Natural-history modified treatment policies (grace periods)

Code

library(causatr)
library(tinytable)
library(tinyplot)

Most interventions assign treatment as a function of covariates — static(1) ignores everything, dynamic(rule) reads the covariate history, shift(δ) nudges the observed value. Natural-history modified treatment policies (G-LMTPs; Díaz, Williams, Morzywołek & Rudolph 2026) are different: the treatment assigned at time t depends on the natural value of treatment — the treatment a patient would take absent the intervention — at t and at earlier periods.

The motivating example is a grace period or delay: “postpone treatment initiation by one period relative to when the patient would naturally start.” To even define that intervention you need the patient’s natural initiation time, which is a counterfactual quantity.

This vignette explains what these policies are, why they cannot be written as dynamic() rules, how causatr estimates them with an augmented-data sequential regression, and how the estimator relates to LMTPs and to natural-value IPW.

The natural value of treatment

Write $A_t$ for treatment at period t and $L_t$ for time-varying covariates. Under a longitudinal regime d, the natural value $A_t(\bar A^d_{t-1})$ is the treatment that would be observed at t if the intervention were stopped just before t, given the regime was followed through t−1. It is the formal version of “what the patient would do on their own.”

A standard LMTP (Díaz et al. 2023) assigns $A^d_t = d_t(A_t, H_t)$ — it may use the contemporaneous natural value $A_t$ and the covariate history $H_t$. shift(), scale_by(), threshold() are of this kind.
A natural-history MTP (G-LMTP) assigns $A^d_t = d_t(\bar A_t, H_t)$ — it may use the entire natural-value history $\bar A_t = (A_1, A_2(A^d_1), \dots, A_t(\bar A^d_{t-1}))$. Grace periods, delays, and dose-escalation caps need this.

What the policies assign

The cleanest way to see the difference is to follow one patient whose natural (untreated-course) treatment path initiates at period 3 and stays on (absorbing): $\bar A = (0, 0, 1, 1, 1)$.

Code

nat <- c(0, 0, 1, 1, 1) # natural / observed treatment path
traj <- rbind(
  data.frame(t = 1:5, A = nat, policy = "natural (observed)"),
  data.frame(t = 1:5, A = rep(1, 5), policy = "static(1)"),
  data.frame(t = 1:5, A = c(0, 0, 0, 1, 1), policy = "grace_period(1)"),
  data.frame(t = 1:5, A = rep(nat[1], 5), policy = "carry_forward()")
)
with(traj, tinyplot(
  A ~ t | policy,
  type = "b", pch = 19,
  ylab = "treatment", xlab = "period",
  main = "Assigned treatment under each policy"
))

Figure 1: Treatment assigned to one patient (natural initiation at t = 3) under four policies. `static(1)` ignores the natural value; `grace_period(1)` delays the natural initiation by one period; `carry_forward()` freezes everyone at their baseline level (here 0).

static(1) and carry_forward() are flat — they do not read the natural value. grace_period(1) tracks the natural path but shifted one period later: that delay is the estimand. Crucially, the delay is defined relative to the patient’s own (counterfactual) initiation time, which is exactly what makes it a natural-history policy.

Why this is not a `dynamic()` rule

It is tempting to write the delay as dynamic(\(data, trt) data$lag1_A) — “set this period’s treatment to last period’s.” This is wrong, and it fails silently. causatr’s ICE recursion conditions on the observed lagged treatment (lag1_A), but under the delay the natural value at t differs from the observed value the moment the policy has perturbed the trajectory (treatment-state feedback). The rule runs without error and returns a number — just the wrong causal estimand.

Here is the gap on a simulated absorbing-treatment process with covariate feedback (so the natural history under the regime genuinely diverges from the observed history):

Code

set.seed(1)
n <- 2500L
tau <- 4L
L0 <- rnorm(n)
A <- matrix(0L, n, tau)
L <- matrix(0, n, tau)
Lprev <- L0
Aprev <- integer(n)
for (t in seq_len(tau)) {
  Lt <- if (t == 1L) 0.5 * L0 + rnorm(n) else 0.5 * Lprev + 0.8 * Aprev + rnorm(n)
  At <- ifelse(Aprev == 1L, 1L, rbinom(n, 1L, plogis(-1 + 0.5 * Lt)))
  A[, t] <- At; L[, t] <- Lt; Lprev <- Lt; Aprev <- At
}
Y <- 1 + 0.8 * rowSums(A) + 0.4 * L[, tau] - 0.5 * L0 + rnorm(n)
d <- data.frame(
  id = rep(seq_len(n), each = tau), time = rep(seq_len(tau), n),
  L0 = rep(L0, each = tau), A = as.vector(t(A)), L = as.vector(t(L)),
  Y = NA_real_
)
d$Y[d$time == tau] <- Y

