Numerator Models and the Marginal Structural Model

Why the Numerator Model Matters

Stabilized weights are often introduced as a way to make inverse-probability weights less variable. That is useful, but it is not the whole story: the numerator also specifies a treatment law in the weighted population. Consequently, its history determines which associations with treatment are retained and helps determine the interpretation of a weighted outcome model.

Consider decision times \(t=0,\ldots,K\). Let \(A_t\) be treatment at time \(t\), and let \(H_t\) denote the observed history just before that treatment. The history includes prior treatments and measured covariates available at that time. Write \(g_t(a\mid H_t)=P(A_t=a\mid H_t)\) for the true treatment probability, and let \(q_t(a\mid Q_t)\) be a normalized numerator treatment law, where \(Q_t\) is a specified part or function of \(H_t\). For discrete treatment, each law sums to one over possible values of \(a\).

For the results below, assume consistency, sequential exchangeability given \(H_t\), and positivity: whenever a treatment value is considered under the target regime, its true probability \(g_t(a\mid H_t)\) is positive almost surely in the relevant histories. Assume also that each \(q_t\) is a normalized probability law and that the weights are defined on the support being considered. These conditions concern different issues: exchangeability and denominator positivity support causal identification, while numerator support determines which weighted comparisons can be formed.

The Treatment Law Induced by the Full Weight

Define the weighted probability law \(P^W\) by \(P^W(B)=E[SW_K I(B)]\), for events \(B\), when \(E[SW_K]=1\). Under the conditions above, this normalization holds: iterated conditioning on each pre-treatment history gives \(E[q_t(A_t\mid Q_t)/g_t(A_t\mid H_t)\mid H_t]=1\), and applying this successively from the last decision time back to the first gives \(E[SW_K]=1\).

The full product in \(SW_K\) matters to this result. To prove it, set \(F_{t+1:K}=\prod_{s=t+1}^K q_s(A_s\mid Q_s)/g_s(A_s\mid H_s)\), with an empty product equal to one. For each \(s\), normalization of \(q_s\) gives \(E[q_s(A_s\mid Q_s)/g_s(A_s\mid H_s)\mid H_s]=1\). Backward iterated conditioning therefore gives \(E[F_{t+1:K}\mid H_t,A_t]=1\): at each future decision time, condition on its pre-treatment history, integrate its treatment factor to one, and then continue to the preceding decision time. This uses the sequential structure in which the history before a future treatment contains the earlier treatments and observed covariates.

Now write \(SW_K=SW_{t-1}\{q_t(A_t\mid Q_t)/g_t(A_t\mid H_t)\}F_{t+1:K}\), where \(SW_{t-1}\) is measurable given \(H_t\). For any treatment value \(a\), the conditional weighted mass is \(E[SW_K I(A_t=a)\mid H_t]=SW_{t-1}q_t(a\mid Q_t)\), because the future product has conditional expectation one and \(P(A_t=a\mid H_t)=g_t(a\mid H_t)\). Summing this expression over \(a\) gives \(E[SW_K\mid H_t]=SW_{t-1}\). Taking the ratio proves the claimed conditional law wherever this denominator is positive, which is precisely the relevant support under \(P^W\). If \(H_t=(Q_t,R_t)\), that conditional law does not depend on \(R_t\), establishing the stated conditional independence. \(\square\)

Thus, a component of \(H_t\) omitted from \(Q_t\) no longer predicts \(A_t\) conditional on \(Q_t\) in the weighted population. A component included in \(Q_t\) may still predict treatment, according to the numerator law. “May” is important: a variable’s presence in \(Q_t\) permits dependence but does not guarantee it. Nor does conditional independence imply marginal independence; dependence can arise through \(Q_t\).

Preserving Treatment-History Dependence

A common numerator choice lets treatment depend on prior treatment history, for example \(Q_t=\bar A_{t-1}\), where \(\bar A_{t-1}=(A_0,\ldots,A_{t-1})\). This removes dependence on other components of \(H_t\) conditional on prior treatment, but it does not generally make treatment independent of prior treatment. That retained dependence is deliberate: it allows the weighted treatment process to preserve a specified treatment-history law.

This is the link to the marginal structural model (MSM). An MSM describes a marginal mean of potential outcomes as a function of treatment history, such as \(E[Y^{\bar a}]\), where \(Y^{\bar a}\) is the outcome under regime \(\bar a\). When the numerator depends only on prior treatments, the numerator probability of a fixed treatment trajectory is a constant across individuals. That constant cancels from the weighted mean for the trajectory, leaving the usual marginal potential-outcome mean.

To verify the result, on the event \(\bar A_K=\bar a_K\), the numerator part of the weight equals \(c(\bar a_K)\), so the numerator of the ratio is \(c(\bar a_K)E[I(\bar A_K=\bar a_K)Y/\prod_{t=0}^K g_t(a_t\mid H_t)]\). Sequential exchangeability and consistency give the inverse-probability identity \(E[I(\bar A_K=\bar a_K)Y/\prod_{t=0}^K g_t(a_t\mid H_t)]=E[Y^{\bar a_K}]\): iterated conditioning cancels each treatment probability against its corresponding indicator, and consistency identifies the observed outcome with the regime-specific outcome among those following the regime. Applying the same cancellation to the denominator gives \(E[I(\bar A_K=\bar a_K)/\prod_{t=0}^K g_t(a_t\mid H_t)]=1\). The denominator of the displayed ratio is therefore \(c(\bar a_K)\), and cancellation proves the claim. The explicit requirement \(c(\bar a_K)>0\) ensures the ratio exists. \(\square\)

Matching the Numerator to the Outcome Model

The cancellation above relies on \(c(\bar a_K)\) being constant across individuals. If a numerator depends on an additional variable, its product along a regime can vary across individuals. The stabilized weighted outcome contrast can then average \(Y^{\bar a_K}\) with different weights on that variable rather than equal the original population mean. In that setting, the outcome-model target and the population over which it is interpreted must be stated explicitly. Including numerator-history variables in a conditional outcome model, or standardizing predictions to a specified distribution, may be appropriate, but neither step is automatic.

The practical rule is therefore not “include as much history as possible” or “always use the shortest numerator.” First identify the outcome estimand. Then choose a numerator law whose retained treatment associations are compatible with that estimand and with the outcome model. A numerator that retains prior-treatment dependence is compatible with an MSM indexed by treatment history when the numerator probability for each history is constant across individuals; broader numerator histories require additional care about the target.

Diagnostics and Common Pitfalls

• Interpreting stabilization as randomization. The weighted treatment law is \(q_t(\cdot\mid Q_t)\), not necessarily a constant probability. Treatment can remain associated with numerator-history variables.
• Ignoring future factors. To derive a conditional law under the full product \(SW_K\), future weight factors cannot simply be dropped. Their conditional expectation is one by sequential conditioning and normalization of each numerator law.
• Assuming every regime is estimable because \(g_t\) is positive. A zero numerator probability makes the stabilized regime-specific ratio undefined. Check numerator support as well as denominator positivity.
• Calling a weighted contrast marginal without checking its averaging distribution. If the numerator product varies across individuals, the cancellation to the original marginal mean need not hold.

Useful diagnostics include tabulating the numerator probabilities by the history variables they use, checking that the intended regimes have positive numerator support, and examining whether treatment remains associated with omitted history variables after weighting, conditional on \(Q_t\). These checks address the specified weighted treatment law; they do not by themselves establish exchangeability, correct treatment models, or a valid outcome model.