Bayesian Scalar-on-Function Regression with `refundBayes::sofr_bayes`

2026-03-03

Introduction

This vignette provides a detailed guide to the sofr_bayes() function in the refundBayes package, which fits Bayesian Scalar-on-Function Regression (SoFR) models using Stan. The function is designed with a syntax similar to mgcv::gam, making it accessible to users familiar with the frequentist approach while providing full Bayesian posterior inference.

The methodology follows the tutorial by Jiang, Crainiceanu, and Cui (2025), Tutorial on Bayesian Functional Regression Using Stan, published in Statistics in Medicine.

Install the refundBayes Package

The refundBayes package can be installed from GitHub:

library(remotes)
remotes::install_github("https://github.com/ZirenJiang/refundBayes")

Statistical Model

The SoFR Model

Scalar-on-Function Regression (SoFR) models the relationship between a scalar outcome and one or more functional predictors (curves or trajectories observed over a continuum), along with optional scalar covariates.

For subject $i = 1, \ldots, n$, let $Y_i$ be the scalar outcome, $\mathbf{Z}_i$ be a $p \times 1$ vector of scalar predictors, and $\{W_i(t_m), t_m \in \mathcal{T}\}$ for $m = 1, \ldots, M$ be a functional predictor observed at $M$ time points over a domain $\mathcal{T}$. The SoFR model assumes that the distribution of $Y_i$ belongs to an exponential family with mean $\mu_i$, and the linear predictor $\eta_i = g(\mu_i)$ has the following structure:

$$\eta_i = \eta_0 + \int_{\mathcal{T}} W_i(t)\beta(t)\,dt + \mathbf{Z}_i^t \boldsymbol{\gamma}$$

where:

• $\eta_0$ is the overall intercept,
• $\beta(\cdot) \in L^2(\mathcal{T})$ is the unknown functional coefficient that characterizes the effect of the functional predictor on the outcome,
• $\boldsymbol{\gamma}$ is a $p \times 1$ vector of scalar regression coefficients,
• $\mathcal{T}$ is the domain of the functional predictor. This domain is not restricted to $[0,1]$; it is determined by the actual time points in the data (e.g., $\mathcal{T} = [1, 1440]$ for minute-level 24-hour activity data, or $\mathcal{T} = [0, 100]$ for a generic index set).

The integral $\int_{\mathcal{T}} W_i(t)\beta(t)\,dt$ is approximated using a Riemann sum over the observed time points.

Basis Expansion and Penalized Splines

The functional coefficient $\beta(t)$ is represented nonparametrically using a set of $K$ pre-specified spline basis functions $\psi_1(t), \ldots, \psi_K(t)$:

$$\beta(t) = \sum_{k=1}^K b_k \psi_k(t)$$

With this expansion, the linear predictor becomes:

$$\eta_i = \eta_0 + \mathbf{X}_i^t \mathbf{b} + \mathbf{Z}_i^t \boldsymbol{\gamma}$$

where $\mathbf{X}_i = (X_{i1}, \ldots, X_{iK})^t$ is a $K \times 1$ vector with entries:

$$X_{ik} = \sum_{m=1}^M L_m W_i(t_m) \psi_k(t_m)$$

Here $L_m = t_{m+1} - t_m$ are the Riemann sum integration weights and $t_m$ are the time points at which the functional predictor is observed.

The Role of tmat, lmat, and wmat

The Riemann sum approximation $\sum_{m=1}^M L_m W_i(t_m)\psi_k(t_m)$ to the integral $\int_{\mathcal{T}} W_i(t)\psi_k(t)\,dt$ is constructed directly from three user-supplied matrices in the data argument:

• tmat (an $n \times M$ matrix): contains the time points $t_m$ at which the functional predictor is observed. The $(i, m)$-th entry equals $t_m$. The range of values in tmat determines the domain of integration $\mathcal{T}$. For example, if the functional predictor is observed at minutes $1, 2, \ldots, 1440$ within a day, then tmat has entries ranging from $1$ to $1440$ and $\mathcal{T} = [1, 1440]$. There is no requirement that the domain be rescaled to $[0, 1]$.

• lmat (an $n \times M$ matrix): contains the integration weights $L_m = t_{m+1} - t_m$ for the Riemann sum approximation. The $(i, m)$-th entry equals $L_m$. For equally spaced time points with spacing $\Delta t$, every entry of lmat equals $\Delta t$. For unevenly spaced time points, lmat reflects the varying widths of the integration intervals. These weights, together with tmat, fully specify how the numerical integration is performed and over what domain.