fit <- causat(
  d, outcome = "Y", treatment = "A",
  confounders = ~ L0, confounders_tv = ~ L,
  id = "id", time = "time", estimator = "gcomp", history = Inf
)
#> Warning: `model_fn` not specified; defaulting to `stats::glm`.
#> ℹ Set `model_fn` explicitly (e.g. `model_fn = stats::glm` or `model_fn = mgcv::gam`).

# Correct: natural-history engine.
mu_grace <- contrast(
  fit, list(delay1 = grace_period(1L)),
  ci_method = "bootstrap", n_boot = 100L
)$estimates$estimate[1]

# Wrong: a dynamic() rule that reads the OBSERVED lag.
naive <- dynamic(function(data, trt) {
  v <- data$lag1_A; v[is.na(v)] <- 0L; v
})
mu_naive <- contrast(
  fit, list(naive = naive), ci_method = "bootstrap", n_boot = 2L
)$estimates$estimate[1]

c(grace_period = mu_grace, naive_dynamic_lag = mu_naive)
#>      grace_period naive_dynamic_lag 
#>          2.281738          1.003515

On this data generating process the true one-period-delay mean is ≈ 2.30; the augmented engine recovers it to within ~1%, while the naive dynamic(lag1_A) rule is off by tens of percent. The tests/testthat/test-glmtp.R suite pins this against the forward-simulation truth and replicates the Díaz et al. (2026) Section-6 delay result.

How causatr estimates it: augmented data

The standard sequential-regression g-formula breaks for these policies: at a backward integration step it would need to set one past treatment to two values at once — the value it conditions on (in $P(a_{t+1}\mid a_t)$) and the value the policy feeds forward (in $d_{t+1}(\dots, a_t)$).

The fix (the paper’s augmented data) is to decouple the two. Each observation is replicated once per possible natural-treatment-history sequence $\bar s_t$, which is carried as a label through the backward recursion:

original row	natural-history label $\bar s_t$	response used
patient i	$(0, 0)$	$q_{t+1}((0,0), A_{t+1,i}, H_{t+1,i})$
patient i	$(0, 1)$	$q_{t+1}((0,1), A_{t+1,i}, H_{t+1,i})$
patient i	$(1, 0)$	$q_{t+1}((1,0), A_{t+1,i}, H_{t+1,i})$
patient i	$(1, 1)$	$q_{t+1}((1,1), A_{t+1,i}, H_{t+1,i})$

A separate outcome model is fit per label (so the conditioning value and the policy-input value never collide), then the policy is applied at prediction time. causatr’s parametric plug-in is $\sqrt n$-consistent under correct specification, with ID-cluster bootstrap inference. The treatment lag columns keep their observed values (they are the conditioning history); only the label carries the counterfactual natural history — that decoupling is the whole trick.

Worked example

grace_period(window) delays initiation; carry_forward() holds everyone at their baseline level. Both need a longitudinal g-computation fit with a discrete treatment, and use the ID-cluster bootstrap.

Code

res <- contrast(
  fit,
  interventions = list(
    delay1 = grace_period(1L),
    locf = carry_forward(),
    natural = NULL
  ),
  reference = "natural",
  ci_method = "bootstrap", n_boot = 200L
)
tt(res$estimates)

intervention	estimate	se	ci_lower	ci_upper
delay1	2.281738	0.03665249	2.209901	2.353576
locf	2.059251	0.04147649	1.977958	2.140543
natural	2.946063	0.03767625	2.872219	3.019907

Figure 2: Counterfactual means under a one-period delay and carry-forward, vs the natural course, with bootstrap 95% CIs.

Code


est <- res$estimates
with(est, tinyplot(
  x = factor(intervention, levels = intervention), y = estimate,
  ymin = ci_lower, ymax = ci_upper,
  type = "pointrange", ylab = expression(E(Y^d)), xlab = ""
))

Figure 3: Counterfactual means under a one-period delay and carry-forward, vs the natural course, with bootstrap 95% CIs.

Delaying initiation lowers the counterfactual mean relative to the natural course (fewer treated person-periods); carry-forward (hold at baseline) lowers it further.

