Statin Use With Time-Varying Lipid Measurements

Why Repeated LDL Measurements Change the Analysis

Consider a cardiovascular cohort in which statin use is recorded repeatedly, LDL cholesterol is measured before treatment decisions, and myocardial infarction or cardiovascular death can occur during follow-up. A plausible sequence is that statin use lowers later LDL cholesterol, while LDL cholesterol predicts both the next statin decision and the event. LDL is therefore a time-varying confounder affected by prior treatment. Adjusting for each LDL measurement as if it were an ordinary baseline confounder can block part of the treatment effect while also failing to handle the later treatment confounding it creates.

The target here is the contrast between risks under two sustained strategies: use a statin at every treatment decision, and do not use a statin at any decision. The focus is the case-specific construction of the risk estimator. As in other target-trial analyses, the strategy, eligibility, outcome, and follow-up must be defined before estimation.

The timing matters: LDL at \(t\) is measured before \(A_t\), and \(D_{t+1}\) occurs after that decision during the interval. Thus, for example, \(L_1\) may have been changed by \(A_0\), but it is available to predict \(A_1\) and the next event. A person who experiences an event during interval \(t\) contributes that event and has no later at-risk treatment decisions in the analysis.

Assumptions and the Sequential Weight

For identification, assume consistency (the observed outcome and covariate history under the observed treatment sequence equal those under that sequence), no interference between cohort members, and sequential exchangeability: among people with the same observed history and still at risk at \(t\), treatment assignment is independent of the relevant future counterfactual outcomes under the strategies being compared. Write the observed decision history as \(H_t=(C,\bar L_t,\bar A_{t-1})\). Exchangeability is required at each decision time, not merely at baseline.

Also require positivity: for histories that occur among people who would be at risk under a target strategy, the probability of receiving that strategy's next treatment must be positive. In a sustained statin analysis, this means that, wherever the target risk set has support, there are people with the relevant history who both do and do not receive statins as needed by the strategy. For uncensored analyses, assume complete outcome ascertainment through follow-up. With loss to follow-up, additionally assume sequential exchangeability and positivity for remaining observed, conditional on the measured history, and account for censoring with inverse-probability-of-censoring weights.

The weight upweights adherent people whose observed treatment sequence was unlikely given their measured history. For the interval-\(t\) hazard, it uses treatment probabilities only through \(t\): no event case must have a treatment history after the event. If censoring occurs, multiply by the corresponding inverse probability of remaining observed through the interval, conditional on the observed past. Treatment and censoring models should use information available before the decision or censoring event they predict.

Estimate Interval Hazards, Then Cumulative Risk

For each interval, use the weighted event total divided by the weighted at-risk total among people who followed the strategy up to that interval. In a sample of \(n\) people, the estimator is

This is an interval-specific hazard estimator. Define the competing-death hazard \(\widehat m_t^{\bar a}\) using the same weighted risk-set denominator, with \(M_{i,t+1}\) in the numerator: $$ \widehat m_t^{\bar a} = \frac{\displaystyle\sum_{i=1}^{n}R_{it}I(\bar A_{it}=\bar a_t)W_{it}^{\bar a}M_{i,t+1}} {\displaystyle\sum_{i=1}^{n}R_{it}I(\bar A_{it}=\bar a_t)W_{it}^{\bar a}}. $$ Each event is counted in the interval in which it occurs, with the treatment weight through that interval. People who have an event are removed from later risk sets, as they should be; they are not excluded from the earlier interval in which their event occurred. The strategy-specific risk through interval \(K-1\) is then estimated by

where \(Y_K^{\bar a}=1\) denotes an event by the end of follow-up under strategy \(\bar a\). This construction is appropriate for a first-event outcome with interval risk sets. It does not require treatment histories after an event.

To justify the hazard result, consider the weighted at-risk total at time \(t\). Its indicator requires observed adherence through \(t\) and survival event-free through \(t\). Consistency makes the observed event and risk status for such adherent people equal to their counterfactual status under \(\bar a_t\). At each decision \(k\le t\), exchangeability permits comparison of people with the same measured history; multiplying by \(1/P(A_k=a_k\mid H_k,R_k=1)\) cancels the selection induced by receiving \(a_k\). Iterating this argument over \(k=0,\ldots,t\) makes the weighted event and risk-set totals proportional to their corresponding counterfactual totals under \(\bar a\). Their ratio is therefore the counterfactual interval hazard. Positivity ensures each denominator probability is nonzero. In a large sample, the empirical weighted ratio converges to that population ratio when the relevant expectations are finite and the denominator limit is positive. \(\square\)

For the cumulative-risk result, let \(h_t^{\bar a}\) and \(m_t^{\bar a}\) be the counterfactual cardiovascular-event and competing-death hazards among people alive and free of the cardiovascular event at the start of interval \(t\) under \(\bar a\). The probability of remaining alive and free of the cardiovascular event through interval \(t\), given that status at its start, is \(1-h_t^{\bar a}-m_t^{\bar a}\). Thus the probability of a cardiovascular event in interval \(t\) is the probability of remaining alive and free of either event through prior intervals, multiplied by \(h_t^{\bar a}\). Summing these interval probabilities gives $$ P(Y_K^{\bar a}=1) = \sum_{t=0}^{K-1} \left\{ \prod_{j=0}^{t-1} (1-h_j^{\bar a}-m_j^{\bar a}) \right\} h_t^{\bar a}. $$ Replacing the hazards by their consistent estimates yields the displayed estimator. \(\square\)

What to Check in a Statin Analysis

The design and models must reflect the clinical timing. Check that LDL was measured before the treatment decision it is used to predict, that the cardiovascular event definition is applied consistently, and that each event is attributed to the correct interval. Review the estimated treatment probabilities within clinically relevant histories: near-zero probabilities can produce very large weights, while probabilities of zero violate positivity outright. Also inspect weighted risk-set totals and event counts at each time. A hazard with a zero or extremely small denominator is not reliably estimated merely because software returns a number.

Useful diagnostics include distributions and upper quantiles of the weights, the fraction of person-intervals with extreme weights, and whether weighting improves balance in measured prior LDL and other treatment predictors. Weight truncation may reduce variance but changes the estimator and can introduce bias; report the rule and assess sensitivity to it. Model misspecification in the treatment probabilities can leave confounding, and weight diagnostics cannot verify exchangeability for unmeasured factors.

These approaches target related causal quantities only when their assumptions and estimands align. Weighting is particularly transparent about how observed treatment histories are reweighted, but it does not eliminate the need for good measurement, adequate overlap, or a defensible causal design. LDL measurement error, unmeasured adherence or clinical indications, changes in treatment versions, and informative loss to follow-up can all undermine interpretation. The estimated contrast is a marginal comparison of strategy-specific risks in the target cohort, not automatically the effect for every clinical subgroup.

Common Pitfall

• Wrong reasoning: “To estimate risk under sustained statin use, keep only people with a complete statin history through the end of follow-up and compare their event proportions.”
• Why it fails: People who experience early events cannot have later treatment histories. Requiring complete histories excludes them and conditions on surviving long enough to provide those histories.
• Correction: At each interval, include people who are at risk and adherent through that interval, weight their event and risk-set contributions using treatment probabilities through that interval, and combine the estimated hazards into cumulative risk.