A flexible class of latent variable models for the analysis of antibody response data

TL;DR

This work proposes a latent variable modelling framework in which the immune status of each individual is represented along a continuum of latent seroreactivity, ranging from minimal to strong immune activation, which provides greater flexibility in capturing age-related changes in antibody distributions while preserving the full information content of quantitative measurements.

Abstract

Existing approaches to modelling antibody concentration data are mostly based on finite mixture models that rely on the assumption that individuals can be divided into two distinct groups: seronegative and seropositive. Here, we challenge this dichotomous modelling assumption and propose a latent variable modelling framework in which the immune status of each individual is represented along a continuum of latent seroreactivity, ranging from minimal to strong immune activation. This formulation provides greater flexibility in capturing age-related changes in antibody distributions while preserving the full information content of quantitative measurements. We show that the proposed class of models can accommodate a large variety of model formulations, both mechanistic and regression-based, and also includes finite mixture models as a special case. We also propose a computationally efficient $L_2$-based estimator as an alternative to maximum likelihood estimation, which substantially reduces computational cost, and we establish its consistency. Through a case study on malaria serology, we demonstrate how the flexibility of the novel framework enables joint analyses across all ages while accounting for changes in transmission patterns. We conclude by outlining extensions of the proposed modelling framework and its relevance to other omics applications.

Figures (10)

• Figure 1: Example of a hypothesized distribution of the latent immune activation in the general population. The latent variable $T \in [0,1]$ represents a continuous seroreactivity scale from no activation ($t=0$) to full activation ($t=1$), with most individuals near the extremes and fewer in intermediate activation states.
• Figure 2: Example of Beta distributions for $T$ and resulting distribution for $Y$ based on the model in \ref{['eq:lbg']}. Here, $\mu_0=-4$, $\mu_1=4$ and $\sigma_{0}^2=\sigma_{1}^2=1$. The value of the Beta distribution parameters are defined in the legend.
• Figure 3: Examples of latent-state mixture models using Beta distributions. Each row represents a distinct epidemiological scenario characterised by different levels of disease transmission intensity: low transmission (top panels, predominantly seronegative individuals), high transmission (middle panels, predominantly seropositive individuals), and moderate transmission (bottom panels, predominantly intermediate immune states). The left-hand panels show the density of the latent immune response state $T$, while the right-hand panels depict the induced marginal distributions of the outcome variable $Y$.
• Figure 4: Left panel: Histograms of the empirical distributions of log AMA1 concentrations within each age group and fitted densities (blue lines). Middle panel: Estimates of the Beta distribution parameters $\alpha$, $\beta$ and its mean $\alpha/(\alpha+\beta)$ across age groups. Right panel: Fitted Beta distributions of the latent variable $T$ for each age group.
• Figure 5: Model validation results for AMA1. Left: The plot shows the envelope and the median histogram compared with the empirical distribution of observed AMA1 antibody concentrations (black). Right: Fitted probability of $\pi(a)$ of two-components Beta mixture, as a function of age; the vertical dashed line correspond to the estimated change point parameter $\tau$.

Theorems & Definitions (4)

• Theorem 4.1
• proof
• Lemma B.1
• proof