. 2024 Oct 17;43(28):5380–5400. doi: 10.1002/sim.10246

TABLE 1.

Algorithm for estimating Estimand 2 at time $t$ .

Step 0a	Define a set of interventions ${\overline{A}}_{t}$ , with $∣ {\overline{A}}_{t} ∣ = n_{a}$ and ${\overline{A}}_{t}^{(j)}$ is the $j^{th}$ element of ${\overline{A}}_{t}$ , $j = 1, \dots, n_{a}$ .
Step 0b	Set ${\tilde{Y}}_{t} = Y_{t}$ .
For $s = t, \dots, 1$
Step 1a	Estimate the conditional density $g (A_{s} \| H_{s})$ , see also Footnote 1.
Step 1b	Estimate the conditional density $g (A_{s} \| A_{s - 1}, H_{s - 1})$ , see also Footnote 1.
Step 2	Set $a_{s} = a_{s}^{(1)}$ , where $a_{s}^{(1)}$ is the $s^{th}$ element of intervention ${\overline{A}}_{t}^{(1)} \in {\overline{A}}_{t}$ .
Step 3a	Plug in $a_{s}^{(1)}$ into the estimated densities from step 1, to calculate $\hat{g} (a_{s}^{(1)} \| h_{s})$ and $\hat{g} (a_{s}^{(1)} \| a_{s - 1}^{(1)}, h_{s - 1})$ .
Step 3b	Calculate the weights $w_{s} (a_{s}^{(1)}, c)$ from (15) based on the estimates from 3a.
	If $\hat{g} (a_{s}^{(1)} \| a_{s - 1}^{(1)}, h_{s - 1}) \leq c$ , estimate $g (a_{s}^{(1)} \| a_{s^{}}^{(1)}, h_{s^{}})$ as required by the definition of (15) for $s^{*} = s - 2, \dots, 0$ .
Step 4	Estimate $E (w_{s} (a_{s}^{(1)} c) {\tilde{Y}}_{s} \| A_{s}, H_{s})$ , see also Footnote 2.
Step 5	Predict ${\tilde{Y}}_{s - 1} = \hat{E} ({\hat{w}}_{s} (a_{s}^{(1)} c) {\tilde{Y}}_{s} \| {\overline{A}}_{s} = {\overline{a}}_{s}^{(1)}, l_{s})$ based on the fitted model from step 4 and the given intervention ${\overline{a}}_{s}^{(1)}$ .
For $t = 0$
Step 1a	Estimate the conditional density $g (A_{0} \| L_{0})$ , see also Footnote 1.
Step 1b	Estimate the conditional density $g (A_{0})$ , see also Footnote 1.
Step 2	Set $a_{0} = a_{0}^{(1)}$ .
Step 3a	Calculate $\hat{g} (a_{0}^{(1)} \| l_{0})$ and $\hat{g} (a_{0}^{(1)})$ .
Step 3b	Calculate the weights $w_{0} (a_{0}^{(1)}, c)$ . If $\hat{g} (a_{0}^{(1)}) \leq c$ , then $m_{w, t} ({\overline{a}}_{t}^{(1)}, c)$ is undefined.
Step 4	Estimate $E ({\tilde{Y}}_{0} \| A_{0}, L_{0})$ .
Step 5	Calculate ${\hat{m}}_{w, t} ({\overline{a}}_{t}^{(1)}, c) = {\hat{E}}_{w (c)} (Y_{t}^{{\overline{a}}_{t}^{(1)}}) = {(\sum_{i = 1}^{n} w_{0, i})}^{- 1} ({\hat{w}}_{0} {(a_{0}^{(1)}, c)}^{T} {\tilde{Y}}_{- 1})$ ; that is, obtain the estimate of Estimand 2 at ${\overline{a}}_{t}^{(1)}$ through calculating the weighted mean of the iterated outcome under the respective intervention $a_{0}^{(1)}$ at t=0.
Then
Step 6	Repeat steps 2–5 for the other interventions ${\overline{A}}_{t}^{(j)}$ , $j = 2, \dots, n_{a}$ . This yields an estimate of estimand 2 at $t$ .
Step 7	Repeat steps 1–6 on $B$ bootstrap samples to obtain confidence intervals.

Note: ¹The conditional treatment densities can be estimated with (i) parametric models, if appropriate, like the linear model, (ii) nonparametric flexible estimators, like highly‐adaptive LASSO density estimation [32], (iii) a “binning strategy” where a logistic regression model models the probability of approximately observing the intervention of interest at time $t$ , given one has followed the strategy so far and given the covariates, (iv) other options, like transformation models or generalized additive models of location, shape and scale [33, 34]. Items (i)–(iii) are implemented in our package mentioned below. ²The iterated weighted outcome regressions are recommended to be estimated data‐adaptively, because the weighted outcomes are often non‐symmetric. We recommend super learning for it [4], and this is what is implemented in the package mentioned below.