Relationship to LMTPs and natural-value IPW

Natural-value interventions have a lineage. The table below places the causatr G-LMTP engine alongside the standard LMTP framework and the original IPW treatment.

	Young, Hernán & Robins (2014)	LMTP (Díaz et al. 2023)	G-LMTP (Díaz et al. 2026)
Policy reads	natural value (foundational)	contemporaneous natural value $A_t$ + $H_t$	natural-value history $\bar A_t$ + $H_t$
Estimation	IPW (density ratio $g^d/g$)	g-comp / IPW / doubly-robust TMLE/SDR	augmented-data sequential regression
In causatr	IPW engine (`shift`/`scale_by`/`ipsi`)	IPW engine	gcomp engine (`grace_period`/`carry_forward`)

Two practical points:

lmtp is not a valid oracle here. The lmtp package applies its shift once to the observed data, computing the standard LMTP — which differs from the natural-history target whenever the policy has perturbed the trajectory. The correct reference is forward Monte-Carlo simulation of the natural-history regime (the paper’s Proposition 1), which the test suite uses.
Why G-LMTP is gcomp-only in causatr. A Young-style IPW estimator for a history-dependent natural-value policy needs the ratio of the whole-trajectory natural-history density to the observed density; positivity is fragile and the variance explodes. The augmented g-computation route sidesteps this — which is why grace periods route through the gcomp engine, while the contemporaneous natural-value MTPs (shift/scale/ipsi) use the IPW density-ratio engine.

Scope and limits

Discrete treatment only. The augmentation enumerates the natural-history support, which is finite only for a discrete exposure. Continuous treatments are rejected (shift() / scale_by() are the continuous MTPs); the augmented engine handles binary and ordinal/count.
Longitudinal g-computation only. grace_period() / carry_forward() are rejected under IPW / AIPW / matching and for point treatments.
Bootstrap variance. The augmented-data sandwich is a planned future chunk; ci_method = "sandwich" aborts and points to the bootstrap.
Nonlinear ordered policies (e.g. a dose-escalation cap) are estimated consistently once the dose enters the per-step model flexibly — see Flexible dose models below.

Flexible dose models for ordered treatments

By default the treatment enters every per-step outcome model as a bare numeric main effect. For a binary treatment that is exactly saturated, but for an ordered / ordinal dose a single linear slope can misspecify a non-monotone or kinked counterfactual response — the canonical case being a dose-escalation cap, whose policy is piecewise-linear in the dose. The bare-numeric plug-in then carries a small but persistent asymptotic bias even though the augmented recursion is exactly correct.

treatment_form lets the dose enter the per-step model via a transformed term, while the policy still sets the numeric dose column — only the model’s design term changes:

fit <- causat(
  dose_data,
  outcome = "Y", treatment = "dose",
  confounders = ~ L0, confounders_tv = ~ L,
  id = "id", time = "t", estimator = "gcomp",
  treatment_form = ~ factor(dose)        # or ~ splines::ns(dose, df = 3)
)
contrast(fit, interventions = list(d = grace_period(1L)),
         ci_method = "bootstrap")

With ~ factor(dose) each dose level carries its own coefficient, so a kinked capped-dose response is recovered to sampling error rather than fit through one slope. Lag terms expand automatically (factor(dose) → factor(lag1_dose)). The flexible term is longitudinal-g-computation-only and benefits standard ICE regimes (static / shift / dynamic) as much as natural-history policies. Under factor(dose) an intervention must keep the dose within the observed levels (a new level makes prediction fail); splines::ns(dose, df) extrapolates smoothly.

References

Díaz I, Williams NT, Morzywołek P, Rudolph KE (2026). Modified treatment policies that depend on the natural history of treatment. arXiv:2605.24167.

Díaz I, Williams N, Hoffman KL, Schenck EJ (2023). Non-parametric causal effects based on longitudinal modified treatment policies. Journal of the American Statistical Association 118:846–857.

Young JG, Hernán MA, Robins JM (2014). Identification, estimation and approximation of risk under interventions that depend on the natural value of treatment using observational data. Epidemiologic Methods 3:1–19.

