Why Blood-Pressure Feedback Matters
Antihypertensive treatment is often reconsidered as a patient’s blood pressure changes. Kidney function can also affect whether a clinician starts, continues, or intensifies a drug. Both measures may themselves be affected by earlier treatment. This creates a feedback problem: current blood pressure and kidney function predict the next treatment and stroke risk, while previous treatment may have changed those measurements.
A regression that simply adjusts for the latest blood pressure and kidney function can block part of the effect of earlier treatment, while failing to resolve the time-varying confounding in the treatment decision. In this case study, inverse-probability weighting instead constructs weighted comparisons of sustained strategies. The target is a marginal risk: the risk of stroke in the population if everyone followed one specified strategy, compared with the risk if everyone followed another.
Set-Up and Identification Assumptions
Consider two treatment decisions. Let \(L_0\) denote baseline information, including baseline blood pressure and kidney function. Let \(A_0\) indicate antihypertensive treatment at the first decision. Next, \(L_1\) records blood pressure and kidney function measured before the second decision, and \(A_1\) indicates treatment then. Let \(Y\) indicate stroke by the end of follow-up. Write \(H_0=L_0\) and \(H_1=(L_0,A_0,L_1)\) for the observed histories available at the two decisions.
For clarity, compare the sustained strategies \(g=(1,1)\), treatment at both decisions, and \(g=(0,0)\), no treatment at either decision. Write \(Y^g\) for the stroke outcome that would occur under strategy \(g\). The same framework can define other prespecified strategies, such as treatment only when blood pressure exceeds a threshold, provided the strategy is well-defined and supported by the observed data.
Define the observed treatment probabilities \(e_0(a\mid H_0)=P(A_0=a\mid H_0)\) and \(e_1(a\mid H_1)=P(A_1=a\mid H_1)\). For a person who follows \(g=(g_0,g_1)\), the unstabilized adherence weight is the reciprocal of the probability of their observed treatment path, conditional on the histories at each decision.
A person who does not follow the strategy has weight zero for that strategy. Among people who do follow it, small probabilities of the observed path produce large weights. Blood pressure and kidney function enter through the treatment-probability models because they help explain why treatment was chosen at each decision.
Why the Weighted Mean Targets a Strategy Risk
The key identity concerns the outcome \(Y\), not an arbitrary function of the covariate and outcome path. Let \(E[Y^g]\) be the marginal stroke risk under strategy \(g\). Under the assumptions above, the outcome-specific inverse-probability identity is
The corresponding denominator identity is
These identities can be seen by factoring the observed-data distribution into the baseline distribution, the first treatment probability, the covariate-transition distribution, the second treatment probability, and the outcome distribution. Restrict to histories following \(g\). The two inverse treatment-probability factors cancel the treatment factors in that path-specific distribution. Sequential exchangeability and consistency identify the remaining baseline, covariate-transition, and outcome distributions along the specified path with those under intervention \(g\). Integrating the outcome gives \(E[Y^g]\); integrating without the outcome gives 1. This argument justifies the stated outcome and denominator identities, not an identity for every function of the covariate path.
For \(n\) independent people, the practical weighted risk estimate is the Hájek ratio
With the true treatment probabilities, the law of large numbers gives \(n^{-1}\sum_i W_{ig}Y_i\xrightarrow{p}E[Y^g]\) and \(n^{-1}\sum_i W_{ig}\xrightarrow{p}1\), assuming independent, identically distributed observations and integrable weights. The ratio therefore converges in probability to \(E[Y^g]\). In practice, probabilities are estimated; the same interpretation requires adequate models and regularity conditions for that estimation as well as the causal assumptions.
Worked Example: Weighting One Patient’s Treatment Path
A patient has baseline blood pressure \(164\) mmHg and estimated glomerular filtration rate (eGFR) \(58\) mL/min/1.73 m². Their estimated probability of treatment at the first decision is \(0.60\). After treatment, their next recorded blood pressure is \(150\) mmHg and eGFR is \(52\), and their conditional probability of treatment at the second decision is \(0.40\). They receive treatment at both decisions.
For the sustained-treatment strategy \(g=(1,1)\), the probability of this observed treatment path, conditional on the recorded histories, is \(0.60\times0.40=0.24\). The path weight is therefore
This patient contributes \(4.167\) to the weighted denominator and, if a stroke occurred, \(4.167\) to the weighted event total. The weight does not mean that the patient literally represents exactly 4.167 people. It corrects the representation of treatment paths in the weighted comparison. Their weight for the never-treat strategy is zero because their observed path does not follow \(g=(0,0)\).
