Stabilized Weights and Their Numerator Models

Why Stabilize Longitudinal Treatment Weights?

In a longitudinal study, treatment at each visit may depend on a history of measured covariates and earlier treatments. The longitudinal treatment weight corrects for this treatment assignment using the probability of the observed treatment given that history. When several such probabilities are multiplied, the resulting weights can be highly variable. Stabilization replaces each denominator probability with a ratio: the numerator describes a simpler reference treatment mechanism, and the denominator accounts for the measured history that drives treatment assignment.

The numerator is not just a mathematical convenience. It specifies how treatment is allowed to vary in the weighted population. Its predictors should therefore be chosen together with the marginal structural model (MSM) and the causal contrast being reported. A numerator that is too simple or inconsistent with the outcome model can leave unwanted associations in the weighted data; a more elaborate numerator may preserve selected associations and require the MSM to account for them.

Definition: Let $A_t$ denote treatment at visit $t$, $V$ baseline covariates, and $L_t$ covariates measured before treatment at visit $t$. Write $\bar A_t=(A_1,\ldots,A_t)$ and $\bar L_t=(L_1,\ldots,L_t)$. For an individual who received treatment history $\bar A_K$, the stabilized treatment weight through visit $K$ is $$ SW_K=\prod_{t=1}^{K} \frac{q_t(A_t\mid \bar A_{t-1},V)} {g_t(A_t\mid \bar A_{t-1},V,\bar L_t)}, $$ where $g_t$ is the denominator treatment probability and $q_t$ is the numerator treatment probability. The numerator may use a selected subset of information in the denominator history. The unstabilized weight is the same product with numerator probabilities equal to 1.

For binary treatment, each probability is the probability of the treatment value actually observed. Thus, if $A_t=1$, the factor uses $P(A_t=1\mid\cdots)$; if $A_t=0$, it uses $P(A_t=0\mid\cdots)$. For a treatment with several categories, use the probability of the observed category. This tutorial concerns discrete treatment; continuous treatment requires a density rather than a probability.

What Belongs in the Numerator and Denominator?

The denominator model represents treatment assignment conditional on the measured history needed to address confounding. At visit $t$, that history includes earlier treatment, baseline covariates, and the time-varying covariates measured before $A_t$. In particular, $L_t$ belongs in the denominator when it predicts treatment and is needed for sequential exchangeability. Omitting such a variable can leave confounding unaddressed.

The numerator usually includes earlier treatment and may include baseline covariates. Earlier treatment matters because the probability of treatment at a later visit can depend on earlier treatment. Baseline covariates may be included when the intended MSM conditions on them, or when preserving their association with treatment is useful for the chosen analysis. The numerator is generally simpler than the denominator: it does not include the time-varying confounder history that the weighting is intended to remove from treatment assignment.

A useful way to organize the choice is to ask what the weighted treatment mechanism should be. With $q_t(A_t\mid\bar A_{t-1})$, treatment in the weighted population follows a mechanism depending on prior treatment but not on measured covariates. With $q_t(A_t\mid\bar A_{t-1},V)$, the mechanism can still depend on baseline $V$. The latter can be appropriate when the MSM includes $V$; it should not be treated as if weighting had made treatment independent of $V$.

Modeling principle: Specify the denominator to control measured confounding, and specify the numerator to define a manageable reference treatment mechanism compatible with the MSM. A variable in the numerator is not automatically removed from the weighted treatment assignment; if the numerator retains its association with treatment, the outcome model or standardization must handle that feature of the target.

The usual causal identification conditions remain essential. Consistency links observed outcomes to the potential outcomes under the treatment actually received. Sequential exchangeability requires that, at each visit, treatment be independent of future potential outcomes conditional on the measured past. Positivity requires that treatment options relevant to the target have positive probability within the histories represented in the target population. The denominator models must adequately estimate the relevant conditional treatment probabilities. Stabilization does not repair violations of these conditions.

Why the Ratio Changes the Weighted Population

The role of the numerator follows directly from the likelihood ratio. Suppose the observed data density factors into the baseline distribution, covariate distributions over time, and treatment probabilities $g_t$. Multiplying by $SW_K$ cancels the $g_t$ factors and replaces them with $q_t$. In a discrete-data notation, the weighted joint density is proportional to

f(V)\prod_{t=1}^{K} f(L_t\mid V,\bar L_{t-1},\bar A_{t-1}) g_t(A_t\mid V,\bar L_t,\bar A_{t-1}) \frac{q_t(A_t\mid V,\bar A_{t-1})} {g_t(A_t\mid V,\bar L_t,\bar A_{t-1})} = f(V)\prod_{t=1}^{K} f(L_t\mid V,\bar L_{t-1},\bar A_{t-1}) q_t(A_t\mid V,\bar A_{t-1}).

