Rank-Ordered Logit Model With Ties

The rank-ordered logit model with ties[1] is a generalization of the Sequential Logit Model which is, in turn, a generalization of the Multinomial Logit Model.

Key aspects of the model are:

• Its interpretation is identical to that of the Multinomial Logit Model. That is, the parameters have the same interpretation and predictions are made in the same way (e.g., a Choice Simulator can be constructed from the rank-ordered logit model with ties).
• The Outcome Variable is assumed to be a ranking, where ties are permitted (i.e., a partial ranking).

Computation of the log-likelihood

The model assumes that all possible rankings consistent with the an observed ranking containing ties are equally likely. For example, if a respondent that has given the following ranking: C > A = B > D > E (i.e., where A and B are tied), then there are two possible rankings consistent with this data: C > A > B > D > E and C > B > A > D > E.

The likelihood is then computed as the average of all of the possible likelihoods, where the likelihood for a possible ranking is computed using same approach as employed with the Sequential Logit Model.

Software

SAS has a procedure called PHREG that can estimate this model.[2]

Q has a generalized version of this model that estimates latent class and random parameter logit models.

References

1. Allison, P. D. and N. A. Christakis (1994). "Logit Models for Sets of Ranked Items." Sociological Methodology 24: 199-228.
2. SAS (2008). The PHREG Procedure. SAS/STAT® 9.2 User’s Guide. Cary, NC, SAS Institute Inc.