The package returns a fitted object of class fitted_onlinesurr that stores point estimates and bootstrap draws for treatment-effect trajectories and PTE-based summaries.

Data requirements and conventions

fit.surr() expects data in long format with one row per subject-time measurement. Key requirements enforced by the code:

id identifies subjects; there must be at most one observation per subject-time combination.
treat indicates treatment assignment; it is coerced to a factor and is intended to represent two treatment levels.
time must be numeric and equally spaced across observed time points. If time is omitted, the function creates a within-subject index Time assuming the data are already ordered and equally spaced.
The surrogate design must not make treatment a linear combination of surrogate terms; otherwise the conditional model is not identifiable.

Package functions used in this vignette

fit.surr() fits:
- a marginal model producing total treatment effects $\Delta(t)$
- a conditional model (given surrogate) producing residual treatment effects $\Delta_R(t)$
- stores bootstrap draws for the corresponding fixed-effect parameters.
plot.fitted_onlinesurr() plots:
- Local PTE: $\text{LPTE}(t) = 1 - \Delta_R(t)/\Delta(t)$
- Cumulative PTE: $\text{CPTE}(t) = 1 - \sum_{h\le t} \Delta_R(h) / \sum_{h\le t} \Delta(h)$
- Treatment effects $\Delta(t)$ and $\Delta_R(t)$
time_homo_test() tests the hypothesis that the PTE is constant over time (implemented via a max-type statistic and Monte Carlo approximation of the null).

A complete worked example (simulated data)

This section builds a minimal simulated dataset consistent with the checks inside fit.surr().

set.seed(1)

# Dimensions
N <- 100 # subjects
T <- 6 # time points
times <- seq_len(T) # numeric, equally spaced

# Subject IDs and treatment assignment (two levels)
id <- rep(seq_len(N), each = T)
trt <- rep(rbinom(N, 1, 0.5), each = T) # 0/1

time <- rep(times, times = N)

# Simulate surrogate(s) and outcome
# Surrogate is affected by treatment and time
s <-
  0.2 * time + # Trend
  (0.2 + 0.4 * time) * trt + # Treatment effect
  rnorm(N * T, sd = 0.1) # Noise


# Outcome depends on treatment effect, time trend, surrogate, and noise
y <-
  0.2 + # intercept
  0.1 * time + # Trend
  (0.2 + 0.1 * time) * trt + # (direct) treatment effect
  0.6 * s + # Surrogate effect
  rnorm(N * T, sd = 0.1) # Noise

dat <- data.frame(
  id = id,
  trt = trt,
  time = time,
  s = s,
  y = y
)

# Ensure the data are ordered by (id, time) (recommended)
dat <- dat[order(dat$id, dat$time), ]
head(dat)
#>   id trt time         s         y
#> 1  1   0    1 0.2398106 0.4499785
#> 2  1   0    2 0.3387974 0.5209293
#> 3  1   0    3 0.6341120 1.0634402
#> 4  1   0    4 0.6870637 0.8692466
#> 5  1   0    5 1.1433024 1.4113951
#> 6  1   0    6 1.3980400 1.3448466

Fitting the models with `fit.surr()`

fit.surr() requires:

formula: outcome mean model. The function will internally add treatment-by-time fixed effects.
id: subject identifier (unquoted).
treat: treatment variable (unquoted).
surrogate: surrogate structure (as a formula or a string).
time: numeric time variable (unquoted).

library(OnlineSurr)

fit <- fit.surr(
  formula   = y ~ 1, # baseline fixed effects; trt*time terms added internally
  id        = id,
  surrogate = ~s, # surrogate structure
  treat     = trt,
  data      = dat,
  time      = time,
  N.boots   = 2000, # bootstrap draws stored in the fitted object
  verbose   = 0 # hide progress
)

The formulas for the fixed effects and the surrogate structures accept any temporal structure available in the kDGLM package (see its vignette for details). Functions that transform the data are also supported.

In particular, we provide the lagged function, which computes lagged values of its arguments and can be included in a model formula to account for delayed or lingering effects of a predictor over time. We also provide the s function, which generates a spline basis for a numeric variable and can be used to model smooth, potentially non-linear effects without having to specify the basis expansion manually.

library(OnlineSurr)

fit <- fit.surr(
  formula   = y ~ 1, # baseline fixed effects; trt*time terms added internally
  id        = id,
  surrogate = ~ s(s) + s(lagged(s, 1)) + s(lagged(s, 2)), # surrogate structure
  treat     = trt,
  data      = dat,
  time      = time,
  verbose   = 0 # hide progress
)

What `fit.surr()` stores

The returned object has class fitted_onlinesurr and is a list with (at least):

fit$T: number of time points
fit$N: number of subjects
fit$n.fixed: number of fixed-effect coefficients per subject design (reference size)
fit$Marginal$point: point estimates (vector) from the marginal model
fit$Marginal$smp: bootstrap draws (matrix) from the marginal model
fit$Conditional$point: point estimates (vector) from the conditional model
fit$Conditional$smp: bootstrap draws (matrix) from the conditional model

The first T (in practice, the first n.fixed) elements used by plotting/testing methods correspond to the time-indexed treatment-effect parameters.

