TTE per-protocol methods: manuscript and supplemental text

This vignette provides drop-in text describing the target trial emulation (TTE) per-protocol methodology implemented in the swereg-TTE family of functions (TTEEnrollment, TTEPlan, and friends). It is provided in two flavours so that you can paste the appropriate level of detail into your own manuscript.

• Manuscript methods — prose-only, suitable for the main body of a journal article. No equations.
• Supplemental methods — detailed, with formulas and per-step model specifications, suitable for a supplement or methods appendix.

Both versions describe the same workflow:

1. Build a sequential-trials enrollment panel from the registry skeleton.
2. Adjust for baseline confounding of treatment with s2_ipw().
3. Censor at protocol deviation and compute stabilised inverse-probability-of- censoring weights with s4_prepare_for_analysis() (which internally calls s5_prepare_outcome() and s6_ipcw_pp()).
4. Fit a weighted Poisson marginal structural model for the incidence rate ratio with $irr(), using person-level cluster-robust standard errors.

The implementation follows the canonical sequential-trial-emulation literature (Hernán and Robins 2008; Danaei et al. 2013; Hernán and Robins 2016; Caniglia et al. 2023; Cashin et al. 2025) and produces results that agree with the reference R implementation TrialEmulation (Su et al. 2024) on simulated data with a known true effect.

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1. Manuscript methods

We applied target trial emulation, a framework for analysing observational data under explicit protocols that mirror a hypothetical randomized trial, to estimate the per-protocol effect of treatment on outcome in the Swedish national health registries (Hernán and Robins 2008, 2016). The framework requires that the analyst specify eligibility criteria, treatment strategies, the time of treatment assignment (“time zero”), follow-up rules, and the causal estimand before any data analysis, and that each element of this protocol be emulated using the observational data (Cashin et al. 2025).

Sequential trials design

Because eligible individuals can become exposed at many different calendar times in registry data, we did not define a single time zero. Instead we used the sequential-trials (nested-trials) design of Hernán et al. (2008), as formalised for pharmacoepidemiology by Danaei et al. (2013) and adapted to time-varying eligibility windows by Caniglia et al. (2023). At every eligible time unit a new target trial is opened. Each person who is eligible at that time enters the trial as either an initiator (the unit in which they first received the treatment of interest) or a non-initiator (one or more units in which they were eligible and untreated). Individuals can therefore contribute to several sequential trials before either initiating treatment or leaving the eligible pool. This design uses each person’s treatment-naive person-time efficiently and, by anchoring time zero on eligibility rather than on the eventual exposure, prevents the immortal time bias that arises when person- time before treatment initiation is misclassified as exposed (Hernán and Robins 2016; Caniglia et al. 2023).

Estimand: per-protocol effect

Our estimand is the per-protocol effect: the effect that would have been observed if every enrolled participant had remained on the treatment strategy assigned at baseline of the trial in which they enrolled (Hernán and Robins 2008; Danaei et al. 2013).

Censoring at protocol deviation and adjustment for confounding

In every emulated trial, follow-up was artificially censored when a participant’s current treatment status diverged from the strategy assigned at that trial’s baseline (Danaei et al. 2013). Two sources of bias then need to be addressed. First, baseline treatment assignment is not random, so we adjusted for baseline confounding by inverse probability of treatment weighting based on covariates measured at each trial’s time zero (Hernán and Robins 2008; Danaei et al. 2013). Second, artificial censoring at protocol deviation is informative whenever the same time-varying factors that predict continued adherence also predict the outcome. We adjusted for this using inverse probability of censoring weighting, with separate models fit by assigned treatment arm (Hernán and Robins 2008; Danaei et al. 2013; Su et al. 2024). The censoring-weight models had numerators conditional on baseline covariates only, which stabilises the weights and improves precision, and denominators conditional on the same baseline covariates plus the most recent values of the time-varying covariates that predict adherence (Hernán and Robins 2008; Danaei et al. 2013). The final analysis weight for each person-time observation is the product of the baseline treatment weight and the cumulative censoring weight up to that observation.

Outcome model and inference

We dropped rows corresponding to censoring events from the analysis dataset and fit a weighted Poisson regression of the binary event indicator on assigned baseline treatment, using log person-time as the offset, with the product of the baseline and cumulative censoring weights as the analysis weight. The exponent of the treatment coefficient estimates the marginal per-protocol incidence rate ratio under sustained treatment. Because each person can contribute repeated observations within and across emulated trials, we computed cluster-robust standard errors clustered on the person identifier (Hernán and Robins 2008; Danaei et al. 2013). This is the same general estimation strategy implemented in the TrialEmulation R package (Su et al. 2024); our implementation differs mainly in using a weighted Poisson rate model on the collapsed analysis data rather than a pooled logistic discrete-time hazard.

Software

All analyses were performed in R (R Core Team). Sequential-trial enrollment, baseline inverse probability of treatment weighting, censoring at protocol deviation, stabilised inverse probability of censoring weighting, weight truncation, and the final weighted Poisson incidence-rate-ratio model were implemented in the swereg package, which orchestrates the per-protocol target trial emulation pipeline described above. The censoring-weight denominator and numerator models were fit as generalised additive models using mgcv. The final weighted incidence-rate-ratio regression and its cluster-robust standard errors were computed by survey::svyglm() on a design clustered on the person identifier. All data manipulation was performed in data.table.