--- title: "Natural-history modified treatment policies (grace periods)" code-fold: show code-tools: true vignette: > %\VignetteIndexEntry{Natural-history modified treatment policies (grace periods)} %\VignetteEngine{quarto::html} %\VignetteEncoding{UTF-8} --- ```{r} #| include: false knitr::opts_chunk$set(collapse = TRUE, comment = "#>") ``` ```{r} #| message: false library(causatr) library(tinytable) library(tinyplot) ``` Most interventions assign treatment as a function of **covariates** — `static(1)` ignores everything, `dynamic(rule)` reads the covariate history, `shift(δ)` nudges the observed value. **Natural-history modified treatment policies** (G-LMTPs; Díaz, Williams, Morzywołek & Rudolph 2026) are different: the treatment assigned at time *t* depends on the **natural value of treatment** — the treatment a patient *would* take absent the intervention — at *t* **and at earlier periods**. The motivating example is a **grace period** or **delay**: "postpone treatment initiation by one period relative to when the patient would naturally start." To even define that intervention you need the patient's natural initiation time, which is a counterfactual quantity. This vignette explains what these policies are, why they cannot be written as `dynamic()` rules, how causatr estimates them with an augmented-data sequential regression, and how the estimator relates to LMTPs and to natural-value IPW. ## The natural value of treatment Write $A_t$ for treatment at period *t* and $L_t$ for time-varying covariates. Under a longitudinal regime *d*, the **natural value** $A_t(\bar A^d_{t-1})$ is the treatment that *would be observed* at *t* if the intervention were stopped just before *t*, given the regime was followed through *t−1*. It is the formal version of "what the patient would do on their own." - A **standard LMTP** (Díaz et al. 2023) assigns $A^d_t = d_t(A_t, H_t)$ — it may use the *contemporaneous* natural value $A_t$ and the covariate history $H_t$. `shift()`, `scale_by()`, `threshold()` are of this kind. - A **natural-history MTP (G-LMTP)** assigns $A^d_t = d_t(\bar A_t, H_t)$ — it may use the *entire natural-value history* $\bar A_t = (A_1, A_2(A^d_1), \dots, A_t(\bar A^d_{t-1}))$. Grace periods, delays, and dose-escalation caps need this. ## What the policies assign The cleanest way to see the difference is to follow one patient whose natural (untreated-course) treatment path initiates at period 3 and stays on (absorbing): $\bar A = (0, 0, 1, 1, 1)$. ```{r} #| label: fig-trajectories #| fig-cap: "Treatment assigned to one patient (natural initiation at t = 3) under four policies. `static(1)` ignores the natural value; `grace_period(1)` delays the natural initiation by one period; `carry_forward()` freezes everyone at their baseline level (here 0)." #| fig-width: 6.5 #| fig-height: 3 nat <- c(0, 0, 1, 1, 1) # natural / observed treatment path traj <- rbind( data.frame(t = 1:5, A = nat, policy = "natural (observed)"), data.frame(t = 1:5, A = rep(1, 5), policy = "static(1)"), data.frame(t = 1:5, A = c(0, 0, 0, 1, 1), policy = "grace_period(1)"), data.frame(t = 1:5, A = rep(nat[1], 5), policy = "carry_forward()") ) with(traj, tinyplot( A ~ t | policy, type = "b", pch = 19, ylab = "treatment", xlab = "period", main = "Assigned treatment under each policy" )) ``` `static(1)` and `carry_forward()` are flat — they do not read the natural value. `grace_period(1)` tracks the natural path but shifted one period later: that *delay* is the estimand. Crucially, the delay is defined relative to the patient's **own** (counterfactual) initiation time, which is exactly what makes it a natural-history policy. ## Why this is not a `dynamic()` rule It is tempting to write the delay as `dynamic(\(data, trt) data$lag1_A)` — "set this period's treatment to last period's." **This is wrong**, and it fails silently. causatr's ICE recursion conditions on the *observed* lagged treatment (`lag1_A`), but under the delay the natural value at *t* differs from the observed value the moment the policy has perturbed the trajectory (treatment-state feedback). The rule runs without error and returns a number — just the wrong causal estimand. Here is the gap on a simulated absorbing-treatment process with covariate feedback (so the natural history under the regime genuinely diverges from the observed history): ```{r} set.seed(1) n <- 2500L tau <- 4L L0 <- rnorm(n) A <- matrix(0L, n, tau) L <- matrix(0, n, tau) Lprev <- L0 