The Functional Cox Regression Model

The Functional Cox Regression (FCR) model extends the classical Cox proportional hazards model to settings where one or more predictors are functional (i.e., curves or trajectories observed over a continuum).

For subject \(i = 1, \ldots, n\), let \(T_i\) be the event time and \(C_i\) the censoring time. The observed data consist of \([Y_i, \delta_i, \mathbf{Z}_i, \{W_i(t_m), t_m \in \mathcal{T}\}]\), where:

• \(Y_i = \min(T_i, C_i)\) is the observed follow-up time,
• \(\delta_i\) is the censoring indicator with \(\delta_i = 0\) if the event is observed (\(Y_i = T_i\)) and \(\delta_i = 1\) if the event is censored (\(Y_i < T_i\)),
• \(\mathbf{Z}_i\) is a \(p \times 1\) vector of scalar predictors,
• \(\{W_i(t_m), t_m \in \mathcal{T}\}\) for \(m = 1, \ldots, M\) is a functional predictor observed over a domain \(\mathcal{T}\).

The domain \(\mathcal{T}\) 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). See the section on tmat, lmat, and wmat below for details.

The model assumes a proportional hazards structure:

\[h_i(t) = h_0(t) \exp(\eta_i)\]

where \(h_0(t)\) is the baseline hazard function, and \(\eta_i\) is the linear predictor defined as:

\[\eta_i = \eta_0 + \int_{\mathcal{T}} W_i(s)\beta(s)\,ds + \mathbf{Z}_i^t \boldsymbol{\gamma}\]

Here \(\eta_0\) is the intercept, \(\beta(\cdot)\) is the functional coefficient, and \(\boldsymbol{\gamma}\) is the vector of scalar regression coefficients. The integral is approximated by a Riemann sum using the time point matrix tmat and the integration weight matrix lmat (see below).

The log-likelihood for this model has the form:

\[\ell(\mathbf{Y}, \boldsymbol{\delta}; h_0, \boldsymbol{\eta}) = \sum_{i=1}^n \left[ (1 - \delta_i)\left\{\log h_0(y_i) + \eta_i - H_0(y_i)\exp(\eta_i)\right\} + \delta_i\left\{-H_0(y_i)\exp(\eta_i)\right\} \right]\]

where \(H_0(t) = \int_0^t h_0(u)\,du\) is the cumulative baseline hazard function.

Modeling the Baseline Hazard with M-Splines

Unlike the classical Cox model, which leaves the baseline hazard unspecified and uses partial likelihood, the Bayesian approach requires a full likelihood specification. The fcox_bayes() function models the baseline hazard function \(h_0(t)\) using M-splines:

\[h_0(t) = \sum_{l=1}^L c_l M_l(t; \mathbf{k}, \tau)\]

where \(M_l(t; \mathbf{k}, \tau)\) denotes the \(l\)-th M-spline basis function with knots \(\mathbf{k}\) and degree \(\tau\), and \(\mathbf{c} = (c_1, \ldots, c_L)^t\) are the spline coefficients. The constraints \(c_l \geq 0\) and \(\sum_{l=1}^L c_l > 0\) ensure that the baseline hazard is non-negative and non-trivial. M-splines are a family of non-negative, piecewise polynomial basis functions that integrate to one over their support.

The cumulative baseline hazard function is modeled using I-splines, which are the integrated forms of M-splines:

\[H_0(t) = \sum_{l=1}^L c_l I_l(t; \mathbf{k}, \tau)\]

where \(I_l(t; \mathbf{k}, \tau) = \int_0^t M_l(u; \mathbf{k}, \tau)\,du\). Because M-splines are non-negative, the resulting I-spline cumulative hazard is non-decreasing by construction.