• wmat (an $n \times M$ matrix): contains the functional predictor values. The $i$-th row contains the $M$ observed values $W_i(t_1), \ldots, W_i(t_M)$ for subject $i$.

In the formula s(tmat, by = lmat * wmat, bs = "cc", k = 10), the mgcv infrastructure uses tmat to construct the spline basis $\psi_k(t_m)$ at the observed time points, and the by = lmat * wmat argument provides the element-wise product $L_m \cdot W_i(t_m)$ that enters the Riemann sum. This means the basis functions are evaluated on the scale of tmat, and the integration weights in lmat ensure that the discrete sum correctly approximates the integral over the actual domain $\mathcal{T}$ — regardless of whether it is $[0, 1]$, $[1, 1440]$, or any other interval.

Note that for all subjects, the time points are assumed to be identical so that $t_{im} = t_m$ for all $i = 1, \ldots, n$. Thus every row of tmat is the same, and every row of lmat is the same. The matrices are replicated across rows to match the mgcv syntax, which expects all terms in the formula to have the same dimensions.

Smoothness Penalty

To induce smoothness on $\beta(t)$, a quadratic penalty on the spline coefficients is applied. The penalty is based on the integrated squared second derivative of $\beta(t)$:

$$\int \{\beta''(t)\}^2\,dt = \mathbf{b}^t \mathbf{S} \mathbf{b}$$

where $\mathbf{S} = \int \boldsymbol{\psi}''(t)\{\boldsymbol{\psi}''(t)\}^t\,dt$ is the penalty matrix. In the Bayesian framework, this penalty is equivalent to placing a multivariate normal prior on the spline coefficients:

$$p(\mathbf{b}) \propto \exp\left(-\frac{\mathbf{b}^t \mathbf{S} \mathbf{b}}{\sigma_b^2}\right)$$

where $\sigma_b^2$ is the smoothing parameter that controls the smoothness of $\beta(t)$, and is estimated from the data.

Full Bayesian Model

The complete Bayesian SoFR model is:

$$\begin{cases} \mathbf{Y} \sim \text{Exponential Family}(\boldsymbol{\eta}, a) \\ \boldsymbol{\eta} = \eta_0 \mathbf{J}_n + \tilde{\mathbf{X}}_r^t \tilde{\mathbf{b}}_r + \tilde{\mathbf{X}}_f^t \tilde{\mathbf{b}}_f + \mathbf{Z}^t \boldsymbol{\gamma} \\ \tilde{\mathbf{b}}_r \sim N(\mathbf{0}, \sigma_b^2 \mathbf{I}) \\ \eta_0 \sim p(\eta_0);\; \tilde{\mathbf{b}}_f \sim p(\tilde{\mathbf{b}}_f);\; \boldsymbol{\gamma} \sim p(\boldsymbol{\gamma}) \\ \sigma_b^2 \sim p(\sigma_b^2);\; a \sim p(a) \end{cases}$$

where $\tilde{\mathbf{X}}_r$ and $\tilde{\mathbf{X}}_f$ are the correspondingly transformed design matrices, and $p(\cdot)$ denotes non-informative priors (uniform or weakly informative). The smoothing variance $\sigma_b^2$ is assigned an inverse-Gamma prior $IG(0.001, 0.001)$.

The sofr_bayes() Function

Usage

sofr_bayes(
  formula,
  data,
  family = gaussian(),
  joint_FPCA = NULL,
  intercept = TRUE,
  runStan = TRUE,
  niter = 3000,
  nwarmup = 1000,
  nchain = 3,
  ncores = 1
)

Arguments

Argument	Description
formula	Functional regression formula, using the same syntax as mgcv::gam. Functional predictors are specified using the s() term with by = lmat * wmat to encode the Riemann sum integration (see Example below).
data	A data frame containing all scalar and functional variables used in the model.
family	Distribution of the outcome variable. Currently supports gaussian() and binomial(). Default is gaussian().
joint_FPCA	A logical (TRUE/FALSE) vector of the same length as the number of functional predictors, indicating whether to jointly model FPCA for each functional predictor. Default is NULL, which sets all entries to FALSE (no joint FPCA).
intercept	Logical. Whether to include an intercept term in the linear predictor. Default is TRUE.
runStan	Logical. Whether to run the Stan program. If FALSE, the function only generates the Stan code and data without sampling. This is useful for inspecting or modifying the generated Stan code. Default is TRUE.
niter	Total number of Bayesian posterior sampling iterations (including warmup). Default is 3000.
nwarmup	Number of warmup (burn-in) iterations. These samples are discarded and not used for inference. Default is 1000.
nchain	Number of Markov chains for posterior sampling. Multiple chains help assess convergence. Default is 3.
ncores	Number of CPU cores to use when executing the chains in parallel. Default is 1.

