From Sepsis Records to a Causal Question
A sepsis analysis can contain repeated measurements of blood pressure, lactate, organ dysfunction, antimicrobial treatment, vasopressor decisions, and follow-up status. The central challenge is that these variables evolve together: current severity can affect the next treatment decision and predict mortality, while earlier treatment can itself affect current severity. A conventional regression that simply adjusts for every measured severity variable can therefore fail to estimate the effect of a longitudinal treatment strategy.
The aim here is to specify and estimate a marginal contrast between two strategies for managing septic shock. The analysis is hypothetical and observational; the strategies must be judged clinically and ethically before applying them to real data.
For concreteness, suppose the decision times are 0, 6, and 12 hours after eligibility. The target population is adults meeting a prespecified septic-shock definition at time 0. One strategy, \(g_E\), follows a protocol to initiate norepinephrine promptly when the specified hypotension threshold is met and to titrate toward a mean arterial pressure (MAP) target of 65 mmHg. The comparator, \(g_F\), follows a fluid-first protocol and initiates norepinephrine only after hypotension persists despite the protocol-specified fluid assessment. Each strategy must define the threshold, time window, contraindications, and permitted exceptions in advance. These are dynamic strategies: the action at each time can depend on current measured history.
Let \(Y^g\) be the day-28 death outcome that would be observed if, from eligibility onward, the person followed strategy \(g\). The primary marginal contrast is the risk difference \(E[Y^{g_E}]-E[Y^{g_F}]\); the corresponding risks are \(E[Y^{g_E}]\) and \(E[Y^{g_F}]\). A negative risk difference favors \(g_E\) on this outcome scale.
Specifying the Longitudinal Analysis
The time ordering is part of the scientific specification, not just a data-management detail. At each decision time, record variables measured before treatment: for example, MAP, lactate, vasopressor history, fluid balance, organ-support status, infection source, and recent laboratory results. Treatment is then recorded, followed by subsequent clinical measurements. A severity value measured after a decision cannot be used as a confounder of that same decision.
For the strategy-specific analysis, each eligible person can be represented under both strategies, or “cloned.” A clone is artificially censored at the first decision when the observed care departs from its assigned strategy. Natural loss to follow-up is a separate source of censoring. In either case, continued observation and adherence can depend on measured clinical history, so simply analyzing the remaining adherent, observed patients can select a nonrepresentative group.
These assumptions are substantive. In particular, exchangeability is not guaranteed by fitting a flexible model: unmeasured severity, clinician judgment, or treatment preference may still confound the comparison. Positivity also requires support for the strategy in the histories where it calls for action. A strategy rarely or never followed for a particular clinical history cannot be reliably evaluated by reweighting.
Strategy and Censoring Weights
For an individual following strategy \(g\) through decision \(t\), define \(q_t(H_t)\) as the conditional probability of receiving the action specified by \(g_t(H_t)\), given the observed history and prior adherence. Define \(r_t(H_t,A_t)\) as the conditional probability of remaining uncensored through the next interval, given the relevant measured history and treatment. In a cloned analysis, these probabilities are estimated among clones still adherent and at risk at the corresponding time.
An unstabilized weight through the final decision has the form below. The indicator is one only if the clone followed the assigned strategy at every decision and remained observed; otherwise its contribution to that strategy's endpoint analysis is zero.
In practice, censoring may be recorded at different points within an interval, so the product must follow the actual time ordering. The denominator probabilities are estimated from treatment-adherence and censoring models. Stabilized weights can be used in a marginal structural model, with numerator probabilities chosen to preserve the intended target population; their exact form must match the analysis and estimand. The examples below use unstabilized weights for a direct regime-specific risk calculation.
Worked Example: One Clone’s Weight
A clone is assigned to \(g_E\). At the three decision times, the estimated probabilities of following the assigned action, conditional on the histories observed, are \(0.80\), \(0.75\), and \(0.90\). The probabilities of remaining uncensored through the corresponding intervals are \(0.95\), \(0.90\), and \(0.80\). The clone follows the strategy and is observed through day 28.
