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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```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
One ordinal dependent variable	41
Data level of predictor variables	41
Normal distribution of the dependent variable	41
No complete or quasi-complete separation	41
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

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