Latent single-index models for ordinal data
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We propose a latent semi-parametric model for ordinal data in which the single-index model is used to evaluate the effects of the latent covariates on the latent response. We develop a Bayesian sampling-based method with free-knot splines to analyze the proposed model. As the index may vary from minus infinity to plus infinity, the traditional spline that is defined on a finite interval cannot be applied directly to approximate the unknown link function. We consider a modified version to address this problem by first transforming the index into the unit interval via a continuously cumulative distribution function and then constructing the spline bases on the unit interval. To obtain a rapidly convergent algorithm, we make use of the partial collapse and parameter expansion and reparameterization techniques, improve the movement step of Bayesian splines with free knots so that all the knots can be relocated each time instead of only one knot, and design a generalized Gibbs step. We check the performance of the proposed model and estimation method by a simulation study and apply them to analyze a real dataset.
KeywordsFree-knot splines Generalized Gibbs sampler Ordinal data Parameter expansion and reparameterization Single-index model
The work was supported by the Natural Science Foundations of China (Nos. 11471272 and 11661074) and the Natural Science Foundation of Fujian Province of China (No. 2013J01019). The authors are grateful to the anonymous referees, the Associate Editor, and the Editor for valuable suggestions for improving the manuscript.
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