Ordinal Regression

Overview

Ordinal regression is used with ordinal dependent (response) variables, where the independents may be categorical factors or continuous covariates. Ordinal regression models are sometimes called cumulative logit models. Ordinal regression typically uses the logit link function, though other link functions are available. Ordinal regression with a logit link is also called a proportional odds model, since the parameters (regression coefficients) of the independent variable are independent of the levels (categories) of the ordinal dependent variable, and because these coefficients may be converted to odds ratios, as in logistic regression.

Ordinal regression creates multiple prediction equations. For an ordinal dependent variable with k categories, k -1 equations will be created, each with a different intercept but all with the same b coefficients (slopes) for the predictor variables. That is, ordinal regression requires assuming that the effect of the independents is the same for each level of the dependent. In practice, researchers often consider it sufficiently "the same" if the slopes do not cross. The "test of parallel lines assumption" tests this critical assumption, which should not be taken for granted. In SPSS, select Analyze, Regression, Ordinal.

See also the separate Statistical Associates "blue book" volume on generalized linear models. Ordinal regression is a special case of generalized linear modeling (GZLM). Identical parameter and model fit estimates can be obtained using the GZLM procedure, but options vary somewhat between PLUM (the ordinal regression procedure discussed here, standing for "polytomous universal model") and GZLM. Other coverage of ordinal regression is found in the separate Statistical Associates "Blue Book" on "Probit Regression and Response Models," which covers ordinal signal-response models.

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Below is the unformatted table of contents.

ORDINAL REGRESSION Overview 6 Key Terms and Concepts 7 Variables 7 The dependent variable 7 Factors 8 Covariates 9 Models 10 The location model 10 The default model 10 Interactions 11 Nested effects 11 Model Options 11 Statistics and saved variables for ordinal logistic regression 13 Output selections 13 SPSS Options 13 Example 14 The parallel lines test 15 Tests and effect size measures for model goodness of fit 16 Parameter estimates for the predictor variables and intercept 19 Odds ratios 21 Other output 25 Ordinal Regression in SAS 30 Overview 30 SAS syntax for ordinal regression 30 SAS output for ordinal regression 32 Testing the global null hypothesis 32 The analysis of maximum likelihood estimates table 32 Type 3 Analysis of Effects 33 Odds ratio estimates 34 Score test for the proportional odds assumption 34 R-square 35 Association of predicted probabilities and observed responses 35 Model fit statistics 36 Ordinal Regression for Signal-Response Models (Probit Link) 37 Overview 37 Assumptions 38 Parallel lines assumption 38 Adequate cell count 39 One ordinal dependent variable 41 Data level of predictor variables 41 Normal distribution of the dependent variable 41 Adequate sample size 41 No complete or quasi-complete separation 41 Frequently Asked Questions 42 Why not use multinomial logistic regression instead of ordinal (logit) regression? 42 Why not use ordinary least-squares regression instead of ordinal (logit) regression? 42 Why not use ANOVA instead of ordinal (logit) regression? 42 Why do SPSS and Stata parameter estimates differ, and what is "parameterization"? 43 Does the direction of coding of the ordinal dependent matter? 43 What is the SPSS syntax for ordinal regression models? 43 Bibliography 45