# Two-part models with stochastic processes for modelling longitudinal semicontinuous data: computationally efficient inference and modelling the overall marginal mean

## TL;DR

This paper introduces a computationally efficient method for fitting two-part models with stochastic processes to longitudinal semicontinuous data, enabling maximum likelihood estimation and improved flexibility over traditional models.

## Contribution

The authors develop a novel approach that transforms high-dimensional integrations into a multivariate normal CDF, allowing efficient maximum likelihood estimation for complex two-part models.

## Key findings

- Efficient computation of marginal likelihood using multivariate normal CDF.
- Application to psoriatic arthritis data demonstrating practical utility.
- Enhanced model flexibility over standard two-part models.

## Abstract

Several researchers have described two-part models with patient-specific stochastic processes for analysing longitudinal semicontinuous data. In theory, such models can offer greater flexibility than the standard two-part model with patient-specific random effects. However, in practice the high dimensional integrations involved in the marginal likelihood (i.e. integrated over the stochastic processes) significantly complicates model fitting. Thus non-standard computationally intensive procedures based on simulating the marginal likelihood have so far only been proposed. In this paper, we describe an efficient method of implementation by demonstrating how the high dimensional integrations involved in the marginal likelihood can be computed efficiently. Specifically, by using a property of the multivariate normal distribution and the standard marginal cumulative distribution function identity, we transform the marginal likelihood so that the high dimensional integrations are contained in the cumulative distribution function of a multivariate normal distribution, which can then be efficiently evaluated. Hence maximum likelihood estimation can be used to obtain parameter estimates and asymptotic standard errors (from the observed information matrix) of model parameters. We describe our proposed efficient implementation procedure for the standard two-part model parameterisation and when it is of interest to directly model the overall marginal mean. The methodology is applied on a psoriatic arthritis data set concerning functional disability.