Biologic Escalation in Rheumatoid Arthritis

Why Biologic Escalation Requires a Longitudinal Comparison

In rheumatoid arthritis, clinicians may consider biologic treatment at more than one decision point. Disease activity measured at an earlier visit can influence whether treatment is started or continued at a later visit. It can also predict subsequent functional status. If earlier treatment changes that disease activity, then disease activity is both a consequence of prior treatment and a predictor of later treatment and outcome. This is the treatment-confounder feedback that motivates a marginal structural model (MSM).

The outcome here is the follow-up Health Assessment Questionnaire Disability Index (HAQ-DI), on its standard scale from 0 to 3, with higher values indicating greater disability. The target is not a treatment effect for a patient with a particular observed disease-activity history. It is a population mean outcome under a specified escalation strategy, and the difference between such means.

A strategy can be static, prescribing the same treatment at a decision for everyone, or dynamic, using measured history to determine treatment. For example, one might compare biologic treatment at both decisions with a strategy that defers treatment initially and starts it at the second decision only if disease activity is high. A strategy must specify the action for every history it covers; otherwise its target is ambiguous.

Identification and the Treatment-Path Weight

For a two-decision illustration, let \(H_0=V\) be the first-decision history and \(H_1=(V,A_0,L_1)\) the second-decision history. Write \(p_0(a\mid H_0)=P(A_0=a\mid H_0)\) and \(p_1(a\mid H_1)=P(A_1=a\mid H_1)\). The treatment-path weight for a person whose observed actions follow strategy \(g\) is

The indicator is zero when the observed path departs from the strategy. Among people following the prescribed path, the inverse probabilities counteract the overrepresentation of histories that commonly receive the prescribed treatment and the underrepresentation of histories for which it is unusual. In practice, the probabilities are estimated from treatment-assignment models using the measured histories available at each decision.

The central identity is that the weighted observed outcome recovers the mean under the strategy. The history \(L_1\) must be included in the second treatment model: it predicts the second treatment and outcome, even though it may itself have been affected by \(A_0\).

Proof. Consider the observed-data distribution factored in temporal order. At the first decision, conditional on \(H_0\), the chance of the prescribed action is \(p_0\{g_0(H_0)\mid H_0\}\). Multiplication by the first inverse-probability factor cancels that chance, leaving the distribution of baseline histories without selection by the prescribed first action. By consistency and sequential exchangeability, among those histories the subsequent distribution of \(L_1\) under the prescribed first action is the corresponding counterfactual distribution under \(g_0\).

At the second decision, conditional on \(H_1\), the prescribed action occurs with probability \(p_1\{g_1(H_1)\mid H_1\}\). The second inverse-probability factor cancels this probability. Sequential exchangeability then permits the conditional distribution of the final outcome among those following the prescribed second action to be interpreted as the distribution under that action. Consistency identifies the observed outcome on the prescribed path with \(Y^g\). Integrating these successive conditional distributions gives the intervention distribution of \(Y^g\); hence \(E[W^gY]=E[Y^g]\). Positivity ensures both inverse factors are defined. \(\square\)

Worked Weight Calculations

Estimation, Diagnostics, and Limits

For \(n\) independent individuals, the Hájek estimator is \(\widehat{\mu}_g^{\,H}=\sum_{i=1}^n W_i^gY_i/\sum_{i=1}^n W_i^g\), provided the denominator is positive. Under the identification assumptions, this estimates the strategy-specific marginal mean. If the treatment probabilities are estimated, their models should respect the time ordering and include measured predictors of treatment and outcome available at each decision. The resulting uncertainty calculation should account for estimation of those probabilities.

Justification. With known probabilities and two decisions, the lower bound implies \(0\leq W_i^g\leq\epsilon^{-2}\). Since HAQ-DI is bounded between 0 and 3, both \(W_i^g\) and \(W_i^gY_i\) have finite expectations. The weak law of large numbers gives \(n^{-1}\sum_i W_i^gY_i\xrightarrow{p}E[W^gY]=\mu_g\) and \(n^{-1}\sum_i W_i^g\xrightarrow{p}E[W^g]=1\). The denominator limit is 1 because the treatment probabilities cancel the path indicators successively, with positivity. Since that limit is nonzero, the ratio converges in probability to \(\mu_g\). With consistently estimated probabilities, the same conclusion requires regularity conditions ensuring their estimates and the resulting weighted sample averages converge to the corresponding population quantities; consistency alone does not excuse a badly specified treatment model. \(\square\)

Diagnostics should focus on whether the data support each strategy, not only on whether a fitted model converged. Review the distributions of the estimated treatment probabilities and weights, the number of observed paths consistent with each strategy, and whether important histories have very few patients receiving the prescribed action. A very large weight indicates that a small number of observations may dominate the estimate. The effective sample size, \((\sum_i W_i^g)^2/\sum_i(W_i^g)^2\), helps summarize that concentration: it is small relative to the number of adherers when weights are highly variable. Weight truncation can reduce variance, but it changes the estimator and may introduce bias; it does not create support for unobserved treatment choices.

An ordinary regression that includes \(L_1\) as a conventional adjustment covariate may fail to estimate the desired marginal contrast. Because \(L_1\) can be affected by \(A_0\), conditioning on it can block part of the earlier treatment effect, while omitting it can leave confounding of \(A_1\). The MSM weighting approach instead uses \(L_1\) to model treatment assignment and then targets a marginal outcome. The longitudinal g-formula is a main alternative: it models outcome and covariate processes to calculate strategy-specific means. The approaches rely on different modeling choices, but neither resolves unmeasured confounding or a lack of positivity.

The conclusions are limited to the defined population, treatment strategies, follow-up time, and HAQ-DI outcome. If biologic treatment is recorded too broadly to distinguish clinically relevant versions, if disease activity is measured poorly, or if important confounders are unmeasured, the identifying assumptions may not be credible. If follow-up is incomplete, treatment weights alone do not generally correct selection from dropout. In an application, investigators should report the strategy rules, the fitted treatment models, weight diagnostics, and the implications of weak support alongside the estimated marginal means and contrast.