Why Curvature Matters in Treatment Assignment
In an antihypertensive cohort, treatment may depend on more than whether blood pressure is high. Clinicians may treat patients at both unusually low and high values differently from those near a reference level. Kidney function can likewise influence prescribing through nonlinear clinical decisions. If the treatment model represents these relationships as straight lines, it can assign incorrect treatment probabilities even when the relevant variables are measured and included.
The consequence is not merely a less realistic description of prescribing. Inverse-probability weights are calculated from the estimated treatment probabilities. Incorrect probabilities can therefore leave treatment groups with different distributions of prognostic blood pressure or kidney function, distorting the estimated marginal stroke contrast.
For this case, let \(A\in\{0,1\}\) indicate antihypertensive treatment at baseline, let \(L=(B,K)\) denote pretreatment blood pressure and kidney-function covariates, and let \(Y\) indicate stroke during a specified follow-up period. Write \(Y^a\) for the potential stroke outcome if treatment were set to \(a\). The target is the marginal risk difference \(E[Y^1]-E[Y^0]\) in the cohort.
For a one-time baseline treatment, a common propensity model is logistic regression. A linear specification uses a linear predictor such as \(\alpha+\beta_B B+\beta_K K\). A model allowing curvature might instead include \(B^2\) and \(K^2\), or use spline basis functions for the two variables. Interactions can be included when clinically justified. The functional form is a substantive modeling choice: including a variable by name does not guarantee that the model captures how it relates to treatment.
Worked Example: A Curved Propensity Model Changes Individual Weights
Suppose a fitted model uses standardized blood pressure \(B\) and kidney-function measure \(K\), with \(\operatorname{logit}\{\hat e(B,K)\}=-1.4+1.4B^2+K^2+0.6BK\). Calculate the predicted treatment probabilities and unstabilized weights for three profiles: \((0,0)\), \((1,0)\), and \((1,1)\).
At \((B,K)=(0,0)\), the linear predictor is \(-1.4\), so \(\hat e=1/(1+\exp(1.4))\approx0.198\). The treated weight is \(1/0.198\approx5.05\), and the untreated weight is \(1/(1-0.198)\approx1.25\).
At \((1,0)\), the predictor is \(-1.4+1.4=0\). Thus \(\hat e=0.5\), and either treatment arm has weight \(2\).
At \((1,1)\), the predictor is \(-1.4+1.4+1+0.6=1.6\). Thus \(\hat e=1/(1+\exp(-1.6))\approx0.832\). The treated weight is about \(1.20\); the untreated weight is \(1/(1-0.832)\approx5.95\).
The same observed covariates enter differently depending on their values: the squared terms allow treatment probability to change nonlinearly, while the interaction changes it when both measures are elevated. These are model-implied probabilities, not proof that this particular specification is appropriate.
When Weighting Identifies the Marginal Risks
The identification argument requires assumptions about the treatment process and outcomes, not a particular regression family. For this baseline-treatment example, assume consistency, conditional exchangeability, and positivity. Consistency means \(Y=Y^A\); conditional exchangeability means \(Y^a\) is independent of \(A\) given \(L\), for each \(a\); and positivity means \(P(A=a\mid L)>0\) for each arm at covariate values in the target population. Also assume no interference between individuals.
Under these assumptions, the inverse-probability-weighted outcome has a simple expectation. The result explains why the propensity model must estimate the conditional treatment probabilities adequately: the identity uses the true \(e_a(L)\), not an arbitrary fitted score.
Proof. Condition first on \(L\). By consistency, on the event \(A=a\), the observed outcome is \(Y^a\). Conditional exchangeability gives \(E[Y^a\mid A=a,L]=E[Y^a\mid L]\). Therefore,
Positivity makes the denominator nonzero. Taking expectations over \(L\) gives the first identity by iterated expectation. Multiplying its integrand by the constant \(P(A=a)\) gives the stabilized-weight identity. \(\square\)
In a sample, a normalized weighted risk estimator is \(\hat\mu_a=\sum_i I(A_i=a)\hat w_iY_i/\sum_i I(A_i=a)\hat w_i\), where \(\hat w_i\) is the stabilized weight for the observed treatment. With the true propensity score, the numerator and denominator target \(P(A=a)E[Y^a]\) and \(P(A=a)\), respectively. Under independent sampling and suitable finite-moment conditions, the law of large numbers then gives the target risk. With estimated scores, this conclusion additionally requires the fitted probabilities to consistently estimate the relevant treatment probabilities, or other valid conditions for the chosen estimator.
Worked Example: A Misspecified Linear Model Leaves Confounding
Consider five equally common covariate profiles. \(B\) is standardized blood pressure and \(K\) is a centered kidney-function measure. The true treatment probability is \(0.8\) at the four profiles with \(B^2+K^2=1\), and \(0.2\) at \((0,0)\). Suppose the potential stroke risks are \(0.1\) at each of the four outer profiles and \(0.3\) at the center under no treatment. Treatment lowers each risk by \(0.05\).
| Profile | \(B\) | \(K\) | \(P(A=1\mid B,K)\) | \(E[Y^0\mid B,K]\) | \(E[Y^1\mid B,K]\) |
|---|---|---|---|---|---|
| 1 | -1 | 0 | 0.8 | 0.1 | 0.05 |
| 2 | 1 | 0 | 0.8 | 0.1 | 0.05 |
| 3 | 0 | -1 | 0.8 | 0.1 | 0.05 |
| 4 | 0 | 1 | 0.8 | 0.1 | 0.05 |
| 5 | 0 | 0 | 0.2 | 0.3 | 0.25 |
The true propensity is represented on these profiles by \(\operatorname{logit}\{e(B,K)\}=\operatorname{logit}(0.2)+[\operatorname{logit}(0.8)-\operatorname{logit}(0.2)](B^2+K^2)\). Since \(\operatorname{logit}(0.8)-\operatorname{logit}(0.2)=\log(16)\), this gives 0.2 at the center and 0.8 at each outer profile. The potential-outcome risk difference is \(-0.05\) in every profile, so its marginal value is also \(-0.05\).
