We review some current approaches to the analysis of the relation between an ordinal response variable and a set of covariates and propose a new regression method for estimating the conditional quantiles of the ordinal response variable. By assuming a continuous latent variable underlying the observed ordinal response variable and utilizing the equivalence property of quantile regression we obtain the estimates of the conditional quantile of the ordinal response variable through the optimization of a piece-wise constant object function. Several issues regarding the proposed ordinal quantile regression model, such as the model identification, interpretation of estimators from the model, estimation of probabilities, are addressed. The simulated annealing algorithm is used for the optimization. The proposed ordinal quantile regression method is demonstrated in a series of simulation studies and is applied to the data from the low birth weight study. Confidence intervals of the parameter estimates are constructed using bootstrap resampling technique. The Goodman-Kruskal Gamma statistic, a measure of predictive power, is reported for the proposed model and used as a criterion to compare the proposed model with existing approaches. In our simulation study the proposed ordinal quantile regression method is shown to be an effective tool for analyzing ordinal response data. It gives a full picture of the latent variable underlying the ordinal response though only the ordered response data were used in the model. The predictive power as measured by the Gamma statistics also higher for the proposed model than existing approaches.
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