This cancellation is the key justification: under correct treatment models, the weighted data behave as if treatment had been assigned according to the numerator mechanism, while the covariate history continues to evolve under the treatments received. The ratio does not erase the covariates from the data or change an individual's potential outcomes. It changes the treatment-assignment mechanism represented by the weighted data.

This derivation also explains why the numerator and outcome model must be considered together. If $q_t$ depends on $V$, the weighted treatment assignment still depends on $V$. A treatment-only MSM may then compare groups with different distributions of $V$, even if the denominator model is correct. An MSM that conditions on $V$, followed by standardization to the chosen baseline population when a marginal contrast is wanted, can account for that dependence.

Worked Examples

Worked Example: Multiplying Two Stabilized Factors

Consider a hypothetical two-visit study with binary treatment. One participant received $A_1=1$, then $A_2=0$. Suppose the fitted numerator probabilities for this observed history are $q_1(A_1=1)=0.40$ and $q_2(A_2=0\mid A_1=1)=0.60$. The corresponding denominator probabilities, conditional on the participant's measured covariate histories, are $g_1(A_1=1)=0.80$ and $g_2(A_2=0\mid A_1=1)=0.30$.

The stabilized weight is the product of the two ratios:

SW_2= \frac{0.40}{0.80}\times\frac{0.60}{0.30} =0.50\times2.00 =1.00.

For comparison, the unstabilized weight is $1/0.80\times1/0.30=4.17$. The stabilized weight is smaller here because the numerator factors offset the denominator factors. A weight of 1.00 does not mean that the participant had no confounding or that the probability of their observed history was 1.00; it is a product of probability ratios. In another history, the stabilized weight could be above or below 1.

Worked Example: Adding a Baseline Covariate to the Numerator

Suppose a hypothetical one-visit study has a binary baseline covariate $V$. The denominator probabilities of treatment are $g(A=1\mid V=0)=0.30$ and $g(A=1\mid V=1)=0.90$. Consider two possible numerator models: a marginal model with $q(A=1)=0.40$, and a baseline-adjusted model with $q(A=1\mid V=0)=0.20$ and $q(A=1\mid V=1)=0.60$. The table gives weights for both possible observed treatment values.

Baseline group	Observed treatment	Marginal-numerator weight	Baseline-adjusted-numerator weight
$V=0$	$A=1$	$0.40/0.30=1.33$	$0.20/0.30=0.667$
$V=0$	$A=0$	$0.60/0.70=0.857$	$0.80/0.70=1.14$
$V=1$	$A=1$	$0.40/0.90=0.444$	$0.60/0.90=0.667$
$V=1$	$A=0$	$0.60/0.10=6.00$	$0.40/0.10=4.00$

For example, a participant with $V=1$ who was untreated receives weight $0.40/0.10=4.00$ under the baseline-adjusted numerator. That factor uses the numerator probability of the observed value $A=0$, namely $1-0.60=0.40$. In both models, the large untreated weight in the $V=1$ group reflects the small denominator probability of being untreated there.

The numerator choice changes the weights, but does not guarantee that one set has lower variability: the largest weights are still substantial under either model. More importantly, the baseline-adjusted numerator allows treatment assignment in the weighted data to vary with $V$; the marginal numerator specifies a treatment probability that does not vary with $V$. The outcome model must match that choice if the goal is a marginal causal contrast.

Worked Example: Why a Numerator Covariate Can Change a Treatment-Only Contrast

Suppose, hypothetically, that $V=0$ and $V=1$ each make up half the target population. The mean potential outcomes are $E(Y^1\mid V=0)=10$, $E(Y^1\mid V=1)=30$, $E(Y^0\mid V=0)=8$, and $E(Y^0\mid V=1)=20$, measured in outcome units. The target-population means are therefore

E(Y^1)=0.50(10)+0.50(30)=20,\qquad E(Y^0)=0.50(8)+0.50(20)=14,

so the marginal causal contrast is $20-14=6$ outcome units. Now suppose the numerator model retains $V$, with $q(A=1\mid V=0)=0.20$ and $q(A=1\mid V=1)=0.60$. In the weighted population, the proportion with $V=1$ among treated participants is $0.50(0.60)/[0.50(0.20)+0.50(0.60)]=0.30/0.40=0.75$. Among untreated participants, it is $0.50(0.40)/[0.50(0.80)+0.50(0.40)]=0.20/0.60=0.333$.