Without weighting, the treated group has total profile probability \(4(0.2)(0.8)+(0.2)(0.2)=0.68\). Its observed stroke-risk numerator is \(4(0.2)(0.8)(0.05)+(0.2)(0.2)(0.25)=0.042\), giving risk \(0.042/0.68\approx0.0618\). The untreated group has probability \(4(0.2)(0.2)+(0.2)(0.8)=0.32\), and risk numerator \(4(0.2)(0.2)(0.1)+(0.2)(0.8)(0.3)=0.064\). Its risk is \(0.064/0.32=0.20\). The unweighted contrast is therefore about \(-0.1382\), not \(-0.05\).
The center profile has higher stroke risk and is less likely to receive treatment. That creates confounding even though the treatment effect is constant across profiles. A linear treatment model with only \(B\) and \(K\) cannot represent the symmetric U-shaped propensity pattern in this example. In the population, symmetry makes its two slope scores zero, while its intercept predicts the overall treatment probability \(0.68\); it consequently assigns the same propensity to every profile and does not correct the selection.
Worked Example: Correct Weights Recover the Marginal Contrast
Continue with the five-profile cohort. The marginal treatment probability is \(P(A=1)=0.68\), so \(P(A=0)=0.32\). The stabilized treated weights are \(0.68/0.8=0.85\) at the four outer profiles and \(0.68/0.2=3.4\) at the center. The stabilized untreated weights are \(0.32/0.2=1.6\) at the outer profiles and \(0.32/0.8=0.4\) at the center.
For treated individuals, the total weighted population mass from the outer profiles is \(4(0.2)(0.8)(0.85)=0.544\), and the center contributes \((0.2)(0.2)(3.4)=0.136\). The total is \(0.68\). The weighted outcome numerator is \(0.544(0.05)+0.136(0.25)=0.0272+0.034=0.0612\). Thus the weighted treated risk is \(0.0612/0.68=0.09\).
For untreated individuals, the outer profiles contribute \(4(0.2)(0.2)(1.6)=0.256\), and the center contributes \((0.2)(0.8)(0.4)=0.064\), for a total of \(0.32\). The weighted outcome numerator is \(0.256(0.1)+0.064(0.3)=0.0256+0.0192=0.0448\), so the weighted untreated risk is \(0.0448/0.32=0.14\). The stabilized weighted risk difference is \(0.09-0.14=-0.05\), matching the marginal contrast.
The example also shows why the weights alter the covariate mix: they increase the relative contribution of the center profile in the treated group and decrease it in the untreated group. Stabilization changes the scale of weights but, within each treatment arm, its constant numerator cancels in the normalized risk.
Model Assessment, Alternatives, and Limitations
A practical analysis should compare clinically credible specifications, not select curvature solely because it improves an in-sample fit. Inspect treatment frequencies across the observed ranges of blood pressure and kidney function; look for curved patterns in treatment assignment; and compare a linear propensity model with models using prespecified quadratic terms or restricted spline bases. Predicted probabilities that fail to follow evident nonlinear patterns, or that are implausible in clinically important regions, are warning signs. Sparse covariate regions may make flexible fits unstable, even when the overall fit looks adequate.
A useful model check is whether the fitted probabilities plausibly reproduce treatment assignment across covariate ranges and combinations, including nonlinear terms. Also inspect the resulting weight distribution and the effective representation of covariate profiles. These checks do not prove exchangeability or correct specification. If alternative reasonable forms give materially different marginal contrasts, report that sensitivity rather than presenting one specification as certain.
The main alternatives serve different purposes. A linear propensity model is compact, but can miss curvature. Adding polynomial terms is straightforward, but high-degree polynomials can behave erratically near the data boundaries. Spline terms allow flexible smooth relationships, but require choices about knots and sufficient data support. Direct outcome regression can model nonlinear outcome relationships instead, while doubly robust estimators combine treatment and outcome models; neither removes the need for defensible assumptions or appropriate functional forms. The earlier material on positivity is relevant because no modeling choice can create information about treatment options absent from the data. Truncating extreme weights is a separate decision and does not repair a misspecified propensity model.
Check Your Understanding
Try each question, then compare your reasoning with the solutions below.
- Why can a propensity model containing blood pressure as a covariate still fail to adjust for blood-pressure confounding?
- In the five-profile example, what is the untreated stabilized weight at the center profile?
- State the assumptions needed for the inverse-probability identity for \(E[Y^a]\).
- Why does a flexible propensity model not solve a positivity problem?
Solutions. (1) Including blood pressure only as a linear term constrains its association with treatment to be linear on the log-odds scale; curvature or interactions may be missed. (2) The center has \(e=0.2\), so its untreated stabilized weight is \(P(A=0)/(1-e)=0.32/0.8=0.4\). (3) Consistency, conditional exchangeability given measured pretreatment covariates, positivity for each treatment arm, no interference, and integrability of the outcome suffice for the stated identity. (4) A more flexible model can represent the observed treatment probabilities but cannot supply data for covariate profiles where one treatment has probability zero.