Bimodal distributions have rarely been studied although they appear frequently in datasets. We develop a novel bimodal distribution based on the triangular distribution and then expand it to the multivariate case using a Gaussian copula. To determine the goodness of fit of the univariate model, we use the Kolmogorov–Smirnov (KS) and Cramér–von Mises (CVM) tests. The contributions of this work are that a simplistic yet robust distribution was developed to deal with bimodality in data, a multivariate distribution was developed as a generalisation of this univariate distribution using a Gaussian copula, a comparison between parametric and semi-parametric approaches to modelling bimodality is given, and an R package called btld is developed from the workings of this paper.

Joint modelling of longitudinal and time-to-event data is a powerful tool for analysing complex medical data. Joint latent class models (JLCMs), in particular, provide additional prognostic insight by clustering subjects into latent subgroups within a mixture of experts framework. Traditionally, the mixing proportions are modelled using a multinomial logistic regression. Recent work in mixture regression has proposed a neural network alternative, implemented through the neural network expectation–maximization algorithm, which offers greater flexibility by capturing nonlinearities that may be missed by logistic regression. In this article, we extend the JLCM by applying the neural network to the modelling the mixing proportions captured in the class-membership submodel. The model estimated using an expectation–maximization-type algorithm. We call this the neural-network JLCM (NN-JLCM). We formally define the method, evaluate its performance through extensive simulations, and apply it to real data. The results show that NN-JLCM improves predictive accuracy and class separation compared with traditional JLCMs, while maintaining competitiveness in terms of overall model fit. The code for the article is available at https://github.com/tharris0924/nn-jlcm_p1 .