Aprev <- integer(n) for (t in seq_len(tau)) { Lt <- if (t == 1L) 0.5 * L0 + rnorm(n) else 0.5 * Lprev + 0.8 * Aprev + rnorm(n) At <- ifelse(Aprev == 1L, 1L, rbinom(n, 1L, plogis(-1 + 0.5 * Lt))) A[, t] <- At; L[, t] <- Lt; Lprev <- Lt; Aprev <- At } Y <- 1 + 0.8 * rowSums(A) + 0.4 * L[, tau] - 0.5 * L0 + rnorm(n) d <- data.frame( id = rep(seq_len(n), each = tau), time = rep(seq_len(tau), n), L0 = rep(L0, each = tau), A = as.vector(t(A)), L = as.vector(t(L)), Y = NA_real_ ) d$Y[d$time == tau] <- Y fit <- causat( d, outcome = "Y", treatment = "A", confounders = ~ L0, confounders_tv = ~ L, id = "id", time = "time", estimator = "gcomp", history = Inf ) # Correct: natural-history engine. mu_grace <- contrast( fit, list(delay1 = grace_period(1L)), ci_method = "bootstrap", n_boot = 100L )$estimates$estimate[1] # Wrong: a dynamic() rule that reads the OBSERVED lag. naive <- dynamic(function(data, trt) { v <- data$lag1_A; v[is.na(v)] <- 0L; v }) mu_naive <- contrast( fit, list(naive = naive), ci_method = "bootstrap", n_boot = 2L )$estimates$estimate[1] c(grace_period = mu_grace, naive_dynamic_lag = mu_naive) ``` On this data generating process the true one-period-delay mean is ≈ 2.30; the augmented engine recovers it to within ~1%, while the naive `dynamic(lag1_A)` rule is off by tens of percent. The `tests/testthat/test-glmtp.R` suite pins this against the forward-simulation truth and replicates the Díaz et al. (2026) Section-6 delay result. ## How causatr estimates it: augmented data The standard sequential-regression g-formula breaks for these policies: at a backward integration step it would need to set one past treatment to **two** values at once — the value it conditions on (in $P(a_{t+1}\mid a_t)$) and the value the policy feeds forward (in $d_{t+1}(\dots, a_t)$). The fix (the paper's augmented data) is to **decouple** the two. Each observation is replicated once per possible natural-treatment-history sequence $\bar s_t$, which is carried as a *label* through the backward recursion: | original row | natural-history label $\bar s_t$ | response used | |---|---|---| | patient *i* | $(0, 0)$ | $q_{t+1}((0,0), A_{t+1,i}, H_{t+1,i})$ | | patient *i* | $(0, 1)$ | $q_{t+1}((0,1), A_{t+1,i}, H_{t+1,i})$ | | patient *i* | $(1, 0)$ | $q_{t+1}((1,0), A_{t+1,i}, H_{t+1,i})$ | | patient *i* | $(1, 1)$ | $q_{t+1}((1,1), A_{t+1,i}, H_{t+1,i})$ | A separate outcome model is fit per label (so the conditioning value and the policy-input value never collide), then the policy is applied at prediction time. causatr's parametric plug-in is $\sqrt n$-consistent under correct specification, with ID-cluster bootstrap inference. The treatment **lag columns keep their observed values** (they are the conditioning history); only the *label* carries the counterfactual natural history — that decoupling is the whole trick. ## Worked example `grace_period(window)` delays initiation; `carry_forward()` holds everyone at their baseline level. Both need a longitudinal g-computation fit with a discrete treatment, and use the ID-cluster bootstrap. ```{r} #| label: fig-estimates #| fig-cap: "Counterfactual means under a one-period delay and carry-forward, vs the natural course, with bootstrap 95% CIs." #| fig-width: 6 #| fig-height: 2.6 res <- contrast( fit, interventions = list( delay1 = grace_period(1L), locf = carry_forward(), natural = NULL ), reference = "natural", ci_method = "bootstrap", n_boot = 200L ) tt(res$estimates) est <- res$estimates with(est, tinyplot( x = factor(intervention, levels = intervention), y = estimate, ymin = ci_lower, ymax = ci_upper, type = "pointrange", ylab = expression(E(Y^d)), xlab = "" )) ``` Delaying initiation lowers the counterfactual mean relative to the natural course (fewer treated person-periods); carry-forward (hold at baseline) lowers it further. ## Relationship to LMTPs and natural-value IPW Natural-value interventions have a lineage. The table below places the causatr G-LMTP engine alongside the standard LMTP framework and the original IPW treatment. | | Young, Hernán & Robins (2014) | LMTP (Díaz et al. 2023) | G-LMTP (Díaz et al. 2026) | |---|---|---|---| | Policy reads | natural value (foundational) | **contemporaneous** natural value $A_t$ + $H_t$ | natural-value **history** $\bar A_t$ + $H_t$ | | Estimation | IPW (density ratio $g^d/g$) | g-comp / IPW / **doubly-robust** TMLE/SDR | **augmented-data** sequential regression | | In causatr | IPW engine (`shift`/`scale_by`/`ipsi`) | IPW engine | **gcomp** engine (`grace_period`/`carry_forward`) | Two practical points: - **`lmtp` is not a valid oracle here.