Return Value

The function returns a list of class "refundBayes" containing the following elements:

Element	Description
stanfit	The Stan fit object (class stanfit). Can be used for convergence diagnostics, traceplots, and additional summaries via the rstan package.
spline_basis	Basis functions used to reconstruct the functional coefficients from the posterior samples.
stancode	A character string containing the generated Stan model code.
standata	A list containing the data passed to the Stan model.
int	A vector of posterior samples for the intercept term $\eta_0$. NULL if intercept = FALSE.
scalar_coef	A matrix of posterior samples for scalar coefficients $\boldsymbol{\gamma}$, where each row is one posterior sample and each column corresponds to one scalar predictor. NULL if no scalar predictors are included.
func_coef	A list of posterior samples for functional coefficients. Each element is a matrix where each row is one posterior sample and each column corresponds to one location on the functional domain.
family	The distribution family used for the outcome.

Formula Syntax

The formula follows the mgcv::gam syntax. The key component for specifying functional predictors is:

s(tmat, by = lmat * wmat, bs = "cc", k = 10)

where:

• tmat: an $n \times M$ matrix of time points. Each row contains the same $M$ observation times (replicated across subjects). The values in tmat define the domain $\mathcal{T}$ over which the functional predictor is observed and the integral is computed. For example, tmat may contain values from $1$ to $100$, from $0$ to $1$, or from $1$ to $1440$ — the integration domain adapts accordingly.
• lmat: an $n \times M$ matrix of Riemann sum weights. The $(i,m)$-th entry equals $L_m = t_{m+1} - t_m$. These weights control the numerical integration and should be consistent with the spacing of the time points in tmat. For equally spaced time points, lmat is a constant matrix; for irregular spacing, it reflects the actual gaps between consecutive time points.
• wmat: an $n \times M$ matrix of functional predictor values. The $i$-th row contains the $M$ observed values for subject $i$.
• bs: the type of spline basis. Common choices include "cr" (cubic regression splines) and "cc" (cyclic cubic regression splines, suitable for periodic data).
• k: the number of basis functions (maximum degrees of freedom for the spline).

Scalar predictors are included as standard formula terms (e.g., y ~ X1 + s(tmat, by = lmat * wmat, bs = "cr", k = 10)).

Example: Bayesian SoFR with Binary Outcome

We demonstrate the sofr_bayes() function using a simulated example dataset with a binary outcome and one functional predictor.

Load Data

## Alternative sample data 

# data.SoFR <- readRDS("data/example_data_sofr.rds")

## Load the example data
set.seed(123)
n <- 100
M <- 50
tgrid <- seq(0, 1, length.out = M)
dt    <- tgrid[2] - tgrid[1]
tmat  <- matrix(rep(tgrid, each = n), nrow = n)
lmat  <- matrix(dt, nrow = n, ncol = M)
wmat  <- t(apply(matrix(rnorm(n * M), n, M), 1, cumsum)) / sqrt(M)
beta_true <- sin(2 * pi * tgrid)
X1 <- rnorm(n)
eta <- 0.5 * X1 + wmat %*% (beta_true * dt)
prob <- plogis(eta)
y <- rbinom(n, 1, prob)
data.SoFR <- data.frame(y = y, X1 = X1)
data.SoFR$tmat <- tmat
data.SoFR$lmat <- lmat
data.SoFR$wmat <- wmat

The example dataset data.SoFR contains:

• y: a binary outcome variable,
• X1: a scalar predictor,
• tmat: the $n \times M$ time point matrix (defines the domain $\mathcal{T}$),
• lmat: the $n \times M$ Riemann sum weight matrix (defines the integration weights over $\mathcal{T}$),
• wmat: the $n \times M$ functional predictor matrix.

