Why Transplant Dosing Is a Longitudinal Causal Question
After an organ transplant, immunosuppressant doses are adjusted as clinicians observe graft function, toxicity, and possible rejection. Those observations can influence the next dose and predict later rejection. For example, a toxicity signal may lead to dose reduction, while signs of graft injury may prompt intensification. Comparing patients who happened to receive different doses can therefore mix the effect of dose with the reasons for changing it.
A marginal structural model analysis starts by defining the treatment strategies and outcome precisely. In this example, consider adults who are eligible for either protocol at baseline. Let \(A_0\) and \(A_1\) indicate the prescribed immunosuppressant intensity in two successive treatment intervals: 1 denotes higher intensity and 0 standard intensity. Let \(X\) denote baseline characteristics. Let \(L_1\) contain the interim history measured before the second dose decision, including toxicity, graft function, and any earlier rejection episode. Let \(Y\) indicate rejection during a prespecified final follow-up interval after the second decision.
These are simplified strategies: a clinical protocol must specify dose ranges, monitoring, and any rescue rules well enough that “higher intensity” or “standard intensity” represents a meaningful intervention. The example also assumes the outcome is defined for everyone at the horizon. If graft loss or death can prevent rejection from being observed, the estimand must say how those events are handled; they cannot simply be ignored.
Identification: When Weighting Recovers the Strategy Risk
Write \(H_0=X\) and \(H_1=(X,A_0,L_1)\). The treatment assignment probabilities relevant to a strategy \(g=(g_0,g_1)\) are \(p_0(g_0\mid X)=P(A_0=g_0\mid X)\) and \(p_1(g_1\mid H_1)=P(A_1=g_1\mid X,A_0,L_1)\). For a patient who followed \(g\), the unstabilized inverse-probability weight is \(W^g=1/\{p_0(g_0\mid X)p_1(g_1\mid H_1)\}\). The weight is large when the observed treatment sequence was unlikely given the patient’s measured history.
Here sequential exchangeability means that, conditional on \(X\), the first treatment is independent of the potential interim and final outcomes under the strategies; conditional on the observed history before the second decision, the second treatment is independent of the corresponding potential final outcomes. Positivity requires both prescribed treatment choices to have nonzero probability at the relevant histories. These assumptions cannot be verified from the observed data alone.
Proof. For clarity, use discrete histories; with continuous variables, the same calculation uses densities and integrals under the corresponding positivity and integrability conditions. Sequential exchangeability and consistency identify the distribution of \(L_1\) and \(Y\) under strategy \(g\) from the observed conditional distributions. In the weighted expectation, summing over the observed data cancels the two treatment-assignment probabilities:
In the second line, the observed joint probability of the treatment sequence contributes the factors \(p_0(g_0\mid x)\) and \(p_1(g_1\mid x,g_0,l_1)\); the indicator restricts the sum to that sequence, and division by those factors removes them. The remaining distribution is the sequential distribution under the intervention. Repeating the same calculation with \(Y\) replaced by 1 gives the second identity, since the intervention distribution over \(X\) and \(L_1\) sums to 1. \(\square\)
Worked Example: Defining the Clinical Protocol and Its History
Suppose a transplant cohort records baseline donor and recipient characteristics, the initial prescribed intensity, toxicity and graft function at the end of the first interval, and the second prescribed intensity. The endpoint is biopsy-confirmed rejection in the final interval. The strategies are higher intensity in both intervals versus standard intensity in both.
A patient with \(A_0=1\), followed by substantial toxicity in \(L_1\), and then \(A_1=0\), did not follow either sustained strategy. That patient’s history is still useful for estimating the second-decision probability: \(L_1\) predicts both \(A_1\) and subsequent rejection, while also being affected by \(A_0\).
The protocol therefore has to measure toxicity and graft status before the second decision, not after it. If those measurements are omitted, the analysis may leave important reasons for changing the dose uncontrolled. Conversely, adjusting for \(L_1\) as an ordinary covariate in a final outcome regression can block part of the effect of \(A_0\). Weighting instead uses \(L_1\) to model treatment assignment while targeting the marginal outcome under the sustained strategy.
Estimating and Comparing Marginal Risks
In a sample of \(n\) patients, define \(I_i(g)=I(A_{0i}=g_0,A_{1i}=g_1)\), and calculate each patient’s weight from the estimated treatment probabilities given the appropriate histories. The normalized weighted risk estimator is
Its denominator normalizes the weighted group, and its numerator sums weighted rejection outcomes. The result estimates a risk, not an odds ratio. A risk difference, \(\widehat{R}(g_H)-\widehat{R}(g_S)\), compares absolute risks in the target baseline population.