A treatment-only comparison in that weighted population would have treated mean $0.25(10)+0.75(30)=25$, and untreated mean $0.667(8)+0.333(20)=12.0$. Its contrast is $25-12.0=13.0$ outcome units, not the target marginal contrast of 6. The discrepancy arises because the two treatment groups have different $V$ distributions under this numerator mechanism, and the outcome means vary with $V$.

An MSM that includes $V$ can represent the treatment and baseline-covariate contributions separately. For the specified population, standardizing its conditional predictions to the 50%-50% distribution of $V$ gives the marginal means 20 and 14, and hence the contrast of 6. Alternatively, a numerator that does not depend on $V$ makes treatment assignment independent of $V$ in the weighted population under the stated model, so a treatment-only contrast can match this marginal target in this example. The appropriate choice depends on the intended MSM and target—not on a rule that every numerator must omit baseline covariates.

Precision, Model Choice, and Practical Checks

Stabilization can improve precision because numerator probabilities can offset part of the variability caused by small denominator probabilities. This is not a guarantee. If some observed treatment histories have very small denominator probabilities, their ratios may remain large even after stabilization. A numerator that is poorly chosen or poorly estimated can also yield unstable weights. Weight distributions and positivity therefore still require scrutiny.

The numerator model is a reference mechanism, not a substitute for the denominator model. It should use information available before the treatment decision, and its conditioning variables should be compatible with the MSM. In longitudinal settings, prior treatment is usually included in the numerator at later visits. Baseline covariates may be included when the MSM retains them. Including a time-varying covariate in the numerator means that the weighted treatment mechanism can still depend on that covariate; if the MSM omits it, the resulting weighted comparisons may not represent the desired marginal contrast.

The likelihood-ratio argument assumes the denominator probabilities are correctly specified and that the observed treatment histories have support under the numerator mechanism. In applied work, both numerator and denominator models are estimated, so the calculated weights are estimates rather than known quantities. A fitted numerator should assign positive probability to treatment values that occur in the target histories, and the denominator should not assign near-zero probability to observed treatment values in histories needed for the target. Choosing a numerator cannot create positivity where the data provide none.

Common Mistakes

Putting only confounders in the numerator. The denominator is where the measured history needed to address treatment confounding belongs. The numerator instead defines the reference treatment mechanism.
Assuming a stabilized weight must be near 1. A ratio product may be large or small. Stabilization can reduce variability, but it does not bound weights or fix near-violations of positivity.
Including baseline covariates in the numerator and then ignoring them in the MSM. If the numerator lets treatment depend on those covariates, a treatment-only weighted contrast may compare groups with different covariate distributions. Include the covariates in the outcome model or use a numerator and analysis aligned with the marginal target.
Interpreting the numerator as changing the causal effect itself. The potential outcomes and causal estimand are defined by the target population and treatment strategies. The numerator changes the weighted treatment mechanism and can change the interpretation of an inadequately specified weighted model.

Key takeaway: A stabilized longitudinal weight is a product of numerator-to-denominator treatment-probability ratios. The denominator adjusts for measured treatment history; the numerator defines the treatment mechanism retained in the weighted data. Choose numerator predictors in concert with the MSM and the population over which the causal contrast will be interpreted.

Check Your Understanding

Use these questions to check whether the probability ratios and their interpretation are clear.

At a visit where a participant receives $A_t=0$, which treatment probability is used in the numerator and denominator factors?
Why can a baseline covariate appear in the numerator while a time-varying confounder appears in the denominator?
If the numerator depends on baseline $V$, what feature of the weighted treatment mechanism does that retain?
Does stabilization guarantee that no weight is large? Explain why or why not.
What must be aligned when choosing numerator predictors: only the denominator model, or also the MSM and target population?

Baseline group	Observed treatment	Marginal-numerator weight	Baseline-adjusted-numerator weight
\(V=0\)	\(A=1\)	\(0.40/0.30=1.33\)	\(0.20/0.30=0.667\)
\(V=0\)	\(A=0\)	\(0.60/0.70=0.857\)	\(0.80/0.70=1.14\)
\(V=1\)	\(A=1\)	\(0.40/0.90=0.444\)	\(0.60/0.90=0.667\)
\(V=1\)	\(A=0\)	\(0.60/0.10=6.00\)	\(0.40/0.10=4.00\)