Best,
Nora

-------------------------------------------
Nora M. Bello, PhD DVM
Assistant Professor
Dept. of Statistics
Kansas State University
-------------------------------------------

There have been many great suggestions on how to model ordinal dependent or outcome variables. The post below also nicely covers the two most typical ways to model an ordinal independent variable or covariate and mentions that both approaches can be tried and the results compared with information criteria such as AIC. I may be wrong about this, but if I remember correctly the quantitative single-parameter model may be nested under the model with K-1 dummy variables representing the levels of the covariate compared with the reference group, in which case you could directly test whether restricting the model to a single quantitative covariate significantly worsens the model fit. The test could be performed with a Wald or a likelihood ratio test, the latter being preferred due to superior performance in small to moderately-sized samples.

There are also other clever ways to parameterize the design matrix for the covariate that you might want to investigate. For instance, SAS has one called "ordinal" or "thermometer", which is described here:

    http://support.sas.com/documentation/cdl/en/statug/63962/HTML/default/viewer.htm#statug_introcom_a0000003337.htm

HTH,

Tor Neilands

-------------------------------------------
Torsten Neilands
Professor of Medicine
UCSF Center for AIDS Prevention Studies
-------------------------------------------



Original Message:
Sent: 07-03-2014 10:30
From: Nikita Tuzov
Subject: Regression Analysis for Ordinal Data

I take it your problem is what to do with ordinal covariates because if it's just about the response you can run a standard ordinal regression. For that, I recommend package VGAM in R and these references:

Agresti, Analysis of ordinal categorical data, 2nd edition, 2010. The supplement can be found online:

http://www.stat.ufl.edu/~aa/ordinal/R_examples.pdf

Now, if there is an ordinal covariate X with K categories, you can convert it to a quantitative one by mapping categories to numbers like 1, 2, ... K. That's the simplest form that will require only one regression parameter. On the other extreme, you may say that it's a purely categorical covariate that  takes (K - 1) parameters (main effects). There can also be "intermidiate" versions where you treat X as quantitative, but also include X^2 or X^3, or both.  Then you can use AIC/AICc to pick the model with the best bias-variance tradeoff.

Regards,
Nik Tuzov