Estimating Total Effects with the Parametric G-formula

Introduction

gformula() estimates the mean outcome, or risk, had everyone followed a given exposure strategy, and so the total effect of one strategy against another, with the parametric g-formula (Westreich et al. 2012; McGrath et al. 2020): it fits a model for every time-varying variable, then simulates each subject forward in time under each intervention.

This vignette covers binary and survival outcomes, dynamic interventions, custom covariate distributions, a published replication (the GvHD analysis of Keil et al. 2014), bootstrap intervals, and working with the results. Data format, spec_model() and the recodes() hooks are introduced in vignette("causalMed-01-overview"); direct and indirect effects are in vignette("causalMed-02-mediation").

library(causalMed)
library(data.table)

Example 1: binary end-of-follow-up outcome

nonsurvivaldata follows 3,000 subjects over five time points. Within each period the exposure comes first and the confounders respond to it (?nonsurvivaldata), so the exposure model is listed first and the confounder models include the current A. The list order should always match the assumed data-generating process.

data("nonsurvivaldata")

# Lag bookkeeping (see the overview vignette for the recode hooks)
init_rc <- recodes(lag1_A = 0, lag1_L1 = 0, lag1_L2 = 0)
in_rc   <- recodes(lag1_A = A, lag1_L1 = L1, lag1_L2 = L2)

# ── 1. Models in temporal order: A → L1 → L2 → Y ──────────────────────────
m_A  <- spec_model(A     ~ V + lag1_A + lag1_L1 + lag1_L2 + time,
                   var_type = "binary",  mod_type = "exposure")
m_L1 <- spec_model(L1    ~ V + A + lag1_L1 + time,
                   var_type = "normal",  mod_type = "covariate")
m_L2 <- spec_model(L2    ~ V + A + lag1_L2 + time,
                   var_type = "binary",  mod_type = "covariate")
m_Y  <- spec_model(Y_bin ~ V + A + L1 + L2,
                   var_type = "binary",  mod_type = "outcome")

models_bin <- list(m_A, m_L1, m_L2, m_Y)

# ── 2. Intervention strategies ─────────────────────────────────────────────
# NULL  = natural course (draw exposure from its fitted model)
# 1 / 0 = always treat / never treat
ints <- list(natural = NULL, always_treat = 1, never_treat = 0)

# ── 3. Run the g-formula ───────────────────────────────────────────────────
fit_bin <- gformula(
  data        = nonsurvivaldata,
  id_var      = "id",
  time_var    = "time",
  base_vars   = "V",
  exposure    = "A",
  models      = models_bin,
  intervention = ints,
  ref_int     = "natural",
  init_recode = init_rc,
  in_recode   = in_rc,
  mc_sample   = 10000,
  R           = 1,        # set R > 1 for bootstrap CIs (see below)
  quiet       = TRUE,
  seed        = 2025
)

# Risk (mean outcome) under each strategy
fit_bin$effect_size
#>    Intervention       Est
#>          <fctr>     <num>
#> 1:      natural 0.2349247
#> 2: always_treat 0.2522388
#> 3:  never_treat 0.1066566

# Contrasts vs the reference (natural course)
fit_bin$estimate
#>              Intervention  Risk_type    Estimate
#>                    <char>     <char>       <num>
#> 1: always_treat - natural Difference  0.01731407
#> 2: always_treat / natural      Ratio  1.07370051
#> 3:  never_treat - natural Difference -0.12826810
#> 4:  never_treat / natural      Ratio  0.45400334

effect_size gives the mean outcome under each intervention; estimate gives the risk difference and risk ratio against the reference.