Worked Example: Comparing Weighted Stroke Risks
Suppose a cohort has six people who follow one of the two sustained strategies below, plus two people with discordant paths \(01\) or \(10\). The latter have zero weight for both strategies in this illustration. For each adherent person, the table gives the product of the conditional probabilities of their observed treatment choices and whether stroke occurred. The probabilities depend on the person’s recorded blood pressure and kidney-function history.
| Strategy followed | Path probability | Weight | Stroke \(Y\) |
|---|---|---|---|
| Treat at both: person 1 | 0.40 | 2.50 | 1 |
| Treat at both: person 2 | 0.25 | 4.00 | 0 |
| Treat at both: person 3 | 0.20 | 5.00 | 1 |
| Never treat: person 4 | 0.40 | 2.50 | 0 |
| Never treat: person 5 | 0.20 | 5.00 | 1 |
| Never treat: person 6 | 0.24 | 4.167 | 0 |
For sustained treatment, the weighted event total is \(2.50+5.00=7.50\), and the total weight is \(2.50+4.00+5.00=11.50\). Thus
For never treatment, the event total is \(5.00\), and the total weight is \(2.50+5.00+4.167=11.667\). Thus
The estimated risk difference, treat at both minus never treat, is \(0.6522-0.4286=0.2236\), or about 22.4 percentage points. The estimated risk ratio is \(0.6522/0.4286\approx1.52\). These are illustrative calculations, not evidence that treatment increases stroke risk: the cohort is tiny, and causal interpretation depends on the assumptions and fitted probabilities. The unweighted risks among these adherers would be \(2/3\) and \(1/3\), respectively, so weighting changes their relative contributions.
Diagnostics and Practical Choices
Before interpreting a weighted contrast, inspect the treatment models and the weights. In this setting, a warning sign is that few people with certain combinations of prior blood pressure, eGFR, and treatment history have a positive chance of following a strategy. Other useful checks include comparing the distributions of blood pressure and kidney function across treatment groups before and after weighting, examining the largest weights and their histories, and reporting the effective sample size. For weights \(W_i\), a common summary is \((\sum_i W_i)^2/\sum_i W_i^2\); a value far below \(n\) indicates that a small number of observations dominate the weighted estimate.
Worked Example: What Weight Truncation Changes
In the preceding table, cap weights at \(4\) for illustration. For sustained treatment, the capped weights are \(2.5,4,4\), giving risk \((2.5+4)/(2.5+4+4)=6.5/10=0.65\). For never treatment, the capped weights are \(2.5,4,4\), giving risk \(4/(2.5+4+4)=4/10.5\approx0.381\).
The truncated risk difference is \(0.65-0.381=0.269\), compared with \(0.224\) before truncation. Capping reduced the influence of the largest weights, but changed the estimate; it is not a repair that automatically restores identification or removes bias. Report the rule and, where feasible, compare results under prespecified truncation choices with the untruncated analysis.
A useful alternative is the parametric g-formula, which models the covariate and outcome processes and predicts outcomes under each strategy. It handles treatment-confounder feedback by simulating the covariate distribution under each intervention, but relies on correct models for those processes. A marginal structural model uses weights to estimate marginal contrasts; its validity depends on exchangeability, positivity, and adequate treatment-probability estimation. A conventional outcome regression that adjusts for post-baseline blood pressure and kidney function is not automatically an equivalent alternative: those measures may mediate earlier treatment effects as well as confound later treatment decisions.
Common Pitfalls and Limits
- Using a treatment model as a substitute for causal assumptions. A good-fitting model cannot control unmeasured confounding or make a strategy supported when its treatment probability is zero.
- Ignoring the ordering of measurements. \(L_1\) must be measured before \(A_1\) to be a history variable for that decision. A measure taken after the decision cannot be treated as its cause without a different causal model.
- Overinterpreting extreme weights. Large weights can signal weak positivity, model problems, or both. Truncation trades variance for potential bias and changes the analysis.
- Calling an adherer-only comparison a causal estimate. Restricting to observed people who followed a strategy is generally selective. Weighting is intended to correct this selection under its assumptions.
The strategy contrast is a population-level effect of assigning sustained policies, not the effect for every individual. It also depends on how “treatment” is defined, the follow-up period, stroke definition, and handling of censoring or competing events. If loss to follow-up occurs, additional censoring weights may be needed under their own assumptions; treatment weights alone do not address informative censoring.
Check Your Understanding
Answer each question, then compare with the brief solutions.
- Why can prior blood pressure be a confounder of \(A_1\) and stroke even if it was affected by \(A_0\)?
- A person follows \(g=(1,1)\), with treatment probabilities \(0.5\) and \(0.25\) at their two histories. What is their adherence weight?
- What does a weighted risk difference of \(0.08\) for treat at both versus never treat mean?
- Why does an extremely large treatment weight raise a positivity concern?
- When might the parametric g-formula be preferred, and what assumption remains important?
Solutions. (1) Prior blood pressure predicts the later treatment decision and stroke, and its value may reflect the effect of \(A_0\); this is time-varying confounding affected by prior treatment. (2) \(1/(0.5\times0.25)=8\). (3) Under the identification assumptions, the estimated marginal stroke risk under sustained treatment is 8 percentage points higher than under never treatment over the specified follow-up. (4) It indicates that the observed treatment path had very low estimated probability for that history, so a few observations can dominate and some histories may lack practical overlap. (5) It may be attractive when covariate and outcome processes can be modeled credibly and risks under interventions are to be simulated; correct specification of the relevant models, as well as exchangeability and positivity, remains important.