Bias assessment and correction for Levin's population attributable fraction in the presence of confounding

Abstract

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.

Keywords: Levins formula; Miettinens formula; Population attributable fraction.

Fig. 1

Plots for absolute and relative error in Levin’s formula when $R{R}_U=1.5$ and $\pi =0.5$ over a range for the confounding ratio $C$, assuming no effect-size modification on the relative risk scale. A: absolute bias, $R{R}_C\ge 1$, B: absolute bias, no restriction on $R{R}_C$, C: relative bias, $R{R}_C\ge 1$, D: relative bias, no restriction on $R{R}_C$. In the attached plots, the bias for the ‘adjusted’ version, $PA{F}_L=\frac{\pi (R{R}_C-1)}{1+\pi (R{R}_C-1)}$ of Levin’s formula is given in blue, while biases for the unadjusted version,$PA{F}_{L,U}=\frac{\pi (R{R}_U-1)}{1+\pi (R{R}_U-1)}$ are given in red. The upper blue horizontal lines are the limiting absolute bias: $1-\frac{\pi R{R}_U}{1+\pi (R{R}_U-1)}$ (for plots A and B) and relative bias: $1+\frac{1-\pi }{\pi R{R}_U}$ for $PA{F}_L$ as $C\to \infty$ (for plots C and D). Note that the absolute bias of $PA{F}_L$ is 0 at both $C=1/R{R}_U$ and $C=1$, while the relative bias for $PA{F}_L$ converges to $\frac{1+\pi (R{R}_U-1)}{R{R}_U}$ as $C$ approaches $1/R{R}_U$ (as tagged on the y-axis of plot C). The absolute and relative biases of the unadjusted version of Levin’s formula are more erratic. In particular, the relative bias function for $PA{F}_{L,U}$ is negative when $R{R}_C<1$ or equivalently when $C<\frac{1}{R{R}_U}$ indicating that $PA{F}_{L,U}$ shows the incorrect direction of association in this scenario; in contrast the relative bias for $PA{F}_L$ is always larger than 0 for all $C>0$ and converges to 0 as $C\to 0$