Continuous-Treatment Blip Functions

Why continuous-dose blips need careful parameterization

A treatment such as medication dose, radiation level, or daily activity often takes many possible values rather than just “treated” and “untreated.” In that setting, a blip function describes the causal contrast between doses at a treatment decision, conditional on the history available before that decision. Its parameterization determines which dose comparisons the model can represent. A linear dose term imposes a constant marginal effect; a polynomial can allow curvature, but does not automatically make the model more credible.

The earlier material on structural nested mean models and g-estimation established the role of blips and of sequentially removing treatment effects from an outcome. Here the focus is on continuous doses: how to define the dose contrast, how to use linear and polynomial terms, and how the resulting model supplies g-estimation restrictions over several treatment times.

Definition and assumptions

The contrast is between interventions, not between patients who happened to receive different doses. In longitudinal models, a blip describes the increment attributed to one treatment decision after the model’s recursive blipping convention has accounted for other treatment decisions. It is not generally the same as a regression coefficient from an unadjusted outcome-on-dose regression.

The usual causal interpretation requires consistency, no interference, and sequential exchangeability: conditional on \(H_t\), treatment assignment is independent of the relevant future counterfactual outcomes. Identification also requires positivity over the dose comparisons being modeled. For continuous treatment, positivity concerns conditional density and support: there must be adequate treatment variation near the doses of interest for histories where contrasts are interpreted. A particular dose has probability zero under a continuous distribution, so positivity is not the requirement that every exact dose occur with positive probability. The structural mean model must also be correctly specified for the contrasts being estimated, and the relevant moments must be finite.

Linear and polynomial dose contrasts

Write \(x=a-a_0\) for dose relative to a reference dose. A linear blip is

Here \(\beta_t(h)\) is the conditional change in the modeled mean outcome per one-unit increase in dose, under the structural model. If dose effect modification is included, \(\beta_t(h)\) can depend on measured history. For example, with \(q(h)=(1,L)\), the model \(\beta_t(h)=\psi_{t0}+\psi_{t1}L\) gives a linear dose effect whose slope varies with \(L\).

A quadratic model can represent curvature:

The finite contrast from \(a_0\) to \(a\) is then \(\psi_{t1}(h)x+\psi_{t2}(h)x^2\). Its derivative with respect to dose is \(\psi_{t1}(h)+2\psi_{t2}(h)x\), which is a different quantity. A polynomial coefficient alone is not the effect of moving between two arbitrary doses.

More generally, let \(d_t(h,a)\) be a vector of dose and history basis terms satisfying \(d_t(h,a_0)=0\). Then

For example, \(d_t(h,a)=(x,Lx,x^2,Lx^2)^\mathsf{T}\) allows both the linear and quadratic dose components to vary with \(L\). The model is linear in its unknown parameters even though its dose-response curve is nonlinear.

Longitudinal blipping and g-estimation

For treatment decisions \(t=1,\ldots,K\), let \(Y\) be the final outcome. Under an additive structural nested mean model, define a sequence of recursively blipped outcomes by starting with \(R_{K+1}(\psi)=Y\) and, for \(t=K,\ldots,1\), setting

The final \(R_1(\psi)\) has the modeled treatment contributions removed. At the true parameter value, the structural restriction is that its conditional mean does not vary with the current treatment once the appropriate pre-treatment history is held fixed. The corresponding conditional statement is

This is a mean restriction, not a claim that the blipped outcome is independent of treatment in its entire distribution. It supplies a population g-estimation equation. For a dose basis \(d_t(H_t,A_t)\), define its conditionally centered version

Then the mean restriction implies \(E[Z_tR_t(\psi_0)]=0\). In applications, a collection of such moments across treatment times can be used to estimate \(\psi\), provided the resulting equations identify the parameters. This tutorial uses the mean restriction to explain continuous-dose g-estimation; it does not specify a particular variance estimator or an optimal choice of instruments.

Proof. By iterated expectation and the conditional mean restriction,

The second equality follows because \(Z_t\) is a function of \(H_t,A_t\), while the conditional mean of \(R_t(\psi_0)\) given \(H_t,A_t\) is the \(H_t\)-only quantity \(E[R_t(\psi_0)\mid H_t]\). By definition, \(E[Z_t\mid H_t]=0\). Therefore the final expectation is zero. Integrability ensures these expectations exist. \(\square\)

For polynomial blips, the centered basis includes the dose terms and their conditional means. With \(x_t=A_t-a_0\) and \(d_t=(x_t,x_t^2)^\mathsf{T}\), the two components of \(Z_t\) are \(x_t-E[x_t\mid H_t]\) and \(x_t^2-E[x_t^2\mid H_t]\). Using only the first component generally does not identify the quadratic parameter: its moment does not, by itself, supply a separate restriction for the second dose term.

Reference doses and model checks

The reference dose is a parameterization choice, not a claim that the reference is clinically optimal or common. Changing the reference should not change a correctly represented contrast. For example, if a dose-response function is \(f(a)\), then the contrast relative to \(a_0\) is \(f(a)-f(a_0)\). Re-referencing changes the intercept representation, but not the contrast between any two doses.

Proof. Subtracting the two function values cancels the constant: \(f(a)-f(a_0)=\psi_1a+\psi_2a^2-\psi_1a_0-\psi_2a_0^2\). Grouping terms yields \(\psi_1(a-a_0)+\psi_2(a^2-a_0^2)\). Writing \(x=a-a_0\), the same expression is \((\psi_1+2\psi_2a_0)x+\psi_2x^2\), because \(a=a_0+x\). Thus the coefficients in the centered basis may differ, but the implied contrast is identical. \(\square\)

Several diagnostics matter especially for continuous doses. First, inspect dose distributions across important histories: narrow or disjoint support makes some contrasts unsupported or highly dependent on functional-form extrapolation. Second, examine whether the dose basis has enough conditional variation to distinguish its parameters. For a quadratic model, dose variation must distinguish \(x\) from \(x^2\); if observed doses occupy only a very narrow range, these terms can be nearly indistinguishable. Third, assess whether the fitted curve implies implausible changes near the edges of observed support. Scaling dose units can improve numerical conditioning, but the reported contrast must still be translated back to the original units.

A frequent error is to interpret \(\psi_1\) in a quadratic model as the effect of a one-unit dose increase at every dose. The derivative is \(\psi_1+2\psi_2x\), and a finite one-unit contrast from \(x\) to \(x+1\) is \(\psi_1+\psi_2(2x+1)\). Another error is to include \(a\) and \(a^2\) but forget to subtract their values at the reference dose; that model does not automatically give a zero blip at \(a_0\). Centering the basis as \(x,x^2\) avoids that problem.

A continuous-dose SNMM directly models dose contrasts through its blip functions. The parametric g-formula instead models the longitudinal outcome and treatment process to calculate regime-specific means; it can be useful when the target is a full regime mean, but entails modeling those components. Marginal structural models use weighting to represent marginal treatment contrasts and are an alternative when that marginal target is desired. These approaches do not remove the need to consider dose support or to define the causal contrast clearly.

Polynomial blips remain limited by their functional form. A quadratic is not a universal dose-response curve, and extrapolation beyond the observed dose range is particularly difficult to justify. Sequential exchangeability can fail through unmeasured time-varying confounding, and positivity can fail even when a fitted polynomial produces a smooth curve. A numerically solved moment equation is not, by itself, evidence that these assumptions or the blip model are adequate.