The Asymptotic Covariance Matrix of the Odds Ratio Parameter Estimator in Semiparametric Log-bilinear Odds Ratio Models

TL;DR

This paper derives the asymptotic covariance matrix of the maximum likelihood estimator for the odds ratio parameter in semiparametric log-bilinear models, providing tools for inference and sample size determination.

Contribution

It introduces invariant representations of the covariance matrix of the odds ratio estimator across various sampling schemes and asymptotic conditions, enhancing inference in semiparametric models.

Findings

Covariance matrix invariance across sampling schemes

Asymptotic power calculations for hypothesis tests

Sample size recommendations for desired power

Abstract

The association between two random variables is often of primary interest in statistical research. In this paper semiparametric models for the association between random vectors X and Y are considered which leave the marginal distributions arbitrary. Given that the odds ratio function comprises the whole information about the association the focus is on bilinear log-odds ratio models and in particular on the odds ratio parameter vector {\theta}. The covariance structure of the maximum likelihood estimator {\theta}^ of {\theta} is of major importance for asymptotic inference. To this end different representations of the estimated covariance matrix are derived for conditional and unconditional sampling schemes and different asymptotic approaches depending on whether X and/or Y has finite or arbitrary support. The main result is the invariance of the estimated asymptotic covariance matrix of {\theta}^ with respect to all above approaches. As applications we compute the asymptotic power for tests of linear hypotheses about {\theta} - with emphasis to logistic and linear regression models - which allows to determine the necessary sample size to achieve a wanted power.