Example 2: survival (time-to-event) outcome

For a survival outcome, a mod_type = "survival" model gives the discrete-time hazard, which is accumulated into the cumulative incidence \(1 - \prod_t (1 - h_t)\): the reported quantities are risks of the event by the end of follow-up. The data need one row per subject per period at risk, with no rows after the event (?survivaldata).

data("survivaldata")

m_A2 <- spec_model(A ~ V + lag1_A + lag1_L + time,
                   var_type = "binary", mod_type = "exposure")
m_L  <- spec_model(L ~ V + A + lag1_L + time,
                   var_type = "normal", mod_type = "covariate")
m_Y2 <- spec_model(Y ~ V + A + L + time,
                   var_type = "binary", mod_type = "survival")  # <-- survival

models_surv <- list(m_A2, m_L, m_Y2)

fit_surv <- gformula(
  data        = survivaldata,
  id_var      = "id",
  base_vars   = "V",
  exposure    = "A",
  time_var    = "time",
  models      = models_surv,
  intervention = list(natural = NULL, never = 0, always = 1),
  ref_int     = "natural",
  init_recode = recodes(lag1_L = 0, lag1_A = 0),
  in_recode   = recodes(lag1_L = L, lag1_A = A),
  mc_sample   = 10000,
  R           = 1,
  quiet       = TRUE,
  seed        = 2025
)

fit_surv$effect_size   # Cumulative incidence by strategy
#>    Intervention       Est
#>          <fctr>     <num>
#> 1:      natural 0.6453720
#> 2:        never 0.4012772
#> 3:       always 0.7868445
fit_surv$estimate      # Risk contrasts
#>        Intervention  Risk_type   Estimate
#>              <char>     <char>      <num>
#> 1:  never - natural Difference -0.2440949
#> 2:  never / natural      Ratio  0.6217765
#> 3: always - natural Difference  0.1414724
#> 4: always / natural      Ratio  1.2192106

Dynamic (threshold) interventions

A dynamic intervention sets the exposure by a rule applied to each subject’s simulated values, written with dyn_int() and evaluated in the simulated data at every time step.

The rule runs at the exposure model’s position in the list. Variables listed earlier (and the exposure’s own natural-course draw) hold current-period values; variables listed later, here L1, still hold the previous period’s values. A rule for an exposure decided at the start of each period therefore uses the previous period’s covariates:

fit_dyn <- gformula(
  data        = nonsurvivaldata,
  id_var      = "id",
  time_var    = "time",
  base_vars   = "V",
  exposure    = "A",
  models      = models_bin,
  intervention = list(
    natural = NULL,
    treat_if_prev_L1_pos = dyn_int(as.numeric(lag1_L1 > 0))
  ),
  ref_int     = "natural",
  init_recode = init_rc,
  in_recode   = in_rc,
  mc_sample   = 10000,
  R           = 1,
  quiet       = TRUE,
  seed        = 2025
)

fit_dyn$effect_size
#>            Intervention       Est
#>                  <fctr>     <num>
#> 1:              natural 0.2349247
#> 2: treat_if_prev_L1_pos 0.2223531
fit_dyn$estimate
#>                      Intervention  Risk_type    Estimate
#>                            <char>     <char>       <num>
#> 1: treat_if_prev_L1_pos - natural Difference -0.01257164
#> 2: treat_if_prev_L1_pos / natural      Ratio  0.94648651

Any expression over the columns in scope is allowed, e.g. dyn_int(as.numeric(A > 0 & lag1_L1 > median(lag1_L1))), where A is the natural-course draw that the rule overrides.

Custom covariate distributions

var_type = "custom" covers distributions without a built-in type, such as a zero-inflated normal (a point mass at zero, Gaussian otherwise), here as a two-part model. custom_fit may return any object that custom_sim knows how to use, here a list of two fits:

# Part 1: is the value positive?  Part 2: its level, among the positives.
zin_fit <- function(formula, data, ...) {
  y <- model.response(model.frame(formula, data = data))
  fit_any <- glm(update(formula, I(. > 0) ~ .), family = binomial(), data = data)
  fit_pos <- lm(formula, data = data[y > 0, ])
  list(any = fit_any, pos = fit_pos)
}

zin_sim <- function(model, newdt, ...) {
  p_any  <- predict(model$any, newdata = newdt, type = "response")
  m_pos  <- predict(model$pos, newdata = newdt)
  is_pos <- rbinom(nrow(newdt), 1, p_any)
  pos    <- pmax(rnorm(nrow(newdt), m_pos, sigma(model$pos)), 0)
  is_pos * pos
}

m_zin <- spec_model(
  X ~ A + L + time,
  var_type   = "custom",
  mod_type   = "covariate",
  custom_fit = zin_fit,
  custom_sim = zin_sim,
  truncate   = FALSE   # zin_sim controls its own range
)

custom_fit(formula, data, ...) is called once, when the models are fitted; custom_sim(model, newdata) is called at every time step and returns one value per row. custom_fit must be defined at the top level (or be a namespace-qualified function) so that it can be found when the models are fitted. Choosing and checking the distribution is the analyst’s responsibility.