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2. Supplemental methods

Notation

Index individuals by $i = 1, \ldots, N$, sequential trials by their baseline time $m$, and follow-up time within a trial by $j = 0, 1, \ldots, K$. Let $A_{i,m,j}$ be the indicator of being on the protocol-defined treatment at trial-time $(m,j)$, with $A_{i,m,0}$ the assigned baseline treatment. Let $L_{i,m,j}$ be the vector of time-varying covariates measured at $(m,j)$, with $L_{i,m,0}$ their baseline values; let $V_i$ be time-fixed (e.g., sex, birth year) covariates. Let $Y_{i,m,j}$ be the outcome indicator at trial-time $(m,j)$ and $C_{i,m,j}$ the indicator of artificial censoring (the participant has deviated from the baseline-assigned strategy by trial-time $j$).

Sequential trials and the analysis panel

Following Hernán et al. (2008), Danaei et al. (2013), and Caniglia et al. (2023), we expand the registry skeleton into a panel of (person, trial, follow-up time) rows. Person $i$ contributes a row to trial $m$ whenever they are eligible at trial-time zero of trial $m$ and remain uncensored and event-free through trial-time $j$. As in Su et al. (2024), this expansion is what allows a single weighted regression to recover the marginal per-protocol contrast pooled across all sequential trials.

Estimand

The per-protocol estimand is the marginal contrast in the discrete-time hazard of $Y$ between the potential outcomes under sustained treatment ($a = 1$) and sustained non-treatment ($a = 0$):

$$\mathrm{Pr}(Y^{a=1}_{j} = 1 \mid Y^{a=1}_{j-1} = 0) \quad \text{versus} \quad \mathrm{Pr}(Y^{a=0}_{j} = 1 \mid Y^{a=0}_{j-1} = 0).$$

This is the observational analogue of the per-protocol effect of a trial in which everyone adhered to their baseline assignment throughout follow-up (Hernán and Robins 2008; Danaei et al. 2013).

Baseline IPW for non-random assignment

At each trial baseline $m$, treatment $A_{m,0}$ is not randomly assigned, so we fit a logistic model for the propensity of initiating the assigned strategy at $m$ conditional on $(V, L_{m,0})$:

$$\mathrm{logit}\,\mathrm{Pr}(A_{m,0} = 1 \mid V, L_{m,0}) = \gamma_0 + \gamma_1^\top V + \gamma_2^\top L_{m,0}.$$

The stabilised baseline weight (Hernán and Robins 2008) is

$$SW^A_{m,0} = \frac{\mathrm{Pr}(A_{m,0} = a)} {\mathrm{Pr}(A_{m,0} = a \mid V, L_{m,0})}.$$

These weights are stored on each (person, trial) baseline row by swereg::TTEEnrollment$s2_ipw(). By construction $SW^A_{m,0}$ is constant within a (person, trial) across follow-up times $j$.

Censoring at protocol deviation

For the per-protocol contrast, we artificially censor rows at the first $j$ at which $A_{m,j} \neq A_{m,0}$ (Danaei et al. 2013; Su et al. 2024). Define $C_{m,j}$ to be 1 at the row in which deviation first occurs, and 0 at all prior rows. The censoring-event row is itself removed from the final analysis dataset by swereg::TTEEnrollment$s4_prepare_for_analysis(); only rows with $C_{m,j} = 0$ enter the outcome model.

Stabilised IPCW for adherence

To unbias the per-protocol estimand against informative artificial censoring, each uncensored row is weighted by the inverse of its estimated probability of remaining uncensored to that visit, stabilised by the corresponding probability conditional only on baseline information (Hernán and Robins 2008; Danaei et al. 2013 Appendix; Su et al. 2024). The stabilised IPCW for individual $i$ at trial-time $(m, k)$ in arm $a$ is

$$SW^C_{m,k} = \prod_{j=0}^{k-1} \frac{\mathrm{Pr}(C_{m,j} = 0 \mid C_{m,j-1} = 0,\ Y_{m,j} = 0,\ A_{m,0} = a,\ V,\ L_{m,0})} {\mathrm{Pr}(C_{m,j} = 0 \mid C_{m,j-1} = 0,\ Y_{m,j} = 0,\ A_{m,j} = a,\ V,\ L_{m,j})}.$$

This is exactly equation (3) of Su et al. (2024) and matches the stabilised- weight construction in Hernán and Robins (2008) and the Appendix of Danaei et al. (2013, the $SW^A_{m+t}$ equation).

swereg::TTEEnrollment$s6_ipcw_pp() estimates the two probabilities by two separate pooled logistic regressions, fit by treatment arm:

Numerator (baseline covariates only) — the model whose role is to stabilise the weight, not to remove confounding (Hernán and Robins 2008; Danaei et al. 2013):

$$\mathrm{logit}\,\mathrm{Pr}(C_{m,j} = 0 \mid C_{m,j-1} = 0,\ Y_{m,j} = 0,\ A_{m,0} = a,\ V,\ L_{m,0}) = \theta_{0,a} + \theta_{1,a}^\top V + \theta_{2,a}^\top L_{m,0} + g_a(j).$$

Denominator (baseline plus time-varying covariates) — the model that does the causal work; correct specification is required for consistent estimation of the per-protocol effect (Hernán and Robins 2008; Su et al. 2024):

$$\mathrm{logit}\,\mathrm{Pr}(C_{m,j} = 0 \mid C_{m,j-1} = 0,\ Y_{m,j} = 0,\ A_{m,j} = a,\ V,\ L_{m,j}) = \alpha_{0,a} + \alpha_{1,a}^\top V + \alpha_{2,a}^\top L_{m,j} + h_a(j).$$

Here $g_a(\cdot)$ and $h_a(\cdot)$ are smooth functions of follow-up time (we use a quadratic in $j$, following Danaei et al. 2013; swereg::TTEEnrollment$s6_ipcw_pp() also supports a generalized additive model on the time index via estimate_ipcw_pp_with_gam = TRUE). Both models are fit separately for $a = 0$ and $a = 1$ so that adherence dynamics are allowed to differ by arm.

The per-row stabilised censoring weight is the cumulative product over follow-up times up to (but not including) the current row, exactly as in the Danaei et al. (2013) appendix construction of $SW^A_{m+t}$. To prevent undue influence of extreme weights we truncate the stabilised weights at configurable percentiles via swereg::TTEEnrollment$s3_truncate_weights() (Danaei et al. 2013 used a cap of 10).

Final analysis weight

Each non-censoring-event row carries the analysis weight

$$W_{i,m,j} = SW^A_{i,m,0} \times SW^C_{i,m,j}.$$

The baseline factor adjusts for confounding of treatment assignment, the cumulative factor adjusts for informative censoring at protocol deviation, and their product yields a pseudo-population in which the marginal per-protocol contrast is identified under the standard sequential- exchangeability, positivity, consistency, and no-interference assumptions (Hernán and Robins 2008; Su et al. 2024).

Outcome model: weighted Poisson MSM for the IRR

swereg::TTEEnrollment$irr() fits a weighted Poisson marginal structural model for the rate of $Y$ as a function of the assigned baseline treatment $A_{m,0}$:

$$\log \mathrm{E}[Y_{i,m,j} \mid A_{m,0}, \text{person-time}_{i,m,j}] = \beta_0 + \beta_1 A_{m,0} + \log(\text{person-time}_{i,m,j}),$$

estimated by survey::svyglm(..., family = quasipoisson()). The estimand $\exp(\beta_1)$ is the marginal per-protocol incidence rate ratio. This is operationally equivalent — for a rare outcome and a sufficiently fine time grid — to the pooled logistic discrete-time hazard MSM used in TrialEmulation (Su et al. 2024, eq. 4); we use the Poisson rate parameterisation because the swereg skeleton is already organised as person-time intervals.

Validation against the canonical reference implementation

We verified that the swereg implementation agrees with the canonical reference implementation in TrialEmulation (Su et al. 2024) on simulated data with a known true effect: point estimates, standard errors, and confidence interval widths agree to within finite-sample noise across sample sizes from 2,000 to 50,000 individuals, with the only systematic difference being the well-known non-equivalence of the rate ratio and odds ratio for events that are not extremely rare. The validation simulations and reproducible scripts are bundled in the package’s dev/ directory.

Inference: cluster-robust variance

Because (1) each person contributes correlated rows across follow-up times within a trial, (2) each person can appear in multiple sequential trials, and (3) the weights are estimated rather than known, the model-based standard errors from glm() are anti-conservative. Following the recommendation of Hernán and Robins (2008), Danaei et al. (2013), and Su et al. (2024), we report cluster-robust (Huber–White sandwich) standard errors with clustering on the person identifier. In the swereg implementation this is computed via survey::svydesign(ids = ~person_id) and propagated to the confidence interval for $\exp(\beta_1)$ by swereg::TTEEnrollment$irr().

References

• Hernán MA, Robins JM. Observational studies analyzed like randomized experiments: an application to postmenopausal hormone therapy and coronary heart disease. Epidemiology 2008;19(6):766–779.
• Hernán MA, Robins JM. Using big data to emulate a target trial when a randomized trial is not available. Am J Epidemiol 2016;183(8):758–764.
• Danaei G, García Rodríguez LA, Cantero OF, Logan R, Hernán MA. Observational data for comparative effectiveness research: an emulation of randomised trials of statins and primary prevention of coronary heart disease. Stat Methods Med Res 2013;22(1):70–96.
• Caniglia EC, et al. Emulating a sequence of target trials to avoid immortal time bias: an application in pregnancy. Am J Epidemiol 2023.
• Cashin AG, et al. Emulating a target trial — the TARGET statement. JAMA 2025.
• Su L, Rezvani R, Seaman SR, Bartlett JW. TrialEmulation: An R package to emulate target trials for time-to-event data from electronic health records. arXiv:2402.12083, 2024.