Understanding and interpreting generalized ordered logit models

ABSTRACT

When outcome variables are ordinal rather than continuous, the ordered logit model, aka the proportional odds model (ologit/po), is a popular analytical method. However, generalized ordered logit/partial proportional odds models (gologit/ppo) are often a superior alternative. Gologit/ppo models can be less restrictive than proportional odds models and more parsimonious than methods that ignore the ordering of categories altogether. However, the use of gologit/ppo models has itself been problematic or at least sub-optimal. Researchers typically note that such models fit better but fail to explain why the ordered logit model was inadequate or the substantive insights gained by using the gologit alternative. This paper uses both hypothetical examples and data from the 2012 European Social Survey to address these shortcomings.

KEYWORDS:

• Generalized ordered logit model
• ordered logit model
• partial proportional odds
• proportional odds assumption
• proportional odds model

Acknowledgments

The author wishes to thank Richard Campbell, Sarah Mustillo, J. Scott Long, and the anonymous reviewers and the editor for their many helpful comments.

Notes

¹ The ordered probit model is a popular alternative to the ordered logit model. The terms “Parallel Lines Assumption” and “Parallel Regressions Assumption” apply equally well for both the ordered logit and ordered probit models. However the ordered probit model does not require nor does it meet the proportional odds assumption.

² According to Google Scholar, Williams (2006), which introduced the gologit2 program for Stata, has been cited more than 800 times since its publication. Similarly, various papers by Hedeker (e.g. Hedeker & Mermelstein, 1998) on the similar “stages of change” models have been cited hundreds of times.

³ Small differences are typically found because the gologit model estimates all the parameters simultaneously whereas the separate logistic regressions estimate them one cumulative logit at a time.

⁴ When survey weights are used, several conventional measures of model fit—BIC, AIC, and Likelihood Ratio Chi-square—are not appropriate. Similarly, the Brant test is not appropriate either. However, the Wald tests used by gologit2 to test the proportional odds assumption can still be used. We used gologit2 with the autofit option set to .025; that is, the assumption is rejected if the observed deviations from it would only be expected to 25 times out of 1,000 if the assumption is true. This is consistent with Williams’ (2006) advice that the default .05 level of significance may not be stringent enough when multiple variables are being tested.

⁵ Craemer (2009) offers excellent examples of how to format tables that present results from partial proportional odds models. We have adapted his approach here.

⁶ Such computations and comparisons can of course be done even when the proportional odds assumption is not violated. But, the approach may be especially useful with more complicated models where multiple effects for the same variables are estimated.

⁷ The ideal situation, of course, is when BIC, AIC, and likelihood ratio tests all lead to the same conclusion. However, as Dziak, Coffman, Lanza, and Li (2012) point out, this will not always be the case. Based on their simulations, they argue that AIC often risks choosing too large a model while BIC sometimes leads to selecting a model that is too parsimonious. But ultimately, when criteria conflict, they recommend going with what seems most important in a given situation: Is it worse to have a model that is too parsimonious, or a model that is not parsimonious enough?