A published example: preventing GvHD (Keil et al. 2014)

The package ships gvhd, the person-day bone marrow transplant data of the parametric g-formula illustration of Keil et al. (2014). The code below reproduces that analysis: the risk of death by day 1825 had graft-versus-host disease (GvHD) never occurred, against the natural course. It combines several features:

• five models: relapse → platelet recovery → GvHD (exposure) → censoring → death (hazard). Here the covariates are measured before the day’s exposure, so they come first, as in the paper;
• absorbing states: each state is modelled only among those not yet in it (subset =), and out_recode keeps it at 1 afterwards;
• a censoring model;
• restricted cubic splines of age and day, and day counters, built with the three recode hooks;
• daily data: 137 subjects over 1,825 days.

The models follow Appendix 2 of the paper (?gvhd). First the time-fixed transforms: restricted cubic splines of age (knots 17, 25.4, 30, 41.4) and of day (knots 83.6, 401.4, 947, 1862.2), and their squares:

data("gvhd")

gvhd <- within(gvhd, {
  agesq    <- age^2
  agecurs1 <- (age > 17.0) * (age - 17.0)^3 -
              ((age > 30.0) * (age - 30.0)^3) * (41.4 - 17.0) / (41.4 - 30.0)
  agecurs2 <- (age > 25.4) * (age - 25.4)^3 -
              ((age > 41.4) * (age - 41.4)^3) * (41.4 - 25.4) / (41.4 - 30.0)
  daysq    <- day^2
  daycurs1 <- (day > 83.6)   * ((day - 83.6)   / 83.6)^3 +
              (day > 1862.2) * ((day - 1862.2) / 83.6)^3 * (947.0 - 83.6) -
              (day > 947.0)  * ((day - 947.0)  / 83.6)^3 * (1862.2 - 83.6) / (1862.2 - 947.0)
  daycurs2 <- (day > 401.4)  * ((day - 401.4)  / 83.6)^3 +
              (day > 1862.2) * ((day - 1862.2) / 83.6)^3 * (947.0 - 401.4) -
              (day > 947.0)  * ((day - 947.0)  / 83.6)^3 * (1862.2 - 401.4) / (1862.2 - 947.0)
})

The five models. Each state’s model applies only among those not yet in it (subset = ...m1 == 0), and the death hazard interacts the day spline with gvhd:

models_gvhd <- list(
  spec_model(relapse ~ all + cmv + male + age + gvhdm1 + daysgvhd + platnormm1 +
               daysnoplatnorm + agecurs1 + agecurs2 + day + daysq + wait,
             var_type = "binary", mod_type = "covariate", subset = relapsem1 == 0),
  spec_model(platnorm ~ all + cmv + male + age + agecurs1 + agecurs2 + gvhdm1 +
               daysgvhd + daysnorelapse + wait,
             var_type = "binary", mod_type = "covariate", subset = platnormm1 == 0),
  spec_model(gvhd ~ all + cmv + male + age + platnormm1 + daysnoplatnorm +
               relapsem1 + daysnorelapse + agecurs1 + agecurs2 + day + daysq + wait,
             var_type = "binary", mod_type = "exposure", subset = gvhdm1 == 0),
  spec_model(censlost ~ all + cmv + male + age + daysgvhd + daysnoplatnorm +
               daysnorelapse + agesq + day + daycurs1 + daycurs2 + wait,
             var_type = "binary", mod_type = "censor"),
  spec_model(d ~ all + cmv + male + age + gvhd + platnorm + daysnoplatnorm +
               relapse + daysnorelapse + agesq + wait +
               day * gvhd + daycurs1 * gvhd + daycurs2 * gvhd,
             var_type = "binary", mod_type = "survival")
)

