Dose-Response Analysis of Occupational Exposure

From occupational exposure histories to marginal risks

An occupational pesticide study may record exposure repeatedly, while symptoms or health status measured between visits can affect later exposure and also predict the outcome. A comparison of workers at different observed exposure levels can therefore mix exposure effects with changes in health and work assignment. A marginal structural model uses weighting to address this time-varying confounding and estimate health risks under specified exposure histories.

Here the target is the risk under a sustained exposure regime: set exposure to the same level $a$ at each of $K$ follow-up times, then consider the risk of a binary health outcome by the end of follow-up. The central practical issue is not just fitting a curve. It is deciding which values of $a$ are supported by the cohort well enough for the corresponding contrast to be credible.

Definition: Let $A_t$ be pesticide exposure at time $t$, $L_t$ measured covariates recorded before $A_t$, and $Y$ the binary outcome. Write $\bar a=(a,\ldots,a)$ for the exposure history that sets exposure to $a$ at all $K$ times. The marginal sustained-exposure risk is $\psi(a)=P(Y^{\bar a}=1)$, where $Y^{\bar a}$ is the potential outcome under that history. A dose-response contrast, such as $\psi(a_2)-\psi(a_1)$, compares risks under two such regimes.

This estimand is about the whole sustained history, not merely a worker’s exposure at the final visit. Interpreting a fitted curve as $\psi(a)$ requires that its exposure inputs represent the intervention history of interest. If the model uses an average exposure or a cumulative dose instead, treating that summary as equivalent to a constant exposure regime is an additional structural assumption; it does not follow automatically from the definition.

Identification and weighting

Let $V$ denote baseline covariates and $H_t=(V,\bar L_t,\bar A_{t-1})$ the measured history immediately before exposure at time $t$. Define $g_t(a_t\mid H_t)$ as the conditional density of exposure given that history. For continuous exposure, this is a density, not the probability of receiving one exact value. A basic unstabilized exposure weight is

W_i^A=\prod_{t=1}^{K}\frac{1}{g_t(A_{it}\mid H_{it})}.

If censoring can depend on measured history, include corresponding inverse-probability-of-censoring factors in $W_i$. Stabilized weights replace each numerator $1$ with a specified numerator density, often conditional on baseline variables and past exposure. Stabilization can reduce variability; its choice affects the weighting target under model misspecification. The weight model must represent the relevant exposure and censoring processes adequately.

Identification conditions: The weighted causal interpretation requires consistency (observed outcomes equal the potential outcomes under the observed exposure history), no interference between workers, and sequential exchangeability: conditional on $H_t$, exposure at $t$ is independent of the future potential outcomes under the regimes of interest. It also requires positivity: histories relevant to the target population must have exposure densities that support the regimes being compared. For a continuous sustained regime, this means adequate density support in relevant neighborhoods of the target exposure history, together with the smoothness or structural model assumptions used to estimate the risk there. Correct exposure and censoring weight models are needed for the stated weighting procedure.

These conditions are substantive. For example, exchangeability can fail if a health change that affects both later job assignment and the outcome was not measured. Positivity can fail if workers with certain histories are almost never observed at high exposure. A smooth fitted curve cannot repair either failure simply by returning a numerical prediction.

Result: In a single exposure period, suppose $A$ has conditional density $g(a\mid L)$, consistency holds, and $Y^a$ is conditionally independent of $A$ given $L$. For any integrable function $h$ on the supported exposure range, $$ E\left[\frac{h(A)Y}{g(A\mid L)}\right] =\int h(a)E[Y^a]\,da. $$ This identity explains why inverse-density weighting removes measured confounding and gives a marginal dose-response target.

Proof. By iterated expectation, the left side is

\begin{aligned} E\left[\frac{h(A)Y}{g(A\mid L)}\right] &=E_L\left[\int \frac{h(a)E[Y\mid A=a,L]}{g(a\mid L)} g(a\mid L)\,da\right]\\ &=\int h(a)E_L\{E[Y\mid A=a,L]\}\,da. \end{aligned}

By consistency and conditional exchangeability, $E[Y\mid A=a,L]=E[Y^a\mid L]$ wherever the conditional density supports $a$. Thus $E_L\{E[Y\mid A=a,L]\}=E[Y^a]$, which proves the identity. In a longitudinal study, the product weight applies this adjustment sequentially at each exposure time; iterated conditioning gives the corresponding identity for supported exposure histories under sequential exchangeability. With stabilized weights, the numerator densities change the weighting measure, but a correctly specified marginal structural model can still represent the target risks.

