Latent variable models for the analysis of antibody density data

From non-spatial univariate to spatial multivariate

Emanuele Giorgi

University of Birmingham

Jonas Wallin

Lund University

Outline

  • Part I. One antigen: a continuous latent seroreactivity replaces seropositive / seronegative
  • Part II. Two antigens: three principles for where dependence enters, and the spatial extension

Part I

The dominant paradigm

\[f(y) = \pi_0\, \mathcal{N}(y; \mu_0,\sigma_0^2) + \pi_1\, \mathcal{N}(y; \mu_1,\sigma_1^2)\]

  • Individuals classified into two mutually exclusive states
  • Which threshold to use?
  • Choice should be context dependent
  • Multiple antigens, multiple thresholds?

White, Baudemont et al. (2026). Whose Line Is it Anyway? Defining Seropositivity Cutoffs for Infectious Disease Surveillance. JID. doi.org/10.1093/infdis/jiaf537

A threshold-free approach: Latent variable models

Seroreactivity: degree of antibody reactivity, irrespective of any threshold.

Latent \(T \in [0,1]\)

  • \(T \approx 0\): minimal activity
  • \(T \approx 1\): saturated response
  • \(Y \mid T=t \sim \mathcal{N}\big( (1-t)\mu_0 + t\mu_1,\; (1-t)\sigma_0^2 + t\sigma_1^2 \big)\)

Modelling \(T\): single distribution

Analogy: amount of DNA generated by a cell

  • Mass piles up at two boundary states: 2n (pre-replication) and 4n (post-replication)
  • Cells in S phase occupy a genuine continuum in between


Taken from Alberts B, Johnson A, Lewis J, Morgan D, Raff M, Roberts K, Walter P (2015). Molecular Biology of the Cell, 6th edition. New York: Garland Science.

Modelling \(T\): mixtures

Age dependency

Recall the two ways of modelling \(T\):

\[ 1) \quad T \sim \text{Beta}(\alpha,\beta) \]

\[ 2) \quad T \sim \bigl(1-\pi(a)\bigr)\,\text{Beta}(t;\alpha_1,\beta_1) + \pi(a)\,\text{Beta}(t;\alpha_2,\beta_2) \]


  • Through the shape: \(\alpha(a)=\alpha_0 a^{\gamma}\), \(\beta(a)=\beta_0 a^{\delta}\)

    the single Beta’s own shape parameters vary with age

  • Through the mixing weights: \(\pi(a) = 1-\exp(-\lambda a)\)

    age shifts weight from the low- to the high-seroreactivity component

Application: malaria, Kenyan highlands

Rachuonyo South District, 2011 — IgG against AMA1 and MSP1 (Bousema et al., 2013, 2016).

Strategy: fix conditional parameters \((\mu_0,\mu_1,\sigma_0^2,\sigma_1^2)\) from the youngest age group, then re-estimate only \(T\)’s distribution within each successive age band.

AMA1: age-stratified exploration

  • Left: histograms and fitted densities by age band
  • Middle: Beta shape parameters \(\alpha\), \(\beta\) and their mean across age
  • Right: fitted Beta densities of \(T\) by age band

AMA1: model specification

A change point \(\tau\) separates two regimes, with the mean of \(T\) kept continuous across it.

Below \(\tau\) — mechanistic, with a degenerate mass at \(T=0\):

\[ f_T(t;a) = \begin{cases} 1-\pi(a), & t=0\\[4pt] \pi(a)\,\alpha_2\, t^{\alpha_2-1}, & 0<t<1 \end{cases} \qquad \pi(a) = 1-\exp(-\lambda a) \]

Above \(\tau\) — empirical Beta with logit-linear mean:

\[ T \sim \text{Beta}\big(\mu(a)\phi,\,[1-\mu(a)]\phi\big), \qquad \text{logit}\{\mu(a)\} = \eta_0 + \eta_1\log(a) \]

  • Continuity constraint: \(\eta_0 = \text{logit}(\mu_{\tau^-}) - \eta_1\log(\tau)\)
  • \(\hat\tau = 20.8\) years — mechanistic model appropriate up to about age 20
  • \(\hat\lambda = 0.148\) — roughly half of children have initiated a response by age 5
  • \(\hat\eta_1 < 0\) — modest decline in mean seroreactivity at older ages

AMA profile

AMA1: goodness of fit

  • Envelope of histograms simulated from the fitted model vs. the observed histogram
  • Good agreement across most age bands
  • Slight overstatement of high-antibody individuals in the youngest groups