The recode hooks: init_recode sets day 1 (states and counters at 0, functions of day computed), in_recode updates the functions of day and the one-day lags at the start of each later day, and out_recode advances the counters and carries the absorbing states forward at the end of each day:

init_recode <- recodes(
  daysq    = day^2,
  daycurs1 = (day > 83.6)   * ((day - 83.6)   / 83.6)^3 +
             (day > 1862.2) * ((day - 1862.2) / 83.6)^3 * (947.0 - 83.6) -
             (day > 947.0)  * ((day - 947.0)  / 83.6)^3 * (1862.2 - 83.6) / (1862.2 - 947.0),
  daycurs2 = (day > 401.4)  * ((day - 401.4)  / 83.6)^3 +
             (day > 1862.2) * ((day - 1862.2) / 83.6)^3 * (947.0 - 401.4) -
             (day > 947.0)  * ((day - 947.0)  / 83.6)^3 * (1862.2 - 401.4) / (1862.2 - 947.0),
  relapse = 0, gvhd = 0, platnorm = 0, gvhdm1 = 0, relapsem1 = 0, platnormm1 = 0,
  daysnorelapse = 0, daysnoplatnorm = 0, daysnogvhd = 0,
  daysrelapse = 0, daysplatnorm = 0, daysgvhd = 0)

in_recode <- recodes(
  daysq    = day^2,
  daycurs1 = (day > 83.6)   * ((day - 83.6)   / 83.6)^3 +
             (day > 1862.2) * ((day - 1862.2) / 83.6)^3 * (947.0 - 83.6) -
             (day > 947.0)  * ((day - 947.0)  / 83.6)^3 * (1862.2 - 83.6) / (1862.2 - 947.0),
  daycurs2 = (day > 401.4)  * ((day - 401.4)  / 83.6)^3 +
             (day > 1862.2) * ((day - 1862.2) / 83.6)^3 * (947.0 - 401.4) -
             (day > 947.0)  * ((day - 947.0)  / 83.6)^3 * (1862.2 - 401.4) / (1862.2 - 947.0),
  platnormm1 = platnorm, relapsem1 = relapse, gvhdm1 = gvhd)

out_recode <- recodes(
  daysnorelapse  = ifelse(relapse == 0,  daysnorelapse + 1,  daysnorelapse),
  daysrelapse    = ifelse(relapse == 1,  daysrelapse + 1,    daysrelapse),
  daysnoplatnorm = ifelse(platnorm == 0, daysnoplatnorm + 1, daysnoplatnorm),
  daysplatnorm   = ifelse(platnorm == 1, daysplatnorm + 1,   daysplatnorm),
  daysnogvhd     = ifelse(gvhd == 0,     daysnogvhd + 1,     daysnogvhd),
  daysgvhd       = ifelse(gvhd == 1,     daysgvhd + 1,       daysgvhd),
  # absorbing carry-forward: once a state was 1 yesterday, keep it at 1
  platnorm = ifelse(platnormm1 == 1, 1, platnorm),
  relapse  = ifelse(relapsem1 == 1,  1, relapse),
  gvhd     = ifelse(gvhdm1 == 1,     1, gvhd))

fit_gvhd <- gformula(gvhd,
  id_var    = "id", time_var = "day", exposure = "gvhd",
  base_vars = c("age", "agesq", "agecurs1", "agecurs2", "male", "cmv", "all", "wait"),
  models    = models_gvhd,
  intervention = list(never = 0),     # a natural-course reference is added automatically
  init_recode = init_recode, in_recode = in_recode, out_recode = out_recode,
  mc_sample = 20000, R = 1, quiet = TRUE, seed = 20260703)

fit_gvhd$effect_size
#>    Intervention       Est
#>          <fctr>     <num>
#> 1:      natural 0.6093279
#> 2:        never 0.5881213
fit_gvhd$estimate
#>       Intervention  Risk_type    Estimate
#>             <char>     <char>       <num>
#> 1: never - natural Difference -0.02120662
#> 2: never / natural      Ratio  0.96519671

These are the simulated 5-year risks of death under the natural course and under “never GvHD”, and their contrast, following the specification of Keil et al. (2014); see that paper for the interpretation and the assumptions. With 1,825 daily time steps the simulation takes a few minutes, and bootstrap intervals (R > 1) take much longer.