Fitting and reading a weighted dose-response curve

A common marginal structural model for a binary outcome is a logistic curve in sustained exposure:

\operatorname{logit}\{\psi(a)\} =\beta_0+\beta_1a+\beta_2a^2, \qquad \operatorname{logit}(p)=\log\left(\frac{p}{1-p}\right).

The quadratic term permits curvature on the log-odds scale. A weighted Bernoulli estimating equation for a model based on exposure history $\bar A_i$ is

\sum_{i=1}^{n}W_i X(\bar A_i) \left[Y_i-\operatorname{expit}\{X(\bar A_i)^\mathsf{T}\hat\beta\}\right]=0, \qquad \operatorname{expit}(x)=\frac{1}{1+\exp(-x)}.

Here $X(\bar A_i)$ is the chosen model vector, and the fitted curve is evaluated at the constant history $\bar a$. Inference should account for estimation of the weights, for example with an appropriate robust variance procedure or a bootstrap that re-estimates the weight models. The model coefficients describe the log-odds curve; the principal causal summaries are often predicted risks and risk contrasts.

Worked Example: Predicting risks and a risk difference

Suppose a fitted linear-logit model for sustained exposure $a$, measured in a specified dose unit, is $\operatorname{logit}\{\hat\psi(a)\}=-2.875+0.5a$. Calculate the estimated risks at $a=0.5$ and $a=2.5$, and their risk difference.

At $a=0.5$, the linear predictor is $-2.875+0.5(0.5)=-2.625$. Therefore \[ \hat\psi(0.5)=\frac{1}{1+\exp(2.625)}\approx 0.0675467\approx 0.0675. \]

At $a=2.5$, the linear predictor is $-2.875+0.5(2.5)=-1.625$, so \[ \hat\psi(2.5)=\frac{1}{1+\exp(1.625)}\approx 0.1645165\approx 0.1645. \]

Using the unrounded predictions, the estimated risk difference is $0.1645165-0.0675467=0.0969698$, or approximately $0.0970$, about 9.7 percentage points. Subtracting the displayed rounded risks gives the same result to four decimal places here, but the calculation should be based on the unrounded model predictions.

The risk difference is not generally a regression coefficient. For the logistic model, changing $a$ changes the linear predictor, and the expit transformation then changes the risk nonlinearly. Even if the log odds are linear in exposure, the risk difference depends on both exposure values and the fitted baseline risk.

Derivation: For a linear-logit model with $\eta(a)=\beta_0+\beta_1a$, the risk is $\psi(a)=\operatorname{expit}\{\eta(a)\}$. Since $\operatorname{expit}'(x)=\operatorname{expit}(x)\{1-\operatorname{expit}(x)\}$, the dose derivative is $$ \frac{d\psi(a)}{da}=\beta_1\psi(a)\{1-\psi(a)\}. $$ Thus $\beta_1$ is a log-odds slope, not a constant risk increase per dose unit. A finite risk contrast must be computed as $\operatorname{expit}\{\eta(a_2)\}-\operatorname{expit}\{\eta(a_1)\}$.

Support, diagnostics, and failure modes

A causal contrast is most defensible when both exposure levels are supported among workers with relevant histories. In a longitudinal setting, support for each period separately does not guarantee support for the full sustained history: the joint sequence may be rare even when its individual dose values are common. Examine exposure distributions by important measured histories, the estimated density values, weight tails, and the effective sample size. For weights $W_i$, a common summary is

n_{\mathrm{eff}}=\frac{(\sum_i W_i)^2}{\sum_i W_i^2}.

A small effective sample size indicates that a few observations dominate the weighted fit. Also inspect whether predicted risks are plausible, whether the fitted curve is driven by a small number of workers near a boundary, and whether conclusions change under reasonable model or weight specifications. These checks can reveal instability; they cannot establish exchangeability or prove that unmeasured confounding is absent.

Worked Example: Restricting a contrast to supported doses

Suppose the cohort provides usable support for sustained exposure only from $0.8$ to $2.0$ dose units. Use the model from the previous example to compare $a=1.0$ with $a=1.8$, both within that range.