Identifiability Constraint

The I-spline coefficients \(\mathbf{c}\) and the intercept \(\eta_0\) are not simultaneously identifiable: for any \(a > 0\), the parameter pairs \((\mathbf{c}, \eta_0)\) and \((a\mathbf{c}, \eta_0 - \log a)\) yield the same hazard function. To resolve this, the model imposes the constraint \(\sum_{l=1}^L c_l = 1\) by assigning a non-informative Dirichlet prior \(D(\mathbf{c}; \boldsymbol{\alpha})\) with \(\boldsymbol{\alpha} = (1, \ldots, 1)\) to the coefficients. Consequently, the intercept defaults to FALSE in fcox_bayes().

Functional Coefficient via Penalized Splines

As in the SoFR model, the functional coefficient \(\beta(s)\) is represented using \(K\) spline basis functions \(\psi_1(s), \ldots, \psi_K(s)\):

\[\beta(s) = \sum_{k=1}^K b_k \psi_k(s)\]

The spline coefficients \(\mathbf{b}\) are reparametrised using the spectral decomposition of the penalty matrix \(\mathbf{S}\) (the Wahba-O’Sullivan smoothing penalty) to obtain transformed coefficients \(\tilde{\mathbf{b}} = (\tilde{\mathbf{b}}_r^t, \tilde{\mathbf{b}}_f^t)^t\), where:

• \(\tilde{\mathbf{b}}_r\) (random effects) receive a \(N(\mathbf{0}, \sigma_b^2 \mathbf{I})\) prior,
• \(\tilde{\mathbf{b}}_f\) (fixed effects) receive non-informative priors.

This reparametrisation ensures numerical stability and efficient sampling in Stan.

Full Bayesian Model

The complete Bayesian Functional Cox Regression model is:

\[\begin{cases} \mathbf{Y} \sim \ell(\mathbf{Y}, \boldsymbol{\delta}; h_0, \boldsymbol{\eta}) \\ \boldsymbol{\eta} = \eta_0 \mathbf{J}_n + \tilde{\mathbf{X}}_r^t \tilde{\mathbf{b}}_r + \tilde{\mathbf{X}}_f^t \tilde{\mathbf{b}}_f + \mathbf{Z}^t \boldsymbol{\gamma} \\ \tilde{\mathbf{b}}_r \sim N(\mathbf{0}, \sigma_b^2 \mathbf{I}) \\ \eta_0 \sim p(\eta_0);\; \tilde{\mathbf{b}}_f \sim p(\tilde{\mathbf{b}}_f);\; \boldsymbol{\gamma} \sim p(\boldsymbol{\gamma});\; \sigma_b^2 \sim p(\sigma_b^2) \\ h_0(t) = \sum_{l=1}^L c_l M_l(t; \mathbf{k}, \tau) \\ \mathbf{c} \sim D(\mathbf{c}; \boldsymbol{\alpha}) \end{cases}\]

where most components are shared with the SoFR model, except for the Cox likelihood (first line) and the baseline hazard specification via M-splines with a Dirichlet prior on \(\mathbf{c}\) (last two lines).

Joint Modeling with FPCA (Optional)

When functional predictors are observed with measurement error, the fcox_bayes() function supports a joint modeling approach that simultaneously estimates FPCA scores and fits the Cox regression model. In this case, the model regresses on the latent trajectory \(D_i(t)\) rather than the observed (noisy) function \(W_i(t) = D_i(t) + \epsilon_i(t)\), properly accounting for the uncertainty in the score estimates.

Usage

fcox_bayes(
  formula,
  data,
  cens,
  joint_FPCA = NULL,
  intercept = FALSE,
  runStan = TRUE,
  niter = 3000,
  nwarmup = 1000,
  nchain = 3,
  ncores = 1
)

Arguments

Return Value

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

Formula Syntax

The formula follows the mgcv::gam syntax. The left-hand side is the observed survival time (not a Surv object). 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 defining the domain \(\mathcal{T}\) over which the functional predictor is observed and the integral is computed. The domain adapts to the actual values in tmat (e.g., \([0, 1]\), \([1, 1440]\), etc.).
• lmat: an \(n \times M\) matrix of Riemann sum weights (\(L_m = t_{m+1} - t_m\)), controlling the numerical integration over \(\mathcal{T}\).
• wmat: an \(n \times M\) matrix of functional predictor values.
• bs: the type of spline basis (e.g., "cr" for cubic regression splines, "cc" for cyclic cubic regression splines).
• k: the number of basis functions.