The probability products are \(0.80(0.75)(0.90)=0.54\) for adherence and \(0.95(0.90)(0.80)=0.684\) for observation. Thus \(W^{g_E}=1/(0.54\times0.684)=1/0.36936\), which is approximately \(2.708\).
This clone represents about 2.708 people with its observed history in the weighted analysis. If the clone had departed from the strategy or been lost before the endpoint, its indicator would be zero and it would not contribute an observed endpoint outcome under this regime-specific calculation.
Why the Weighted Risk Targets a Strategy
The weight corrects for the fact that some observed histories are less likely than others to remain adherent and uncensored. The following argument gives the basic identification logic for a finite sequence of decisions. Suppose the conditional adherence and censoring probabilities are the true probabilities, are positive, and the assumptions above hold. Consider a fixed strategy \(g\).
At a decision time, conditional on the history of an adherent clone, the probability of both following \(g\) and remaining observed is \(q_t(H_t)r_t(H_t,A_t)\). Multiplying its indicator by the reciprocal of this probability removes that conditional selection probability in expectation.
Apply the same conditional-expectation argument successively at each decision. The product weight removes the selection associated with the complete observed adherence and censoring path. The resulting expectation is the mean outcome under the intervention that sets each action to \(g_t(H_t)\), with the history distribution generated under that intervention.
Consistency identifies the observed endpoint with the potential endpoint for histories that followed \(g\). Sequential exchangeability permits the observed conditional outcome distributions along those histories to represent the corresponding potential-outcome distributions. Thus \(E[W^gY]=E[Y^g]\). Applying the same argument to the weighted indicator alone gives \(E[W^g]=1\).
For independent, identically distributed individuals with finite expectations, the law of large numbers gives \(n^{-1}\sum_i W_i^gY_i\xrightarrow{p}E[W^gY]\) and \(n^{-1}\sum_i W_i^g\xrightarrow{p}E[W^g]=1\). The denominator limit is nonzero, so the ratio converges in probability to \(E[Y^g]\). \(\square\)
The result explains why a weighted mean is a marginal risk rather than a conditional risk at a fixed severity level. In a model-based implementation, a weighted outcome model can estimate the risk for each strategy, followed by standardization to the target population. With a binary endpoint and a saturated strategy indicator, the fitted weighted means are the regime-specific risks. The risk difference is obtained by subtracting those means; it is not the conditional odds ratio from a logistic model.
Worked Example: Two Weighted Mortality Risks
Consider a small illustrative data set in which the following regime-specific weights already combine adherence and censoring probabilities. The table lists four observed, adherent, uncensored clones for each strategy. The numbers are for calculation practice, not a realistic sample size for a clinical study.
| Strategy | Weighted endpoint contributions | Sum of weights | Weighted deaths |
|---|---|---|---|
| \(g_E\) | \(5.556,\ 6.944,\ 10.000,\ 8.333\) | \(30.833\) | \(5.556+10.000=15.556\) |
| \(g_F\) | \(4.000,\ 5.000,\ 6.000,\ 5.000\) | \(20.000\) | \(5.000\) |
For \(g_E\), the weighted risk is \(15.556/30.833\approx0.5045\). For \(g_F\), it is \(5/20=0.25\). The estimated risk difference, early-strategy risk minus fluid-first-strategy risk, is therefore approximately \(0.5045-0.25=0.2545\), or 25.45 percentage points. In this tiny illustrative calculation, the estimated mortality risk is higher under \(g_E\); the direction is not a claim about sepsis care.
An estimated risk difference is an absolute contrast in the target population. It should be accompanied by uncertainty intervals and interpreted only after assessing design, model fit, positivity, and the clinical meaning of the strategies. A ratio of these risks would answer a different question.
