Pesticide Exposure With Repeated Biomarker Measurements

Why Repeated Biomarkers Change the Analysis

Suppose workers are followed over several seasons. Pesticide exposure in one season can change a later biomarker concentration, and that biomarker can predict both next season’s exposure and a health outcome. The biomarker is then not simply a baseline confounder: it is affected by prior exposure and can confound the effect of subsequent exposure. Directly adjusting for it in an ordinary outcome regression can block part of the earlier exposure effect, while leaving it out can confound the later exposure effect.

A marginal structural model (MSM) addresses this feedback by comparing outcomes under specified exposure histories, using weights based on the probability of the observed exposure history given the measured history. Here the target is a sustained contrast, such as exposure at a high level in every season versus exposure at a low level in every season. The method does not make the exposure randomized; its causal interpretation depends on explicit assumptions and adequate exposure probabilities.

For clarity, this tutorial assumes the biomarker is measured before the next season’s exposure, the outcome is observed at the end of follow-up, and there is no loss to follow-up. The last condition keeps the focus on exposure and biomarker feedback rather than censoring. The regimes and outcome must also be well defined: “high” and “low” exposure should correspond to sufficiently specified interventions, not merely labels for different mixtures of pesticide, task, and protective equipment.

Identification and the Weighting Argument

The required assumptions are consistency, sequential exchangeability, positivity, and appropriate measurement of the history. Consistency says that if a worker’s observed exposure history is \(\bar a\), then the observed outcome equals \(Y^{\bar a}\), and the observed biomarkers equal the corresponding counterfactual biomarkers under that history. It also requires that the intervention represented by each exposure level be sufficiently well defined.

Sequential exchangeability must cover more than the terminal outcome. At each season it must justify learning the counterfactual biomarker transition as well as the counterfactual outcome process from observed data. One sufficient formulation is that, conditional on \(H_t\), the observed exposure \(A_t\) is independent of the entire relevant future counterfactual biomarker and outcome process under each regime being compared. This is a substantive no-unmeasured-confounding assumption: all common causes of current exposure and that future process must be included in the measured history.

Here the statement is required at histories compatible with the regime and for each regime of interest. Positivity requires that each regime’s exposure choice have positive probability at histories that can arise under that regime: \(P(A_t=a_t\mid H_t=h_t)>0\). In practice, probabilities extremely close to zero are also a problem, because they produce unstable weights.

Under these assumptions, the intervention distribution of the biomarker history and outcome can be assembled from observed conditional distributions. The resulting g-formula is

with the final outcome regression conditioned on the full observed exposure and biomarker history. The formula is written for densities; sums replace integrals for discrete variables. Its key feature is that the biomarker transition at each season is evaluated under the exposure specified by the regime, rather than held at the value it happened to take under another exposure history.

Derivation. By the chain rule, the conditional probability of the exposure path \(\bar a\), given the biomarker and baseline histories along that path, is the product of the season-specific probabilities in the denominator. Weighting the observed path by its reciprocal cancels that product. More explicitly, the expectation on the left can be expanded by integrating over \(w\), the biomarker path, and the outcome:

Sequential exchangeability and consistency identify each observed biomarker transition in this expression with the corresponding transition under the intervention; they also identify the conditional outcome distribution. Thus the integral is precisely the g-formula for \(E[Y^{\bar a}]\). This establishes the identity. In a sample, the identity motivates estimating regime means by averages of weighted outcomes among workers following the specified regime. A stabilized weight can improve numerical behavior, but its numerator must be chosen so that the estimand and the weighting argument remain appropriate. \(\square\)

Worked Examples

Diagnostics, Alternatives, and Limitations

In an occupational study, the exposure probabilities should be inspected by season and by relevant biomarker-history strata. Look for histories in which nearly everyone receives the same exposure level, very large weights, and a small effective weighted sample size. These patterns warn that the comparison relies heavily on a few workers and that the data provide little support for one of the sustained regimes. Check that biomarker measurements precede the exposure decisions they are used to model, and assess whether important determinants of exposure and health—such as job task, protective equipment, and prior health—were recorded with sufficient detail.

The inverse-probability weighted MSM is not the only route to a marginal contrast. The parametric g-formula models biomarker transitions and the outcome, then predicts outcomes under each exposure regime and averages over the resulting biomarker distributions. It can be more direct when those models are credible, but it depends on correctly specifying the transition and outcome models. Weighting instead models exposure probabilities; it can be unstable with weak positivity or poorly estimated probabilities. Either approach still requires consistency, adequate measured confounding control, and positivity. Neither makes a poorly defined “high exposure” intervention scientifically precise.

A common mistake is to put every repeated biomarker into a conventional outcome regression and interpret the exposure coefficient as the total effect. That coefficient generally compares workers at the same biomarker value, rather than comparing the full biomarker pathways that would arise under sustained high and sustained low exposure. The correction is to define the marginal regime contrast first, then use an identification strategy—such as the weighted identity above—that accounts for the biomarker’s role in later exposure assignment without fixing it to a common post-exposure value.