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Cumulative Logit Model

General Notes
Also known as:
  • Logit analysis for ordered categories
  • Logistic regression with Ordinal data

Related methods:
  • Multinomial Logit Analysis
  • Logistic Regression

When model assumptions hold (proportional odds), the cumulative logit model may be statistically more powerful than other methods.

From Allison (page 138):

Cumulative probability Fij is the probability that individual ‘i’ is in the ‘jth’ category (rating) or lower. (Categories j = 1,2,3,4,5, etc.)

The explanatory variables (“x”) predict the probability of being in a lower category ('y' rating) rather than a higher category ('y' rating).

Linear predictors for a five-level ordinal scale:
  • logit(P[Y<=1]), logit(P[Y<=2]), logit(P[Y<=3]), logit(P[Y<=4])

General model equation: logit[P(y ≤ j)] = αj - βx (if using R function ‘polr’)
  • (If use function 'vgam', then logit[P(y ≤ j)] = αj + βx due to opposite signs on the coefficients)
  • For the cumulative logit model, the beta coefficients (related to the values of the explanatory variable) are the same for all values of the ordinal response, but each value of the ordinal response has its own intercept (alpha or αj ).

The values of "x" may be continuous, categorical, or if there are several explanatory variables, then a combination.

If the x values are categorical, then dummy variables may used for values of categorical "x".  For instance, if 'x' has four categories, then the dummy variable coding might look like this:

 x coding
 interceptcategory 1
category 2
category 3
 category 1
 1 00
 category 2
 1 10
 category 3
 1 01
 category 4
 1 00

This method of coding makes category 1 the reference level for the other levels.  It is possible in many software packages to define the reference level differently.


The assumptions behind logistic regression apply.
The cumulative logit model assumes proportional odds.


Allison, Paul , “Logistic Regression Using the SAS System”, SAS Institute, 1999, ISBN 0-471-22175-9

Introduction to SAS. UCLA: Statistical Consulting Group.

“Examples of Using R for Modeling Ordinal Data” by Alan Agresti
  • (accessed March 26, 2014)
  • Department of Statistics, University of Florida
  • Supplement for the book Analysis of Ordinal Categorical Data, 2nd ed., 2010 (Wiley)