Understanding and interpreting generalized ordered logit models

(2016) - Richard Williams

Journal: The Journal of Mathematical Sociology
Link:: http://www.tandfonline.com/doi/full/10.1080/0022250X.2015.1112384
DOI:: 10.1080/0022250X.2015.1112384
Links::
Tags:: #paper #Methods #Logit #OrdinalLogit #ProportionalOddsModel
Cite Key:: [@williamsUnderstandingInterpretingGeneralized2016]

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.

Notes

“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 alternativ” (Williams, 2016, p. 7)

“esearchers 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.” (Williams, 2016, p. 7)

“Unfortunately, experience suggests that the assumptions of the ordered logit model are frequently violated (Long & Freese, 2014).” (Williams, 2016, p. 7)

“This model has been known about since at least the 1980s (e.g., McCullagh & Nelder, 1989; Peterson & Harrell, 1990), but recent advances in software (such as the user-written gologit and gologit2 routines in Stata) have made the model much easier to estimate and widely used (Fu, 1998; Williams, 2006).2” (Williams, 2016, p. 7)

“Often it is simply noted that the model fits better and avoids violating the assumptions of the ordered logit model (see, e.g., Cornwell, Laumann, & Shumm, 2008;Do&” (Williams, 2016, p. 7)

“Farooqui, 2011; Kleinjans, 2009; Lehrer, Lehrer, Zhao, & Lehrer, 2007; Schafer & Upenieks, 2015).” (Williams, 2016, p. 8)

“If the assumptions of the ordered logit model are met, then all of the corresponding coefficients (except the intercepts) should be the same across the different logistic regressions, other than differences caused by sampling variability. The assumptions of the model are therefore sometimes referred to as the parallel lines or parallel regressions assumptions (Williams, 2006).” (Williams, 2016, p. 9)

“A test devised by Brant (1990; also see Long & Freese, 2014) is commonly used to assess whether the observed deviations from what the proportional odds model predicts are larger than what could be attributed to chance alone.” (Williams, 2016, p. 9)

“the Generalized Ordered Logit model (Williams, 2006) can be written as” (Williams, 2016, p. 11)

“An unconstrained gologit model and a multinomial logit model will both generate many more parameters than an ordered logit model does. This is because, with these methods, all variables are freed from the proportional odds constraint, even though the assumption may only be violated by one or a few of them.” (Williams, 2016, p. 12)

“However, the interpretation and justification for the gologit model is less straightforward than it is for the ordered logit model.” (Williams, 2016, p. 14)

“Whenever the assumptions of any model appear to be violated, it is tempting to quickly turn to more advanced techniques. However, researchers should first consider simpler alternatives.” (Williams, 2016, p. 14)

“Have key variables been omitted? Do squared terms need to be included in the model?” (Williams, 2016, p. 14)

“As Long and Freese (2006, p. 187) point out, “the ordinal regression model can also be developed as a nonlinear probability model without appealing to the idea of a latent variable.” By way of extension, the simplest thing may just be to also interpret gologit as a nonlinear probability model that lets you estimate the determinants and probability of each outcome occurring. There is no need to rely on the idea of an underlying Y* that accounts for the observed values of Y.” (Williams, 2016, p. 14)

“The proportional odds model assumes that, for each cumulate logit model that can be estimated (e.g., 1 versus 2, 3 4; 1, 2 vs 3, 4; 1, 2, 3 vs 4), the effect of X on Y is the sam” (Williams, 2016, p. 15)

“Fullerton and Dixon (2010) refer to this as asymmetrical effects. In their work they find that several key determinants of attitudes toward government spending on welfare have much stronger effects on opposition to it than on support for it. They further argue (p. 649) that the generalized ordered logit model has key advantages over other techniques when such asymmetries exist: “Traditional OLS models do not allow for the possibility that age, period, and cohort may affect support for education spending but not opposition to it. Similarly, other models used in previous research—such as binary logistic regression—do not allow for this possibility and may even obfuscate it, given that collapsing categories [of an ordinal outcome variable] results in a loss of information.”” (Williams, 2016, p. 15)

“Hedeker and Mermelstein (1998) offer another good example of how the effects of X on Y could differ across the various cumulative logit models. The categories of the dependent variable may represent stages, such as precontemplation, contemplation, and action. An intervention might be effective in moving people from precontemplation to contemplation, but be ineffective in moving people from contemplation to action. If so, the effects of an explanatory variable will not be the same across the K-1 cumulative logits of the model.” (Williams, 2016, p. 15)

“In these cases the underlying latent variable may be the same for all groups; but the thresholds/cut points used may vary. For example, an estimated gender effect could reflect differences in measurement across genders rather than a real gender effect on the outcome of interest. Lindeboom and van Doorslaer (2004; see also Schneider, Pfarr, Schneider, & Ulrich, 2012) note that this has been referred to as state-dependent reporting bias, scale of reference bias, response category cut-point shift, reporting heterogeneity, heterogeneous reporting behavior, and differential item functioning.” (Williams, 2016, p. 16)

“Note that this interpretation is not unique to gologit/ppo models, although such models may make the alternative interpretation stand out more. If the difference in thresholds is constant (index shift), proportional odds will still hold.” (Williams, 2016, p. 16)

“A variable that is ordinal in some respects may not be ordinal or else be differently-ordinal in others. Variables could be ordered either by direction (Strongly Disagree to Strongly Agree) or intensity (Indifferent to Feel Strongly).” (Williams, 2016, p. 17)

“First, it probably works best when relatively few of the variables in the model violate the proportional odds assumption.” (Williams, 2016, p. 18)

“Second, unlike many other methods for the analysis of categorical data, the gologit model can produce negative predicted probabilities.” (Williams, 2016, p. 18)

“Third, when sample sizes are large, even small violations of the proportional odds assumption can be statistically significant.” (Williams, 2016, p. 18)

“Fourth, the researcher must somehow decide which variables should have the proportional odds constraint imposed and which should not. Ideally, researchers should have strong theoretical rationales to guide them. But since such theory rarely exists, empirical means are often used instead. For example, the Brant test can also be used to identify individual variables that violate assumptions.” (Williams, 2016, p. 19)

“The gologit2 routine in Stata (Williams, 2006) uses a stepwise procedure called autofit to identify variables where proportionality constraints should be relaxed” (Williams, 2016, p. 19)

“Finally, there are other alternatives to the ordered logit model. For example, Williams (2009, 2010) notes that the ordered logit model also assumes that errors are homoscedastic, and that differences across groups in residual variability may cause tests of the proportional odds assumption to fail. The heterogeneous choice model addresses those problems by explicitly modeling the causes of the differing variability.” (Williams, 2016, p. 19)