A family of ordered logistic regression models fit by data expansion

Tou Choi and Stephen R Cole

Department of Epidemiology, Bloomberg School of Public Health, Johns Hopkins University, 615 N Wolfe St, Baltimore, MD, USA. E-mail: scole{at}jhsph.edu

Sirs—One of us (SRC) coauthored a paper in the International Journal of Epidemiology that detailed a use of data expansion to fit the continuation-ratio model1 as sketched by Armstrong and Slone,2 as well as a paper in the Annals of Epidemiology that detailed a related use of data expansion to fit proportional-odds models.3 In fact, a family of ordered logistic regression models, including continuation-ratio, proportional-odds and adjacent category models, can be fit using data expansion as detailed briefly below.

Consider an ordered response Yi = 1, 2, ..., K levels for i = 1, 2, ..., N participants. We can write a general ordered regression model as g [Yij] = xßj for j = 1, 2, ..., (K – 1) thresholds, where g [•] is a link function. Setting g [•] to log [Pr(Y > Yj)/Pr(Y = Yj)], log [Pr(Y > Yj)/Pr(Y ≤ Yj)], or log [Pr(Y = Yj+1)/Pr(Y = Yj)] results in the continuation-ratio, proportional-odds or adjacent category models, respectively. The size of the expanded data set is given by {Sigma}i{Sigma}j I [j ≤ y], {Sigma}i{Sigma}j I [1 > 0], and {Sigma}i{Sigma}j I [j less double equals y less double equals j + 1] for the continuation-ratio, proportional-odds and adjacent category models, respectively, where I [•] is the indicator function. SAS code (SAS Institute, Cary NC) to expand an observed data set, where K = 4, into the three expanded data sets needed to fit these three members of this family of ordered logistic regression models is provided in the Appendix. Implementation of analysis using any of the members of this family may follow references.1,3

To the best of our knowledge, use of data expansion to fit the adjacent-category ordered logistic regression model has not been previously shown.4 Data expansion is an extremely flexible quantitative tool that has been applied in varied settings. Specifically, in addition to ordered regression data expansion has been employed when a participant has multiple person-time contributions,5 events,6 competing risks,7 or informants.8


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 Appendix
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data cr po ac;
set _observed data set_; *Where y is a 4-level ordered response;
    do j = 1 to (4–1); *Loop from 1 to (4–1);
        if j < y then d = 1; else d = 0; *Create binary indicator d;
        if j <= y then output cr; *Output to continuation-ratio data set;
        output po; *Output to proportion odds data set;
        if j <= y <= j + 1 then output ac; *Output to adjacent-category data set;
end;


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 Appendix
 References
 
1 Cole SR, Ananth CV. Regression models for unconstrained, partially or fully constrained continuation odds ratios. Int J Epidemiol 2001;30:1379–82.[Abstract/Free Full Text]

2 Armstrong BG, Sloan M. Ordinal regression models for epidemiologic data. Am J Epidemiol 1989;129:191–204.[Abstract]

3 Cole SR, Allison PD, Ananth CV. Estimation of cumulative odds ratios. Ann Epidemiol 2004;14:172–78.[CrossRef][ISI][Medline]

4 Choi T. Estimation of adjacent category odds ratios using data expansion. Epidemiology. Baltimore: Johns Hopkins University, 2003:45.

5 D'Agostino RB, Lee ML, Belanger AJ, Cupples LA, Anderson K, Kannel WB. Relation of pooled logistic regression to time dependent Cox regression analysis: The Framingham Heart Study. Stat Med 1990;9:1501–15.[ISI][Medline]

6 Wei LJ, Lin DY, Weissfeld L. Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. J Am Statist Assoc 1989;84:1065–78.[ISI]

7 Lunn M, McNeil D. Applying Cox regression to competing risks. Biometrics 1995;51:524–32.[ISI][Medline]

8 Pepe MS, Whitaker RC, Seidel K. Estimating and comparing univariate associations with application to the prediction of adult obesity. Stat Med 1999;18:163–73.[CrossRef][ISI][Medline]





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