** The `lmtp` package applies its shift once to the observed data, computing the *standard* LMTP — which differs from the natural-history target whenever the policy has perturbed the trajectory. The correct reference is forward Monte-Carlo simulation of the natural-history regime (the paper's Proposition 1), which the test suite uses. - **Why G-LMTP is gcomp-only in causatr.** A Young-style IPW estimator for a *history-dependent* natural-value policy needs the ratio of the whole-trajectory natural-history density to the observed density; positivity is fragile and the variance explodes. The augmented g-computation route sidesteps this — which is why grace periods route through the gcomp engine, while the contemporaneous natural-value MTPs (shift/scale/ipsi) use the IPW density-ratio engine. ## Scope and limits - **Discrete treatment only.** The augmentation enumerates the natural-history support, which is finite only for a discrete exposure. Continuous treatments are rejected (`shift()` / `scale_by()` are the continuous MTPs); the augmented engine handles binary **and** ordinal/count. - **Longitudinal g-computation only.** `grace_period()` / `carry_forward()` are rejected under IPW / AIPW / matching and for point treatments. - **Bootstrap variance.** The augmented-data sandwich is a planned future chunk; `ci_method = "sandwich"` aborts and points to the bootstrap. - **Nonlinear ordered policies** (e.g. a dose-escalation cap) are estimated consistently once the dose enters the per-step model flexibly — see *Flexible dose models* below. ## Flexible dose models for ordered treatments By default the treatment enters every per-step outcome model as a **bare numeric** main effect. For a binary treatment that is exactly saturated, but for an **ordered / ordinal dose** a single linear slope can misspecify a non-monotone or *kinked* counterfactual response — the canonical case being a *dose-escalation cap*, whose policy is piecewise-linear in the dose. The bare-numeric plug-in then carries a small but persistent asymptotic bias even though the augmented recursion is exactly correct. `treatment_form` lets the dose enter the per-step model via a transformed term, while the policy still sets the *numeric* dose column — only the model's design term changes: ```r fit <- causat( dose_data, outcome = "Y", treatment = "dose", confounders = ~ L0, confounders_tv = ~ L, id = "id", time = "t", estimator = "gcomp", treatment_form = ~ factor(dose) # or ~ splines::ns(dose, df = 3) ) contrast(fit, interventions = list(d = grace_period(1L)), ci_method = "bootstrap") ``` With `~ factor(dose)` each dose level carries its own coefficient, so a kinked capped-dose response is recovered to sampling error rather than fit through one slope. Lag terms expand automatically (`factor(dose)` → `factor(lag1_dose)`). The flexible term is longitudinal-g-computation-only and benefits standard ICE regimes (`static` / `shift` / `dynamic`) as much as natural-history policies. Under `factor(dose)` an intervention must keep the dose within the observed levels (a new level makes prediction fail); `splines::ns(dose, df)` extrapolates smoothly. ## References Díaz I, Williams NT, Morzywołek P, Rudolph KE (2026). Modified treatment policies that depend on the natural history of treatment. *arXiv:2605.24167*. Díaz I, Williams N, Hoffman KL, Schenck EJ (2023). Non-parametric causal effects based on longitudinal modified treatment policies. *Journal of the American Statistical Association* 118:846–857. Young JG, Hernán MA, Robins JM (2014). Identification, estimation and approximation of risk under interventions that depend on the natural value of treatment using observational data. *Epidemiologic Methods* 3:1–19.

original row	natural-history label \(\bar s_t\)	response used
patient i	\((0, 0)\)	\(q_{t+1}((0,0), A_{t+1,i}, H_{t+1,i})\)
patient i	\((0, 1)\)	\(q_{t+1}((0,1), A_{t+1,i}, H_{t+1,i})\)
patient i	\((1, 0)\)	\(q_{t+1}((1,0), A_{t+1,i}, H_{t+1,i})\)
patient i	\((1, 1)\)	\(q_{t+1}((1,1), A_{t+1,i}, H_{t+1,i})\)

The natural value of treatment

What the policies assign

Why this is not a dynamic() rule

How causatr estimates it: augmented data

Worked example

Relationship to LMTPs and natural-value IPW

Scope and limits

Flexible dose models for ordered treatments

References

Why this is not a `dynamic()` rule