At $a=1.0$, $\eta=-2.875+0.5(1.0)=-2.375$, giving $\hat\psi(1.0)=1/\{1+\exp(2.375)\}\approx0.0851$. At $a=1.8$, $\eta=-2.875+0.5(1.8)=-1.975$, giving $\hat\psi(1.8)=1/\{1+\exp(1.975)\}\approx0.1218$. The estimated risk difference is approximately $0.1218-0.0851=0.0367$, about 3.7 percentage points, with the calculation based on unrounded predictions.

The model also produces numbers at $0.5$ and $2.5$, but these are outside the stated support. They are extrapolations, not contrasts supported by this cohort. Their numerical precision does not make them credible causal estimates.

Worked Example: Why weighting matters with measured confounding

For a simple one-period illustration, let $H$ be a binary health-history indicator, with $P(H=0)=P(H=1)=0.5$. Let the chance of high exposure $A=1$ be $0.8$ when $H=0$ and $0.2$ when $H=1$. Suppose the observed outcome risks among high-exposure workers are $0.10$ for $H=0$ and $0.30$ for $H=1$. The crude risk at high exposure is \[ \frac{(0.5)(0.8)(0.10)+(0.5)(0.2)(0.30)} {(0.5)(0.8)+(0.5)(0.2)} =\frac{0.07}{0.5}=0.14. \]

Inverse-probability weighting gives high-exposure workers weights $1/0.8=1.25$ in stratum $H=0$, and $1/0.2=5$ in stratum $H=1$. The weighted risk is \[ \frac{(0.5)(0.8)(1.25)(0.10)+(0.5)(0.2)(5)(0.30)} {(0.5)(0.8)(1.25)+(0.5)(0.2)(5)} =\frac{0.05+0.15}{0.5+0.5}=0.20. \]

The weighted result is the equal-stratum average of the two specified risks, $0.5(0.10)+0.5(0.30)=0.20$. The crude and weighted values differ because high exposure was more common in the lower-risk $H=0$ stratum. This discrete calculation illustrates the balancing logic; it is not itself a continuous dose-response analysis.

Choosing and reporting a dose-response analysis

A weighted parametric curve is not the only option. A flexible spline can reduce dependence on a quadratic shape, but still needs exposure support and adequate data across the curve. A categorized analysis is easier to present, but discards within-category dose information and depends on cut points. An unweighted regression may be simpler, but does not by itself address time-varying confounding affected by prior exposure. Each approach has limitations; none makes unsupported exposure comparisons valid.

Report the sustained regime being modeled, the dose units and follow-up period, the weighting strategy, the fitted risk curve or selected risks, and the range used for causal contrasts. Describe support and weight diagnostics, and distinguish supported estimates from extrapolations. If the weight distribution is highly variable, say so and show how conclusions respond to defensible alternatives; arbitrary trimming may change the target population or estimand.

A frequent mistake is to interpret every fitted value as an identified causal risk. The correction is to separate model evaluation from causal support: the model can return $\hat\psi(a)$ at any numeric input, but causal interpretation requires the stated assumptions and adequate support for that exposure history. In particular, a high-dose versus low-dose contrast should not be reported as a cohort-supported causal effect when one end of the contrast is represented only by model extrapolation.

Check Your Understanding

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

What does $\psi(a)$ represent when exposure is measured at several follow-up times?
Why is an exposure density, rather than the probability of one exact dose, used for continuous exposure weighting?
A cohort supports sustained doses from 1 to 3 units. What is the main concern with reporting a causal contrast between 0.5 and 3 units?
In a weighted logistic dose-response model, what does the exposure coefficient describe, and why is it not generally a risk difference?
Name two diagnostics that can indicate an unstable weighted dose-response fit, and explain what they do not establish.

Solutions. (1) It is the marginal probability of the outcome under the regime that sets exposure to $a$ at every specified time. (2) For a continuous variable, the probability of any one exact value is generally zero; the conditional density describes how exposure values are distributed locally and enters the inverse-density weight. (3) The 0.5 dose lies outside the stated support, so that part of the contrast depends on extrapolation rather than supported comparison. (4) In a linear-logit model the coefficient is a change in log odds per dose unit; the corresponding risk change depends on the starting risk through the expit transformation. (5) Examples include extreme weights, a low effective sample size, and predictions driven by a few boundary observations. Such diagnostics identify possible instability but do not prove exchangeability or rule out unmeasured confounding.