Sensitivity to Treatment-Model Misspecification

Why the Treatment Model Matters

G-estimation uses the treatment assignment process to construct moments that identify blip parameters. In a single-decision setting, these moments commonly compare the blipped-down outcome across treatment groups after accounting for the covariate history that influenced treatment. If the treatment-probability model is wrong, the residual used in that comparison may no longer have conditional mean zero. The resulting estimating equation can therefore target a value different from the true blip parameter, even when the blip function itself is correctly specified.

This issue is distinct from blip-function misspecification, the concern addressed in the earlier material on Diagnostics for Blip-Function Misspecification. Here, take the blip form as correct and ask how much the estimate depends on modeling treatment assignment. The calculations below use one treatment decision to make the mechanism explicit; the same concern applies to sequential g-estimation, where treatment probabilities are modeled conditional on histories at each decision.

Setup and the Misspecified Moment

Let \(H\) be the covariate history immediately before a binary treatment \(A\), and let \(Y\) be the outcome. Assume consistency, positivity, and conditional exchangeability: within levels of \(H\), treatment assignment is independent of the relevant untreated potential outcome. For a correctly specified additive structural mean model, write the blip as \(A\gamma(H;\psi_0)\), where \(\psi_0\) is the true parameter. The blipped-down outcome at a candidate value \(\psi\) is

U_\psi=Y-A\gamma(H;\psi).

At \(\psi=\psi_0\), \(U_{\psi_0}\) has the conditional mean of the untreated potential outcome, denoted \(m(H)=E[U_{\psi_0}\mid H]\). Conditional exchangeability implies that this mean does not depend on \(A\) once \(H\) is fixed. Let the true treatment probability be \(e(H)=P(A=1\mid H)\). If the fitted treatment model converges to \(p^*(H)\), its treatment residual is \(A-p^*(H)\), not necessarily \(A-e(H)\).

Definition: Treatment-model misspecification occurs when the probability model used in g-estimation does not represent the true conditional treatment probability \(e(H)\). In a large sample, a fitted misspecified model may converge to a function \(p^*(H)\neq e(H)\). The population g-estimation equation formed using \(p^*\) can then have a root different from \(\psi_0\); that root is the equation’s pseudo-true parameter.

The next identity isolates the issue. It applies to the usual residual-treatment moment at the true blip value.

Result: Under the conditional mean restriction \(E[U_{\psi_0}\mid H,A]=m(H)\),

E\left[(A-p^*(H))U_{\psi_0}\right] = E\left[(e(H)-p^*(H))m(H)\right].

If \(p^*(H)=e(H)\) almost surely, the moment is zero. If they differ, it is generally not zero; it can still equal zero through cancellation across histories.

Proof. Condition on \(H\). By the stated restriction, \(E[U_{\psi_0}\mid H,A]=m(H)\), so \(E[(A-p^*(H))U_{\psi_0}\mid H]\) equals \(m(H)E[A-p^*(H)\mid H]\). Since \(E[A\mid H]=e(H)\) and \(p^*(H)\) is a function of \(H\), this is \(m(H)(e(H)-p^*(H))\). Taking expectations over \(H\) proves the identity. \(\square\)

How the Misspecified Equation Shifts the Estimate

For a constant additive blip, \(\gamma(H;\psi)=\psi\), suppose \(Y=U_{\psi_0}+A\psi_0\). Then \(U_\psi=U_{\psi_0}+A(\psi_0-\psi)\). Substituting this expression into the population moment yields an explicit root.

Result: If the denominator below is nonzero, the population equation \(E[(A-p^*(H))U_\psi]=0\) has root

\psi^*=\psi_0+ \frac{E[(e(H)-p^*(H))m(H)]} {E[e(H)\{1-p^*(H)\}]}.

This root equals \(\psi_0\) when the numerator is zero. Otherwise, the misspecified treatment model shifts the target.

Derivation. At any fixed \(H\), the binary treatment satisfies \(A^2=A\), so

E[(A-p^*(H))A\mid H] =E[A-p^*(H)A\mid H] =e(H)\{1-p^*(H)\}.

Using the previous identity for the first part of \(U_\psi\), the full moment becomes

E[(A-p^*(H))U_\psi] = E[(e(H)-p^*(H))m(H)] +(\psi_0-\psi)E[e(H)\{1-p^*(H)\}].