MSP1: age-stratified exploration

  • Same three-panel layout as AMA1: histograms, shape parameters, fitted Beta densities
  • \(\alpha\) rises with age as for AMA1; \(\beta\) stabilises after age 10
  • Separation between low- and high-response groups weakens more slowly than for AMA1

MSP1: model specification

A single Beta throughout, with both shape parameters varying with age:

\[ T \sim \text{Beta}\big(\alpha(a),\,\beta(a)\big), \qquad \alpha(a) = \alpha_0\, a^{\gamma}, \qquad \beta(a) = \beta_0\, a^{\delta(a)} \]

with a change point \(\zeta\) in the exponent of \(\beta\):

\[ \delta(a) = \begin{cases} \delta_1, & a \le \zeta\\[4pt] \delta_1+\delta_2, & a > \zeta \end{cases} \qquad \mathbb{E}[T] = \left(1 + \tfrac{\beta_0}{\alpha_0}\, a^{\delta_1+\delta_2-\gamma}\right)^{-1} \]

  • \(\hat\gamma = 0.277 > \hat\delta_1 = 0.110\) — \(\alpha(a)\) grows faster, so \(\mathbb{E}[T]\) rises steadily
  • \(\hat\zeta = 11.6\) years, \(\hat\delta_2 = -0.061 < 0\) — \(\beta(a)\) slows, accelerating the rise
  • Saturation would require \(\delta_1+\delta_2 \approx \gamma\); it does not occur here
  • Contrast with AMA1: a data-driven rather than mechanistic specification

MSP profile

MSP1: goodness of fit

  • Same envelope comparison as for AMA1
  • Model tracks the observed histograms closely across all age bands
  • No systematic bias, unlike AMA1’s youngest bands

LBM vs GMM: how much better is the latent model?

BIC comparison, age-stratified, both fitted with MLE (positive \(\Delta\)BIC favours the latent model):

Antigen Age band \(\Delta\)BIC
AMA1 \(6\)–\(10\) 45
AMA1 \(20\)–\(30\) 700
AMA1 \(45+\) 1394
MSP1 \(<6\) 42
MSP1 \(20\)–\(30\) 259
MSP1 \(45+\) 459
  • The latent Beta model outperforms the GMM in every single age band, for both antigens
  • The advantage for AMA1 grows sharply with age, exceeding 1000 BIC units in older groups
  • MSP1 gains are more modest — the GMM adapts reasonably well here, but the latent model still wins throughout
  • The \(L_2\) histogram approximation reproduces the same ranking as MLE, at a fraction of the computational cost

Part II

Developing a multivariate framework

Dependence in multi-antigen serology operates at two levels:

  • Within an individual — shared infection event, shared immunology
  • Between individuals — shared exposure environment


No existing framework handles cross-antigen correlation, age dependence and spatial dependence together for non-Gaussian, multimodal outcomes.

Notation for two antigens

For individual \(i\) and antigen \(k \in \{1,2\}\):

  • \(\mathbf{Y}_i = (Y_{i,1}, Y_{i,2})^\top\) — the pair of log-antibody concentrations
  • \(\mathbf{T}_i = (T_{i,1}, T_{i,2})^\top \in (0,1)^2\) — the pair of latent seroreactivities
  • \(\mathbf{Z}_i = (Z_{i,1}, Z_{i,2})^\top \in \{0,1\}^2\) — component labels, low or high regime per antigen
  • \(\mathbf{d}_i\) — covariates, with age \(a_i\) written separately where it matters


The question the principles answer: where in this hierarchy does each source of dependence belong?

P1 — Dependence lies in the latent process

  • The observation model describes assay characteristics, not biology

  • Conditional on \(\mathbf{T}_i\), the two antibody measurements are independent:

    \[ p(\mathbf{y}_i \mid \mathbf{T}_i = \mathbf{t}) = \prod_{k=1}^{2} \mathcal{N}\big\{y_{i,k};\, \mu_k(t_{i,k}),\, \sigma_k^2(t_{i,k})\big\} \]

  • Implication: all cross-antigen dependence is carried by the joint density \(g(\mathbf{t}\mid\mathbf{d}_i)\) — nothing is left in the observation model

  • When it fails: antibody cross-reactivity would induce correlation in \(\mathbf{Y}\) unrelated to \(\mathbf{T}\), requiring a residual covariance term instead