Derivation. With known probabilities, the summands in the numerator and denominator are bounded under the stated positivity condition, so the law of large numbers makes their sample averages converge in probability to their expectations. The theorem above gives expectations \(E[Y^g]\) and 1, respectively. With a finite number of history categories and consistent estimated probabilities bounded away from zero, each estimated weight converges to its corresponding true weight; the sample-average difference caused by estimated rather than true weights therefore converges in probability to zero. The denominator converges to 1, so the continuous mapping theorem for a ratio gives the claimed convergence. \(\square\)
Worked Example: Calculating Two Weighted Rejection Risks
For illustration, suppose estimated treatment probabilities and observed patients following each sustained strategy give the following two history groups. The table lists the number of observed strategy followers and rejection events in each group. The probability column is the product of the two conditional probabilities for that strategy.
| Strategy | Interim history | Probability of observed sequence | Patients | Rejections |
|---|---|---|---|---|
| Higher, higher | Favorable | 0.50 × 0.80 = 0.40 | 20 | 4 |
| Higher, higher | Adverse | 0.50 × 0.25 = 0.125 | 10 | 5 |
| Standard, standard | Favorable | 0.50 × 0.60 = 0.30 | 15 | 3 |
| Standard, standard | Adverse | 0.50 × 0.75 = 0.375 | 15 | 6 |
For higher intensity, the group weights are \(1/0.40=2.5\) and \(1/0.125=8\). The weighted denominator is \(20(2.5)+10(8)=50+80=130\). The weighted event total is \(4(2.5)+5(8)=10+40=50\). Thus \(\widehat{R}(g_H)=50/130=0.3846\), or about 38.5%.
For standard intensity, the weights are \(1/0.30=3.333\ldots\) and \(1/0.375=2.666\ldots\). The denominator is \(15(3.333\ldots)+15(2.666\ldots)=50+40=90\). The event total is \(3(3.333\ldots)+6(2.666\ldots)=10+16=26\). Hence \(\widehat{R}(g_S)=26/90=0.2889\), or about 28.9%.
The estimated risk difference is \(0.3846-0.2889=0.0957\), about 9.6 percentage points. This numerical result is illustrative, not a clinical conclusion: it relies on the identification assumptions and on the validity of the treatment probability estimates.
Diagnostics, Alternatives, and Limits
Before interpreting weighted risks, inspect the treatment probabilities and weights separately for each strategy. Useful checks include the minimum and upper quantiles of the probabilities, the largest weights, and how much of the total weight comes from a small number of patients. Also compare measured baseline and interim histories across weighted strategy groups. Balance after weighting is a diagnostic for measured variables, not proof that all confounding has been removed.
A near-zero probability signals weak practical support for the strategy in that history. For example, clinicians may almost never maintain higher intensity after severe toxicity. The corresponding inverse probability becomes very large, allowing a few unusual patients to dominate the estimate and making it unstable. If the probability is zero, positivity fails: the observed data provide no direct information about that treatment decision at that history.
Worked Example: What Weight Truncation Can Change
Consider a one-decision slice of a higher-intensity strategy. In an adverse-history group, two patients received higher intensity when its estimated probability was 0.02; one rejected and one did not. Their weights are each \(1/0.02=50\). In a better-supported group, eight patients received higher intensity with probability 0.50; one rejected and seven did not, so each weight is 2.
Without truncation, the weighted risk is \((50+2)/(2(50)+8(2))=52/116=0.4483\). If weights above 10 are truncated to 10, the risk becomes \((10+2)/(2(10)+8(2))=12/36=0.3333\). Truncation has reduced the influence of the rare adverse-history group, but it has also changed the weighted estimate. It is not a repair that restores information where the data have little support.
A standard outcome regression that includes treatment and measured history can be a useful alternative, but ordinary adjustment for \(L_1\) may not estimate the total effect of the sustained strategy when \(L_1\) is affected by \(A_0\). The longitudinal g-formula is another way to use the same sequential assumptions; rather than reweighting observed strategies, it models the outcome and intermediate history distributions under interventions. These approaches have different modeling vulnerabilities, but none removes the need for adequate measurement of confounders and support for the strategies.
A common mistake is to compare rejection proportions among patients who actually stayed on each dose and call the result a causal effect. That comparison can be biased because interim toxicity or graft status may both influence the next dose and predict rejection. Another mistake is to treat a large weight as evidence that an individual is especially informative. In fact, it indicates that the observed treatment sequence was unusual given the measured history; estimates relying heavily on such observations need careful stability checks.
Even well-behaved weights do not address unmeasured confounding, incorrect treatment timing, poorly defined dose strategies, or outcome misclassification. The causal interpretation is conditional on the study design, measurement, and assumptions—not created by the weighting calculation itself.
Check Your Understanding
Answer each question, then check the solutions below.
- If \(P(A_0=1\mid X)=0.4\) and \(P(A_1=1\mid X,A_0,L_1)=0.5\), what is the weight for a patient who followed the higher-higher strategy?
- Why can interim toxicity be needed in the second-decision treatment model even though it is affected by the first dose?
- In a weighted strategy group, the event total is 18 and the weight total is 60. What is the estimated marginal risk?
- What does a near-zero probability of following a strategy at a particular history imply for the analysis?
Solutions. (1) The joint probability is \(0.4(0.5)=0.2\), so the inverse-probability weight is \(1/0.2=5\). (2) Toxicity can predict both the second dose and rejection, so omitting it can leave confounding of the second decision. Its role in the treatment model does not mean that it should automatically be controlled for as an ordinary covariate in a final outcome regression. (3) The risk is \(18/60=0.30\), or 30%. (4) The inverse probability is very large, so the estimate may be unstable; if the probability is zero, positivity fails and that strategy risk is not identified from the observed data at that history.