Two algorithms for fitting constrained marginal models

TL;DR

This paper compares two algorithms for fitting constrained marginal models to discrete data, showing their equivalence and analyzing their efficiency, while extending the methods to include covariates and L1-penalties.

Contribution

It demonstrates the equivalence of Lagrange multiplier and regression algorithms and extends the regression approach to incorporate covariates and L1-penalties.

Findings

The two algorithms produce identical updates.

Lagrangian method is more efficient for identical observations.

Regression extension allows modeling covariates and penalization.

Abstract

We study in detail the two main algorithms which have been considered for fitting constrained marginal models to discrete data, one based on Lagrange multipliers and the other on a regression model. We show that the updates produced by the two methods are identical, but that the Lagrangian method is more efficient in the case of identically distributed observations. We provide a generalization of the regression algorithm for modelling the effect of exogenous individual-level covariates, a context in which the use of the Lagrangian algorithm would be infeasible for even moderate sample sizes. An extension of the method to likelihood-based estimation under $L_1$-penalties is also considered.