P2 — Exposure and magnitude are separate

  • Exposure determines the probability of mounting a high response

  • Individual immunology determines the magnitude, given exposure

  • Implication: a mixture for \(\mathbf{T}_i\) is the natural choice, since it provides two distinct places to put them

    \[ \mathbf{T}_i \mid \mathbf{Z}_i = \mathbf{z} \;\sim\; \mathcal{N}_{(0,1)^2}\big\{ \mathbf{m}(\mathbf{d}_i;\mathbf{z}),\, \Sigma_T \big\} \]

    exposure → the mixing probabilities \(\pi_i(\mathbf{z})\); magnitude → the component locations \(\mathbf{m}(\mathbf{d}_i;\mathbf{z})\)

  • Consequence: spatial variation, being variation in the force of infection, enters only the mixing probabilities

P3 — Distinct scales, distinct structures

  • Shared infection event, within an individual

    → \(\delta(a_i)\) in \(\Pr(Z_{i,2}=1 \mid Z_{i,1}=z_1)\), the association between component labels

  • Shared within-host processes, within an individual

    → \(\rho_T\), the correlation in \(\Sigma_T\) once the labels are fixed

  • Shared exposure environment, between individuals

    → \(\rho_S\), the cross-correlation of the fields \(\{S_1(\mathbf{x}), S_2(\mathbf{x})\}\)


  • Implication: each mechanism that induce dependence is governed by its own parameter

The latent unit square

Mass concentrates on the diagonal, with an off-diagonal tail.

Correlation between AMA1 and MSP1: Estimates

Quantity Estimate 95% CI
Spatial range, AMA1 473 m (353, 639)
Spatial range, MSP1 437 m (318, 577)
Cross-field \(\rho_S\) 0.587 (0.453, 0.711)
Within-component \(\rho_T\) 0.717 (0.691, 0.750)
Age effect \(\delta_1\) 0.286 (0.211, 0.339)
  • Residential-scale clustering of transmission
  • Substantial spatial co-localisation of the two antigens
  • Cross-antigen association strengthens with age

Bivariate fit: age-stratified validation

Spatial predictions

Does joint modelling help?

Coarsened total variation distance for the joint distribution:

Age band Univariate Joint
\([1,5]\) 0.295 0.189
\((20,40]\) 0.225 0.083
All ages 0.194 0.078
  • Univariate models stay slightly better on the marginals — as expected
  • Joint model → 12.6% smaller bivariate HDR at the same coverage
  • Simulation: most predictions gains are shown in the inferences for \(T\)

Joint modelling matters if we are answering research questions that involve joint properties of antibodies

Discussion: going beyond two antigens

The sequential construction extends directly by adding conditional terms \(\delta_{jk}(a_i)\). The harder questions are epidemiological.

  • Does the pathogen structure matter? Whether the antigens belong to one pathogen or several has direct implications for how cross-antigen association should be modelled

  • Disease-agnostic approach. Impose no prior structure on the correlations and let the data determine the full pattern of dependence

  • Assumption-driven approach. Impose a parsimonious structure informed by biology, for example allowing association only among antigens sharing a pathogen or a transmission route

  • The principles adapt. P3 as stated assumes a shared pathogen and life-cycle stage; for antigens from distinct pathogens the shared-infection pathway no longer applies and \(\delta(a)\) should be left unconstrained

  • Computation is the binding constraint. The approximations used here are adequate for a bivariate outcome but unlikely to scale to tens of antibody responses


Extending to a handful of antigens from the same pathogen is relatively straightforward. A genuinely multivariate formulation would need to be tailored to the epidemiological context.

Summary

  • Latent seroreactivity provides a flexible and biologically intepretable way to model continuous measurements of antibody concentration
  • Three principles keep the multivariate extension interpretable and identifiable
  • The gains of joint modelling are dependent on the research question

THANK YOU!

References

Giorgi, E. and Wallin, J. (2026). A flexible class of latent variable models for the analysis of antibody response data. Biostatistics, Volume 27, Issue 1, January 2026, kxag008, https://doi.org/10.1093/biostatistics/kxag008

Giorgi, E. and Wallin, J. (2026). Bivariate geostatistical latent variable models for the analysis of antibody density data. Pre-print available at https://arxiv.org/abs/2608.27534

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📧 e.giorgi@bham.ac.uk
📍Department of Health Sciences, University of Birmingham, Birmingham, UK