Fit the Bayesian SoFR Model

library(refundBayes)

refundBayes_SoFR <- refundBayes::sofr_bayes(
  y ~ X1 + s(tmat, by = lmat * wmat, bs = "cc", k = 10),
  data = data.SoFR,
  family = binomial(),
  runStan = TRUE,
  niter = 1500,
  nwarmup = 500,
  nchain = 3,
  ncores = 3
)

In this call:

• The formula specifies a binary outcome y with one scalar predictor X1 and one functional predictor encoded via s(tmat, by = lmat * wmat, bs = "cc", k = 10).
• The spline basis is evaluated at the time points stored in tmat, and the integration over the domain $\mathcal{T}$ (determined by tmat) is approximated using the weights in lmat.
• bs = "cc" uses cyclic cubic regression splines, which are appropriate when the functional predictor is periodic (e.g., physical activity measured over a 24-hour cycle).
• k = 10 specifies 10 basis functions. In practice, 30–40 basis functions are often sufficient for moderately smooth functional data on dense grids.
• family = binomial() specifies a logistic regression for the binary outcome.
• The sampler runs 3 chains in parallel, each with 1500 total iterations (500 warmup + 1000 posterior samples).

Visualization

The plot() method for refundBayes objects displays the estimated functional coefficient $\hat{\beta}(t)$ along with pointwise 95% credible intervals:

library(ggplot2)
plot(refundBayes_SoFR)

Extracting Posterior Summaries

Posterior summaries of the functional coefficient can be computed directly from the func_coef element:

## Posterior mean of the functional coefficient
mean_curve <- apply(refundBayes_SoFR$func_coef[[1]], 2, mean)

## Pointwise 95% credible interval
upper_curve <- apply(refundBayes_SoFR$func_coef[[1]], 2,
                     function(x) quantile(x, prob = 0.975))
lower_curve <- apply(refundBayes_SoFR$func_coef[[1]], 2,
                     function(x) quantile(x, prob = 0.025))

The posterior samples in func_coef[[1]] are stored as a $Q \times M$ matrix, where $Q$ is the number of posterior samples and $M$ is the number of time points on the functional domain.

Comparison with Frequentist Results

The Bayesian results can be compared with frequentist estimates obtained via mgcv::gam:

library(mgcv)

## Fit frequentist SoFR using mgcv
fit_freq <- gam(
  y ~ s(tmat, by = lmat * wmat, bs = "cc", k = 10) + X1,
  data = data.SoFR,
  family = "binomial"
)

## Extract frequentist estimates
freq_result <- plot(fit_freq)

The functional coefficient estimates from the Bayesian and frequentist approaches are generally comparable in shape and magnitude, though the Bayesian credible intervals tend to be slightly wider due to accounting for uncertainty in the smoothing parameter.

Inspecting the Generated Stan Code

Setting runStan = FALSE allows you to inspect or modify the Stan code before running the model:

## Generate Stan code without running the sampler
sofr_code <- refundBayes::sofr_bayes(
  y ~ X1 + s(tmat, by = lmat * wmat, bs = "cc", k = 10),
  data = data.SoFR,
  family = binomial(),
  runStan = FALSE
)

## Print the generated Stan code
cat(sofr_code$stancode)

Practical Recommendations

• Number of basis functions (k): For illustrative purposes, k = 10 is often used. In practice, 30–40 basis functions are recommended for moderately smooth functional data observed on dense grids.
• Spline type (bs): Use "cr" (cubic regression splines) for general functional predictors. Use "cc" (cyclic cubic regression splines) when the functional predictor is periodic (e.g., 24-hour activity patterns).
• Number of iterations: Ensure sufficient posterior samples for reliable inference. A common setup is niter = 3000 with nwarmup = 1000, but more complex models may require additional iterations.
• Convergence diagnostics: After fitting, examine traceplots and $\hat{R}$ statistics using the rstan package (e.g., rstan::traceplot(refundBayes_SoFR$stanfit)) to ensure that the Markov chains have converged.
• Joint FPCA: When functional predictors are measured with substantial noise, consider setting joint_FPCA = TRUE for the relevant predictor to jointly estimate FPCA scores and regression coefficients.

References

• Jiang, Z., Crainiceanu, C., and Cui, E. (2025). Tutorial on Bayesian Functional Regression Using Stan. Statistics in Medicine, 44(20–22), e70265.
• Crainiceanu, C. M., Goldsmith, J., Leroux, A., and Cui, E. (2024). Functional Data Analysis with R. CRC Press.
• Crainiceanu, C. M. and Goldsmith, A. J. (2010). Bayesian Functional Data Analysis Using WinBUGS. Journal of Statistical Software, 32(11), 1–33.
• Wood, S. (2001). mgcv: GAMs and Generalized Ridge Regression for R. R News, 1(2), 20–25.
• Carpenter, B., Gelman, A., Hoffman, M. D., et al. (2017). Stan: A Probabilistic Programming Language. Journal of Statistical Software, 76(1), 1–32.