Setting this equal to zero and solving for \(\psi\) gives the stated root. The denominator must be positive for this particular moment to identify a unique root; for example, it is positive if \(0<e(H)\) and \(p^*(H)<1\) on a set of positive probability. This algebra does not assert that every misspecified estimating procedure has this exact scalar form. It shows directly how a common residual-treatment moment can shift.

Worked Example: A One-Stratum Model With a Biased Root

Suppose there is no variation in \(H\), the true treatment probability is \(e=0.60\), the fitted model converges to \(p^*=0.40\), and the untreated outcome has mean \(m=2\). Let the true additive blip be \(\psi_0=3\).

The numerator of the root shift is

$$ (e-p^*)m=(0.60-0.40)(2)=0.40. $$

The denominator is

$$ e(1-p^*)=(0.60)(0.60)=0.36. $$

Therefore,

\psi^*=3+\frac{0.40}{0.36} =3+\frac{10}{9} =\frac{37}{9} \approx 4.11.

The population equation targets approximately 4.11 rather than the true blip of 3. Here, the misspecified residual-treatment moment is nonzero at the true blip: \(E[(A-p^*)U_{\psi_0}]=0.40\). The fitted treatment probability is not merely an auxiliary estimate; its error changes the moment used to identify the blip.

When Errors Cancel—and When They Do Not

A wrong treatment model does not force bias in every particular distribution. The numerator \(E[(e(H)-p^*(H))m(H)]\) may be zero because positive and negative contributions from different histories cancel. Such cancellation is a property of the joint distribution of the treatment-model error and the untreated-outcome mean; it is not evidence that \(p^*(H)\) is a correct propensity model.

Worked Example: Cancellation Across Two Histories

Let \(H\) take two values with probability \(1/2\) each. Suppose the true probabilities are \(e(0)=0.80\) and \(e(1)=0.20\), but the fitted model assigns \(p^*(0)=p^*(1)=0.50\). Let \(m(0)=1\), \(m(1)=4\).

The numerator of the root shift is

\begin{aligned} E[(e-p^*)m] &=\tfrac12(0.80-0.50)(1) +\tfrac12(0.20-0.50)(4)\\ &=\tfrac12(0.30)+\tfrac12(-1.20)\\ &=0.15-0.60=-0.45. \end{aligned}

The denominator is

\begin{aligned} E[e(1-p^*)] &=\tfrac12(0.80)(0.50)+\tfrac12(0.20)(0.50)\\ &=0.20+0.05=0.25. \end{aligned}

Thus \(\psi^*=\psi_0-0.45/0.25=\psi_0-1.80\). The misspecification has a substantial effect because the two history-specific contributions do not cancel. If \(m(0)=m(1)=2\) instead, the numerator would be \(\tfrac12(0.30)(2)+\tfrac12(-0.30)(2)=0\). The root would then equal \(\psi_0\) for this moment, despite the treatment model still being wrong.

Worked Example: Comparing Candidate Treatment Models

Suppose two histories are equally common, with \(e(0)=0.70\), \(e(1)=0.30\), \(m(0)=1\), \(m(1)=3\), and \(\psi_0=2\). Compare three candidate fitted treatment probabilities.

For the constant candidate \(p^*(0)=p^*(1)=0.50\), the numerator is \(\tfrac12(0.20)(1)+\tfrac12(-0.20)(3)=-0.20\), and the denominator is \(\tfrac12(0.70)(0.50)+\tfrac12(0.30)(0.50)=0.25\). Hence the shift is \(-0.20/0.25=-0.80\), giving \(\psi^*=1.20\).

For the candidate \(p^*(0)=0.60\), \(p^*(1)=0.40\), the numerator is \(\tfrac12(0.10)(1)+\tfrac12(-0.10)(3)=-0.10\). The denominator is \(\tfrac12(0.70)(0.40)+\tfrac12(0.30)(0.60)=0.23\). Thus \(\psi^*=2-0.10/0.23\approx1.57\).

For the correctly specified probabilities \(p^*(0)=0.70\), \(p^*(1)=0.30\), the numerator is zero and the root is \(\psi^*=2\). These calculations show how estimates can move across plausible treatment-model specifications. In an applied analysis, the true \(e\) and \(m\) are unknown; this calculation illustrates the mechanism, not a recipe for computing the exact bias from observed data alone.