Bootstrap Confidence Intervals

Set R > 1 for percentile and normal-approximation confidence intervals. The bootstrap resamples whole subjects, keeping each subject’s time points together.

fit_boot <- gformula(
  data        = nonsurvivaldata,
  id_var      = "id",
  time_var    = "time",
  base_vars   = "V",
  exposure    = "A",
  models      = models_bin,
  intervention = list(natural = NULL, always = 1),
  ref_int     = "natural",
  init_recode = init_rc,
  in_recode   = in_rc,
  mc_sample   = 10000,
  R           = 200,       # 200 bootstrap replicates
  quiet       = TRUE,
  seed        = 2025
)

# effect_size now includes Sd, perct_lcl/ucl, norm_lcl/ucl
fit_boot$effect_size
#>    Intervention       Est          Sd perct_lcl perct_ucl  norm_lcl  norm_ucl
#>          <fctr>     <num>       <num>     <num>     <num>     <num>     <num>
#> 1:      natural 0.2349247 0.007879589 0.2182889 0.2468377 0.2194810 0.2503684
#> 2:       always 0.2522388 0.008566108 0.2358453 0.2669507 0.2354495 0.2690280
fit_boot$estimate
#>        Intervention  Risk_type   Estimate          Sd  perct_lcl perct_ucl
#>              <char>     <char>      <num>       <num>      <num>     <num>
#> 1: always - natural Difference 0.01731407 0.002329456 0.01294381 0.0224198
#> 2: always / natural      Ratio 1.07370051 0.009868665 1.05635827 1.0949960
#>      norm_lcl   norm_ucl
#>         <num>      <num>
#> 1: 0.01274842 0.02187972
#> 2: 1.05435829 1.09304274

# the individual per-replicate draws are retained in boot_estimates
# ($interventions and $contrasts), for custom intervals or diagnostics
head(fit_boot$boot_estimates$interventions)
#>    replicate Intervention       Est
#>        <int>       <fctr>     <num>
#> 1:         1      natural 0.2373405
#> 2:         1       always 0.2543564
#> 3:         2      natural 0.2299337
#> 4:         2       always 0.2511462
#> 5:         3      natural 0.2310667
#> 6:         3       always 0.2498870

To run the replicates in parallel, set a plan first, e.g. future::plan(future::multisession).

Working with Results

fit_full <- gformula(
  data        = nonsurvivaldata,
  id_var      = "id",
  time_var    = "time",
  base_vars   = "V",
  exposure    = "A",
  models      = models_bin,
  intervention = list(natural = NULL, always = 1),
  init_recode = init_rc,
  in_recode   = in_rc,
  mc_sample   = 5000,
  R           = 1,
  return_fitted = TRUE,
  quiet       = TRUE,
  seed        = 2025
)

# Names correspond to the response variable of each model
names(fit_full$fitted_models)
#> [1] "A"     "L1"    "L2"    "Y_bin"

# Access a specific model's coefficients
coef(fit_full$fitted_models$A)
#> (Intercept)           V      lag1_A     lag1_L1     lag1_L2        time 
#>  1.29379895  0.53935295  0.17102958  0.20813430  0.25822156  0.01266473

fit_data <- gformula(..., return_data = TRUE)

# One row per MC subject per intervention, at the last time point
head(fit_data$sim_data)

References

• Westreich, D., Cole, S. R., Young, J. G., et al. (2012). The parametric g-formula to estimate the effect of highly active antiretroviral therapy on incident AIDS or death. Statistics in Medicine, 31, 2000–2009.
• Keil, A. P., Edwards, J. K., Richardson, D. B., Naimi, A. I., & Cole, S. R. (2014). The parametric g-formula for time-to-event data: intuition and a worked example. Epidemiology, 25(6), 889–897.
• McGrath, S., Lin, V., Zhang, Z., et al. (2020). gfoRmula: An R package for estimating the effects of sustained treatment strategies via the parametric g-formula. Patterns, 1, 100008.
• Robins, J. M. (1986). A new approach to causal inference in mortality studies with a sustained exposure period. Mathematical Modelling, 7(9–12), 1393–1512.