Scalar predictors are included as standard formula terms. The censoring information is provided separately via the cens argument, not through the formula.

Load and Prepare Data

## Load the example data
Func_Cox_Data <- readRDS("data/example_data_Cox.rds")

## Prepare the functional predictor and censoring indicator
Func_Cox_Data$wmat <- Func_Cox_Data$MIMS
Func_Cox_Data$cens <- 1 - Func_Cox_Data$event

The example dataset contains:

• survtime: the observed survival time \(Y_i = \min(T_i, C_i)\),
• event: a binary event indicator (\(1\) = event occurred, \(0\) = censored),
• 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 \(\mathcal{T}\)),
• MIMS: the \(n \times M\) functional predictor matrix (physical activity data).

Note that the censoring indicator cens is constructed as 1 - event, following the convention where \(\delta_i = 0\) indicates an observed event and \(\delta_i = 1\) indicates censoring.

Fit the Bayesian Functional Cox Model

library(refundBayes)

refundBayes_FCox <- refundBayes::fcox_bayes(
  survtime ~ X1 + s(tmat, by = lmat * wmat, bs = "cc", k = 10),
  data = Func_Cox_Data,
  cens = Func_Cox_Data$cens,
  runStan = TRUE,
  niter = 1500,
  nwarmup = 500,
  nchain = 3,
  ncores = 3
)

In this call:

• The formula specifies the observed survival time survtime as the response, 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.
• The censoring vector cens is supplied separately, with \(0\) = event observed and \(1\) = censored.
• bs = "cc" uses cyclic cubic regression splines, appropriate for periodic functional predictors (e.g., 24-hour activity patterns).
• k = 10 specifies 10 basis functions. In practice, 30–40 basis functions are recommended for moderately smooth functional data on dense grids.
• The intercept argument is not specified and defaults to FALSE, which is appropriate for Cox models due to the identifiability constraint with the baseline hazard.
• 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_FCox)

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_FCox$func_coef[[1]], 2, mean)

## Pointwise 95% credible interval
upper_curve <- apply(refundBayes_FCox$func_coef[[1]], 2,
                     function(x) quantile(x, prob = 0.975))
lower_curve <- apply(refundBayes_FCox$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, which handles Cox proportional hazards models through its cox.ph() family using partial likelihood:

library(mgcv)

## Fit frequentist functional Cox model using mgcv
fit_freq <- gam(
  survtime ~ s(tmat, by = lmat * wmat, bs = "cc", k = 10) + X1,
  data = Func_Cox_Data,
  family = cox.ph(),
  weights = Func_Cox_Data$event
)

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

Note that the frequentist approach uses cox.ph() family with partial likelihood and a Breslow-type nonparametric estimator for the baseline hazard, while the Bayesian approach models the baseline hazard parametrically using M-splines. Despite this difference, simulation studies in Jiang et al. (2025) show that both approaches achieve comparable performance in terms of estimation accuracy and coverage.

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
fcox_code <- refundBayes::fcox_bayes(
  survtime ~ X1 + s(tmat, by = lmat * wmat, bs = "cc", k = 10),
  data = Func_Cox_Data,
  cens = Func_Cox_Data$cens,
  runStan = FALSE
)

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

The generated Stan code includes a functions block that defines custom log-likelihood functions for the Cox model (cox_log_lpdf for observed events and cox_log_lccdf for censored observations), in addition to the standard data, parameters, and model blocks.