# Probabilistic Index Models

*by Olivier Thas*

*Institution:*Ghent University, Belgium

*Date: Tue 24th January 2012*

*Time: 1:00 PM*

*Location: Room 213, Richard Berry Building, University of Melbourne*

*Abstract*: Probabilistic Index Models (PIM) have been recently introduced (Thas et al., 2012) as a semi- parametric framework for modelling the probabilistic index (PI) as a function of covariates. In particular, when (X, Y) and (X∗, Y∗) denote two independent and identically distributed random vectors, the PI is defined as P{Y ≤ Y∗|X, X∗} and the PIM is given by P{Y ≤ Y∗|X, X∗} = g^{−1}(Z^t β), where g is a link function, and Z is a vector that contains elements from X and X∗. When g is the identity link, Z=X−X∗ and X and X∗ are 0/1 dummies coding for two groups, the PI is the population parameter that is the foxus of the Wilcoxon-Mann-Whitney test, but the flexibility of the model allows for many more choices for the covariate vector.

In this presentation we first demonstrate how classical rank tests are related to hypothesis tests in the semiparametric PIM framework. Once this is understood, PIM can be used to generate rank tests for other, more complicated designs for which today no rank tests have been proposed yet.

In a next part of the presentation, we extend the PIM to clustered data. Two different approaches are discussed. We show how meaningful parameters can be estimated, and how their standard deviations can be consistently estimated. This is ongoing research (with Stijn Vansteelandt and Fanghong Zhang).

References

De Neve, J., Thas, O. and Ottoy, J.P. Goodness-of-fit methods for probabilistic index models. Communications in Statistics – Theory and Methods. Submitted.

Thas, O., De Neve, J., Clement, L. and Ottoy, J.P. (2012). Probabilistic Index Models. Journal of the Royal Statistical Society - Series B. To appear (read paper).

*For More Information:* contact: Mihee Lee. email: miheel@unimelb.edu.au