Summaries and inference

Printing a summary

The package provides an S3 summary method summary.fitted_onlinesurr().

t selects the time index.
cumulative=TRUE reports cumulative effects up to time t (when implemented by the method).
cumulative=FALSE reports time-specific quantities at time t only.

summary(fit, t = 6, cumulative = TRUE)
#> Fitted Online Surrogate
#> 
#> Cummulated effects at time 6:
#>         Estimate Std. Error t value   Pr(>|t|)   
#> Delta    9.01931  0.05600   161.06739 0.0000e+00 ***
#> Delta.R  2.47893  0.41384     5.99008 2.0973e-09 ***
#> CPTE     0.72515  0.04604       -         -       
#> 
#> Time homogeneity test: 
#> 
#> Test stat.   Crit. value   p-value     
#>    1.12582       2.45246    0.61942    
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Plotting LPTE, CPTE, and treatment effects

plot() dispatches to plot.fitted_onlinesurr().

plot(fit, type = "LPTE") # Local PTE over time

plot(fit, type = "CPTE") # Cumulative PTE over time

plot(fit, type = "Delta") # Delta and Delta_R over time

Interpretation notes:

LPTE measures, at each time, the proportion of the total treatment effect explained by the surrogate, using the ratio $1 - \Delta_R(t)/\Delta(t)$.
CPTE aggregates effects up to time $t$, using cumulative sums.

Testing time-homogeneity

time_homo_test() provides a max-type test, using a Monte Carlo approximation of the null distribution.

test <- time_homo_test(fit, signif.level = 0.05, N.boots = 50000)
test
#> $T
#> [1] 1.125824
#> 
#> $T.crit
#>      95% 
#> 2.449603 
#> 
#> $p.value
#> [1] 0.61756

Returned components:

T: observed test statistic
T.crit: critical value at the requested significance level
p.value: Monte Carlo p-value

Practical tips and common pitfalls

Time index must be numeric and equally spaced.
If you have missing measurements, include the missing time points with NA outcomes rather than dropping those times, so spacing remains consistent.
One row per subject-time.
If you have duplicates, aggregate first (e.g., average within a time window) or decide which measurement to keep.
Bootstrap size tradeoff.
fit.surr(N.boots=...) controls stored bootstrap draws used for confidence intervals associated with the treatment effect, LPTE and CPTE; time_homo_test(N.boots=...) controls Monte Carlo draws for the null distribution of the time homogeneity test.

See Santos Jr. and Parast (2026) for details about the theoretical aspects of the package.

Session info

sessionInfo()
#> R version 4.5.1 (2025-06-13 ucrt)
#> Platform: x86_64-w64-mingw32/x64
#> Running under: Windows 10 x64 (build 19045)
#> 
#> Matrix products: default
#>   LAPACK version 3.12.0
#> 
#> locale:
#> [1] LC_COLLATE=C                          
#> [2] LC_CTYPE=English_United States.utf8   
#> [3] LC_MONETARY=English_United States.utf8
#> [4] LC_NUMERIC=C                          
#> [5] LC_TIME=English_United States.utf8    
#> 
#> time zone: America/Chicago
#> tzcode source: internal
#> 
#> attached base packages:
#> [1] stats     graphics  grDevices utils     datasets  methods   base     
#> 
#> other attached packages:
#> [1] OnlineSurr_0.0.3
#> 
#> loaded via a namespace (and not attached):
#>  [1] gtable_0.3.6        jsonlite_2.0.0      dplyr_1.1.4        
#>  [4] compiler_4.5.1      Rcpp_1.1.0          tidyselect_1.2.1   
#>  [7] tidyr_1.3.2         jquerylib_0.1.4     scales_1.4.0       
#> [10] yaml_2.3.12         fastmap_1.2.0       ggplot2_4.0.2      
#> [13] R6_2.6.1            labeling_0.4.3      generics_0.1.4     
#> [16] knitr_1.51          rbibutils_2.4.1     tibble_3.3.1       
#> [19] zigg_0.0.2          bslib_0.10.0        pillar_1.11.1      
#> [22] RColorBrewer_1.1-3  rlang_1.1.7         cachem_1.1.0       
#> [25] xfun_0.52           sass_0.4.10         S7_0.2.1           
#> [28] RcppParallel_5.1.10 otel_0.2.0          kDGLM_1.2.14       
#> [31] cli_3.6.5           withr_3.0.2         extraDistr_1.10.0  
#> [34] magrittr_2.0.3      Rdpack_2.6.5        digest_0.6.37      
#> [37] grid_4.5.1          rstudioapi_0.18.0   latex2exp_0.9.8    
#> [40] lifecycle_1.0.5     vctrs_0.6.5         Rfast_2.1.5.1      
#> [43] evaluate_1.0.5      glue_1.8.0          farver_2.1.2       
#> [46] purrr_1.0.4         rmarkdown_2.31      tools_4.5.1        
#> [49] pkgconfig_2.0.3     htmltools_0.5.8.1

Santos Jr., S. V. dos, and Parast, L. (2026). A causal framework for evaluating jointly longitudinal outcomes and surrogate markers: A state-space approach.

OnlineSurr: Fitting marginal/conditional models and computing LPTE/CPTE