Assessing Sensitivity in Practice

A sensitivity analysis compares g-estimates across a defensible set of treatment models. Candidate specifications might differ in functional form, interactions, or smooth terms for pre-treatment covariates. The goal is not to select the model producing the most appealing blip estimate. It is to determine whether substantive conclusions depend strongly on modeling choices that the observed data cannot settle with certainty.

Specify credible alternatives.
Use covariates that are relevant to treatment assignment and available before treatment. Vary functional forms or interactions where there is a substantive reason to do so.

Refit and re-estimate.
For each candidate, fit the treatment model and recompute the g-estimation equation and blip estimate. Keep the outcome and blip specification fixed if the purpose is to isolate treatment-model sensitivity.

Compare estimates and diagnostics.
Report the range of estimates, their uncertainty, and treatment-model diagnostics such as fitted-probability distributions and calibration against observed treatment frequencies.

Interpret dependence, not just fit.
Large changes indicate sensitivity to treatment-model choices. Similar estimates are reassuring about this particular set of alternatives, but do not prove that all relevant treatment-model errors are harmless.

In longitudinal settings, the treatment probability is conditional on the history available at each decision. The same principle applies at each stage: a residual formed with a misspecified conditional probability need not have conditional mean zero. Sensitivity analyses should therefore vary the history-dependent treatment models that enter the relevant sequential g-estimation moments, rather than treating the treatment process as a single baseline regression.

Diagnostics, Alternatives, and Limitations

Useful diagnostics include checking whether fitted probabilities are systematically inconsistent with observed treatment frequencies across covariate strata, whether important nonlinearities or interactions have been omitted, and whether estimates change materially under reasonable alternative specifications. Calibration and residual checks can reveal some problems, but no diagnostic can establish that the treatment model is correct in every relevant conditional detail.

Several issues should not be conflated. Blip-function misspecification concerns the form of \(\gamma(H;\psi)\); treatment-model sensitivity does not repair it. Positivity violations occur when treatment options have zero or extremely small probability in parts of the history space. A flexible model cannot create information where treatment groups are absent, and extreme probabilities can make estimates unstable even if the model is correctly specified. Finally, a treatment model may be correctly estimated within a chosen class yet that class may omit important features of the true assignment mechanism.

Approach	What it addresses	Important limitation
Alternative treatment-model specifications	Dependence of estimates on functional form and included terms	Only probes the alternatives actually considered
Blip-function diagnostics	Whether the structural effect model is adequate	Does not establish that the treatment model is correct
Outcome-model or augmented approaches	Can use additional outcome structure in some procedures	Robustness depends on the specific estimator and its assumptions; it is not automatic for ordinary g-estimation
Randomization or known treatment probabilities	Removes the need to estimate an unknown assignment probability in the relevant design	Usually unavailable in observational treatment assignment

A common mistake is to argue that the g-estimate must be safe because a fitted treatment model predicts treatment reasonably well overall. Overall fit does not guarantee the conditional calibration needed for the g-estimation residual: errors can vary across \(H\), and the moment weights those errors by \(m(H)\). Another mistake is to interpret stability across a few candidate models as proof of no bias. Stability is evidence about robustness to those choices, not a guarantee against unexamined misspecification, unmeasured confounding, or positivity failure.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions below.

State the conditional expectation of \(A-p^*(H)\) given \(H\), and explain why it need not be zero.
In the constant-blip setup, what condition makes the misspecified population root equal to \(\psi_0\), assuming the denominator is nonzero?
Suppose \(e=0.50\), \(p^*=0.30\), \(m=5\), and \(\psi_0=1\). Calculate the population root.
Why does a zero aggregate numerator not establish that the treatment model is correctly specified?
Name one diagnostic limitation of comparing alternative treatment-model specifications.

Solutions. (1) \(E[A-p^*(H)\mid H]=e(H)-p^*(H)\), which is nonzero wherever the fitted limiting probability differs from the true conditional probability. (2) The numerator must vanish: \(E[(e(H)-p^*(H))m(H)]=0\). This can hold by cancellation even if \(p^*\neq e\). (3) The numerator is \((0.50-0.30)(5)=1\), and the denominator is \(0.50(1-0.30)=0.35\). Thus \(\psi^*=1+1/0.35=1+20/7=27/7\approx3.86\). (4) Positive and negative history-specific contributions can cancel in the expectation while the conditional probabilities remain wrong. (5) Stability only covers the candidate models examined; it cannot rule out misspecification shared by all of them or errors in features not considered.