In 1953, Morton Levin introduced a simple approach to estimating population attributable fractions (PAF) depending only on risk factor prevalence and relative risk. This formula and its extensions are still in widespread use today, particularly to estimate PAF in populations where individual data is unavailable. Unfortunately, Levin’s approach is known to be asymptotically biased for the PAF when the risk factor-disease relationship is confounded even if relative risks that are correctly adjusted for confounding are used in the estimator. Here we describe a simple re-expression of Miettinen’s estimand that depends on the causal relative risk, the unadjusted relative risk and the population risk factor prevalence. While this re-expression is not new, it has been underappreciated in the literature, and the associated estimator may be useful in estimating PAF in populations when individual data is unavailable provided estimated adjusted and unadjusted relative risks can be transported to the population of interest. Using the re-expressed estimand, we develop novel analytic formulae for the relative and absolute asymptotic bias in Levin’s formula, solidifying earlier work by Darrow and Steenland that used simulations to investigate this bias. We extend all results to settings with non-binary valued risk factors and continuous exposures and discuss the utility of these results in estimating PAF in practice.

1953 年，Morton Levin 引入了一种简单的方法来估计人群归因分数 (PAF)，仅取决于危险因素的流行率和相对风险。该公式及其扩展至今仍在广泛使用，特别是在无法获得个人数据的人群中估计 PAF。不幸的是，当风险因素与疾病关系混杂时，即使在估计器中使用针对混杂正确调整的相对风险，Levin 的方法也会对 PAF 产生渐近偏差。在这里，我们描述了 Miettinen 估计值的简单重新表达，该表达取决于因果相对风险、未调整的相对风险和人群风险因素患病率。虽然这种重新表达并不新鲜，但它在文献中并未得到充分重视，并且当个体数据不可用时，相关的估计量可能有助于估计人群中的 PAF，前提是可以将估计的调整和未调整的相对风险传递给感兴趣的人群。使用重新表达的被估计值，我们为 Levin 公式中的相对和绝对渐近偏差开发了新颖的分析公式，巩固了 Darrow 和 Steenland 早期使用模拟研究这种偏差的工作。我们将所有结果扩展到具有非二元价值风险因素和持续暴露的环境，并讨论这些结果在实践中估计 PAF 的效用。