Why Warfarin Is a Feedback Problem
Warfarin management is repeatedly adjusted in response to measurements such as the international normalized ratio (INR) and to events such as bleeding. Those same histories can predict later stroke and bleeding. A comparison of patients who happened to receive different doses can therefore mix treatment effects with differences in the clinical histories that drove treatment decisions. The relevant question is often not the effect of one fixed dose, but the marginal risk under a specified strategy for adjusting treatment as history evolves.
The strategies in this tutorial are illustrative statistical policies, not clinical recommendations. A protocol must define its actions and timing precisely; changing a threshold or the handling of a recent bleed changes the intervention being studied.
For simplicity, suppose the cohort is followed through a fixed horizon and outcomes are ascertained for everyone. If death prevents observing an endpoint, it must be handled according to the target estimand—for example, as a competing event in a cumulative-incidence risk—not silently treated as ordinary loss to follow-up. We focus on two treatment decisions, \(A_0\) and \(A_1\), with the intermediate INR and bleeding history \(L_1\) measured before \(A_1\).
Specify the Strategies Before Estimating Their Risks
Consider a binary action at each decision: \(A_t=1\) means continue the protocol’s standard warfarin action, and \(A_t=0\) means reduce or temporarily hold treatment according to the protocol. A strategy should state what to do for every history relevant to the study. For example, define a conservative rule \(g^C\) by \(g^C_0=1\), then at the next decision set \(g^C_1=0\) if the latest INR exceeds 3.0 or there has been a bleed since the previous decision, and set \(g^C_1=1\) otherwise. The thresholds and action labels here are hypothetical.
An alternative rule \(g^L\) might hold or reduce treatment only when INR exceeds 4.0 or there has been a bleed. For a patient with INR 3.5 and no bleed, the conservative rule prescribes \(A_1=0\), while the alternative prescribes \(A_1=1\). For a patient with INR 4.2, both prescribe \(A_1=0\). This comparison makes clear that the intervention is the complete history-dependent rule, not merely “more” or “less” treatment in the abstract.
Worked Example: Applying a Rule to a Patient History
Under \(g^C\), suppose a patient receives \(A_0=1\). At the next visit the INR is 3.5, and no bleed has occurred. Because \(3.5>3.0\), the rule prescribes \(A_1=0\). If instead the INR were 2.8 with no bleed, then \(2.8\) does not exceed 3.0 and the rule prescribes \(A_1=1\). If INR were 2.8 but a bleed had occurred, the “or there has been a bleed” condition applies, so \(A_1=0\).
The action is determined by the history available before the decision. A bleed occurring after \(A_1\) cannot be used to define that earlier action. This ordering is essential when constructing histories and treatment probabilities.
Identification and the Strategy Weight
Let \(p_t(a\mid H_t)=P(A_t=a\mid H_t)\) be the observed treatment probability, estimated from a model using information available by decision \(t\). For a deterministic strategy \(g\), a patient follows the rule through the final decision if \(A_t=g_t(H_t)\) at every \(t\). Such a patient is called adherent to \(g\) for this analysis.
For an adherent patient, define the strategy weight
Nonadherent patients contribute zero to the strategy-specific calculation. In practice, the conditional probabilities are estimated, and the product is evaluated only for patients whose observed actions match the rule. The weight is the inverse probability of following the observed strategy path, conditional on the histories encountered along that path.
To justify this result, factor the observed-data probability along a path into the baseline-history distribution, the conditional treatment probabilities, and the conditional distributions of later histories and outcomes. Under intervention \(g\), each treatment probability is replaced by a point mass at \(g_t(H_t)\); the conditional laws of later history and outcome remain those given the preceding history and prescribed action. In the observed data, restrict to paths with \(A_t=g_t(H_t)\) for every \(t\), and multiply by \(W^g\). Each factor \(P(A_t=g_t(H_t)\mid H_t)\) in the observed path probability cancels its reciprocal in the weight. The remaining path factors are exactly those under intervention \(g\). Exchangeability permits the observed conditional laws along these paths to represent the corresponding laws under \(g\), and consistency equates the observed outcome on an adherent path to \(Y^g\). Thus the weighted numerator is the intervention expectation times the common normalizing factor, and the weighted denominator is that factor. Their ratio gives the displayed mean. \(\square\)
This result does not say that history and treatment are independent among weighted patients. For a deterministic dynamic rule, treatment is prescribed as a function of history; among adherers, the association can be exact. The weights identify the strategy-specific risk by representing the history distribution that would arise under the strategy, including its induced association between INR history and treatment.
Calculating Marginal Stroke and Bleeding Risks
With independent patients and a correctly specified treatment-probability model, a normalized weighted mean estimates the marginal risk for an endpoint. For endpoint \(Y\), the Hájek estimator is
where \(I_i(g)=1\) if patient \(i\) followed \(g\), and zero otherwise. Use the stroke indicator for \(Y\) to estimate stroke risk and the bleeding indicator for \(Y\) to estimate bleeding risk. The two risks need not move in the same direction: a strategy could lower stroke risk while raising bleeding risk.