Diagnostics, Alternatives, and Limits
A fitted weighted analysis should be checked before interpreting its contrast. Examine the distributions of treatment-adherence and censoring probabilities, the resulting weights, and weighted covariate balance at each decision time. Very small probabilities generate large weights and indicate weak empirical support for a strategy. Report summaries such as the maximum and upper percentiles of the weights and the effective sample size, commonly calculated as \((\sum_i W_i)^2/\sum_i W_i^2\). Investigate whether a few observations dominate the result, and assess whether the probability models capture nonlinearities and clinically important interactions.
For the \(g_E\) weights in the previous example, the squared weights sum to approximately \(248.53\), while the weights sum to \(30.833\). The effective sample size is therefore \(30.833^2/248.53\approx3.83\), far below the four observed clones in the unweighted count. In real analyses, a sharp reduction in effective sample size warns that the weighted estimate may be imprecise and sensitive to model choices. Truncating extreme weights can reduce variance but changes the estimating procedure and may introduce bias; it does not repair a structural lack of positivity.
The principal alternative is the parametric g-formula, which models the longitudinal covariate and outcome distributions and then predicts outcomes under each strategy. It avoids inverse weights but relies on correct specification of those models. A conventional regression that adjusts for time-varying severity can be inappropriate when that severity both predicts subsequent treatment and was affected by prior treatment. The weighted marginal structural approach handles this treatment-confounder feedback through the weights, but it remains vulnerable to unmeasured confounding, misspecified probability models, and poor positivity.
Censoring weights only address loss to follow-up explained by the measured history included in their models. They cannot recover outcomes after loss if the relevant causes of dropout are unmeasured. Likewise, estimates apply to the specified eligibility criteria, treatment versions, follow-up horizon, and outcome definition—not automatically to all patients with sepsis or to other clinical protocols. Report the strategy rules, covariate timing, censoring definition, probability models, weight diagnostics, and marginal contrast so that the estimate can be evaluated and reproduced.
Worked Example: Interpreting Weight Diagnostics
Suppose a fitted adherence model assigns probabilities of \(0.60\), \(0.30\), and \(0.05\) to the required actions at three successive decisions for one clone. Suppose its corresponding interval observation probabilities are \(0.90\), \(0.90\), and \(0.80\). The adherence product is \(0.60(0.30)(0.05)=0.009\), and the observation product is \(0.90(0.90)(0.80)=0.648\). The combined probability is \(0.009(0.648)=0.005832\), giving a weight of \(1/0.005832\approx171.5\).
That single clone would carry substantial influence. The \(0.05\) probability also signals that the required treatment decision is uncommon for its measured history. The analyst should inspect that history, assess whether the strategy is clinically supported there, and check model fit and overlap. Truncating the weight may be reported as a sensitivity analysis, but it cannot establish that the desired strategy is identifiable in a history where its probability is effectively zero.
Check Your Understanding
Answer each question, then compare with the brief solutions below.
- At a decision time, why must severity variables used in the treatment model be measured before that treatment decision?
- A clone has adherence probabilities \(0.5\) and \(0.8\), and observation probabilities \(0.9\) and \(0.5\), at two intervals. If it follows the strategy and remains observed, what is its unstabilized weight?
- What does a negative risk difference \(E[Y^{g_E}]-E[Y^{g_F}]\) mean when \(Y\) indicates death?
- Why does a large weight raise concern even if the fitted weighted model converges?
- Name one important limitation shared by the weighted strategy analysis and the parametric g-formula.
Solutions. (1) The treatment model must condition on information available when the decision is made; post-decision severity cannot be a cause of that decision and may itself be affected by treatment. (2) The combined probability is \(0.5(0.8)(0.9)(0.5)=0.18\), so the weight is \(1/0.18\approx5.56\). (3) It means the estimated day-28 death risk under \(g_E\) is lower than under \(g_F\) in the target population, by the absolute amount of the risk difference. (4) A large weight gives one observation disproportionate influence, often reflecting near-positivity violations and unstable estimates. (5) Both can be biased by unmeasured confounding; neither method makes observational treatment assignment exchangeable by itself.