Create a latent factor specification

Define latent factors for capturing unmeasured shared structure between all processes. Factors are observation-level random effects that enter both linear predictors when shared = TRUE (default).

Usage

latent_factor(
  n_factors = 1L,
  prior = NULL,
  shared = TRUE,
  constraint = c("sum_to_zero", "first_zero"),
  scale = TRUE
)

Arguments

n_factors

Integer; number of latent factors. Default is 1. More factors capture more complex unmeasured structure but increase computational cost and risk overfitting.

prior

Prior for factor standard deviations. Default is a PC prior with P(sigma > 1) = 0.01, which shrinks toward simpler models.

shared

Logical; if TRUE (default), latent factors enter both all process linear predictors identically. If FALSE, factors only affect the first process.

constraint

Identifiability constraint for factors:

• "sum_to_zero" (default): Factors sum to zero across observations

• "first_zero": First observation's factor is fixed to zero

scale

Logical; if TRUE (default), factor loadings are standardized to have unit variance before applying sigma.

Value

A tulpa_latent object consumed by ratio / multi-arm model packages built on tulpa (e.g. tulpaRatio); not read by the single-response tulpa() front door (use latent(tgmrf(...)) for a single-response latent Gaussian block).

Details

Why Use Latent Factors?

When modeling multiple processes, they often share unmeasured confounders. For example, in relative abundance data, both the focal species count and total count might be affected by:

• Observer skill (unmeasured)

• Local microhabitat conditions (unmeasured)

• Weather on sampling day (unmeasured)

Without accounting for these shared drivers, estimates can be biased. Latent factors capture this shared structure without requiring the confounders to be measured.

Mathematical Model

For observation i with K latent factors, on each of two model arms (processes 1 and 2):

$$\eta^{(1)}_i = X^{(1)}_i \beta^{(1)} + \sum_{k=1}^{K} f_{ik} \sigma_k + \ldots$$ $$\eta^{(2)}_i = X^{(2)}_i \beta^{(2)} + \sum_{k=1}^{K} f_{ik} \sigma_k + \ldots$$

where:

• \(f_{ik} \sim N(0, 1)\) are standardized factor scores

• \(\sigma_k\) are factor standard deviations with PC prior

• Identifiability: \(\sum_i f_{ik} = 0\) for each k

Because factors enter both linear predictors identically (when shared), they cancel in derived quantities (e.g., ratios, differences):

$$\eta^{(1)}_i - \eta^{(2)}_i$$

This means factors capture shared effects that would otherwise bias derived quantities.

Choosing n_factors

• Start with n_factors = 1 for simple unmeasured confounding

• Use n_factors = 2-3 if you suspect multiple independent confounders

• More than 3 factors is rarely needed and risks overfitting

• The PC prior provides regularization, shrinking unneeded factors toward zero

Relationship to Random Effects

Latent factors differ from random effects in several ways:

• Random effects are grouped (e.g., site-level), factors are observation-level

• Random effects require grouping structure, factors don't

• Factors capture residual correlation not explained by observed predictors

You can use both together: random effects for known grouping, factors for residual unmeasured confounding.

See also

prior_pc() for prior specification; latent() / tgmrf() for a single-response in-formula latent Gaussian block

Examples

# Basic latent factor (single shared factor)
latent_factor()

# Two latent factors
latent_factor(n_factors = 2)

# Custom prior (more regularization)
latent_factor(n_factors = 1, prior = prior_pc(U = 0.5, alpha = 0.01))

# Numerator-only factor (not shared)
latent_factor(n_factors = 1, shared = FALSE)

# The spec is consumed by a ratio / multi-arm model package (tulpaRatio owns
# the two-arm `species | total ~ ...` formula and the negbin/negbin ratio
# family); it is that package's fitter, not the single-response tulpa() front
# door, that reads the `latent_factor()` object.