Worked Example: Estimating Two Risks From Four Adherent Patients
Suppose four patients followed the same two-decision rule. Each had probability 0.5 of receiving the prescribed action at \(t=0\). Conditional on their respective histories at \(t=1\), their probabilities of receiving the prescribed action were 0.5, 0.25, 0.125, and 0.5. Their observed outcomes are:
| Patient | \(P(A_0=g_0\mid H_0)\) | \(P(A_1=g_1\mid H_1)\) | Stroke | Bleeding |
|---|---|---|---|---|
| 1 | 0.5 | 0.5 | 1 | 0 |
| 2 | 0.5 | 0.25 | 0 | 1 |
| 3 | 0.5 | 0.125 | 0 | 0 |
| 4 | 0.5 | 0.5 | 0 | 0 |
The weights are the reciprocals of the products of the two adherence probabilities. Patient 1 has \(W_1=1/(0.5\times0.5)=4\); patient 2 has \(W_2=1/(0.5\times0.25)=8\); patient 3 has \(W_3=1/(0.5\times0.125)=16\); and patient 4 has \(W_4=1/(0.5\times0.5)=4\). Their total weight is \(4+8+16+4=32\).
The weighted stroke risk is \((4\times1+8\times0+16\times0+4\times0)/32=4/32=0.125\). The weighted bleeding risk is \((4\times0+8\times1+16\times0+4\times0)/32=8/32=0.25\). For comparison, the unweighted proportions among these four adherers are \(1/4=0.25\) for each outcome. The difference illustrates how the weighting changes representation; these four records are only an arithmetic illustration, not evidence that either risk is estimated precisely.
Worked Example: Checking Weight Concentration
For the four weights just calculated, the effective sample size is
Although four people followed the strategy, the weight distribution carries information comparable to fewer than three equally weighted observations by this diagnostic. Patient 3 has the largest weight because the prescribed action was relatively unlikely given that patient’s history: \(1/(0.5\times0.125)=16\). This is a warning about weak overlap and potentially unstable risk estimates, not a reason to interpret the weighted estimate as if it came from 2.91 actual patients.
Diagnostics, Alternatives, and Limits
Before interpreting strategy-specific risks, inspect the treatment probabilities and weights at each decision time. Useful checks include the distribution and upper tail of the weights, the effective sample size, and whether histories under the proposed rule have adequate probabilities of receiving the prescribed actions. Also check that INR, bleeding history, and other clinically relevant predictors are measured before the treatment decision they are meant to explain. A model that uses information recorded after the decision can produce misleading probabilities and an invalid analysis.
Very small adherence probabilities create large weights. The resulting estimate can be dominated by a few patients, have high variance, and be sensitive to treatment-model specification. Weight truncation can reduce variance but changes the estimating procedure and may introduce bias; it does not create information where the data lack support. If a rule is unsupported in important histories, consider whether the target strategy can be revised on substantive grounds, and state that the original strategy was not identified from the available data.
An outcome-regression approach models the outcome conditional on treatment and history, then predicts outcomes under the strategy. It can be more efficient when well specified, but it relies on correct outcome-model specification and must handle time-varying confounder feedback appropriately. Strategy weighting instead uses the treatment assignment probabilities to reweight adherent paths. The two approaches rely on different modeling choices; neither rescues unmeasured confounding or a lack of positivity. A marginal structural model can summarize risks across several strategies, but it still needs a clearly defined intervention, valid identification assumptions, and adequate support.
A frequent mistake is to say that these strategy weights make treatment independent of INR or bleeding history. That statement is generally false for a dynamic rule: among adherers, treatment is assigned according to those histories and may remain strongly associated with them. The correction is to interpret the weights as identifying the risk under the specified strategy while retaining the strategy-induced distribution of INR and bleeding history. The treatment rule determines that distribution; the weights do not erase it.
Finally, estimated risks are only as meaningful as their target. The horizon, eligibility criteria, INR thresholds, bleeding definition, treatment-action timing, and handling of death and censoring all affect the estimand. A weighted risk is not automatically a universal effect of “warfarin management”; it is a risk for a particular population under a particular, sufficiently supported strategy.
Check Your Understanding
Answer each question, then compare with the solutions below.
- Under a rule that sets \(A_1=0\) if INR exceeds 3.0 or a recent bleed occurred, what action is prescribed for INR 2.7 with a recent bleed?
- A patient has probabilities 0.4 and 0.5 of receiving the prescribed actions at two decisions and follows the rule at both. What is the strategy weight?
- What additional assumption is needed to identify strategy-specific risks when some patients are lost to follow-up?
- Why is it incorrect to claim that dynamic-strategy weights make INR independent of treatment among adherers?
- For weights 2 and 6 and outcomes 0 and 1, calculate the normalized weighted risk.
Solutions. (1) The recent bleed condition is sufficient, so \(A_1=0\). (2) The weight is \(1/(0.4\times0.5)=5\). (3) One needs an appropriate conditional independent-censoring assumption, together with censoring adjustment and positivity for remaining observed. (4) The rule itself assigns treatment as a function of history, so adherent treatment can remain determined by INR. The weights represent the history distribution under the rule rather than eliminating that relationship. (5) The weighted risk is \((2\times0+6\times1)/(2+6)=6/8=0.75\).