INTRODUCTION

The population-attributable fraction (PAF) has emerged as a cornerstone of epidemiological research, public health planning, and decision-making. The PAF quantifies the proportion of disease incidence in a population that can be prevented by eliminating a particular risk factor [1]. Therefore, it is an essential parameter for devising preventive strategies [1]. However, point estimates of the PAF alone can be misleading as they do not convey the degree of uncertainty inherent in the estimate [2]. The 95% confidence interval (CI) provides a range within which the true PAF is likely to fall, which is important for understanding the reliability of the estimate. Therefore, reporting the PAF with its CI helps public health officials and policymakers understand the potential impact of eliminating a risk factor on disease prevention. Nevertheless, methods for accurately estimating the PAF and its CIs have not yet been established. Most PAF values provide only point estimates, excluding CIs, which are indispensable for understanding the precision and uncertainty of these estimates, and ultimately influence the interpretation and decision making [3-5].

Delta, Greenland, and Monte Carlo simulation methods have been developed as representative techniques for estimating the 95% CIs of PAFs [6-8]. The Delta method utilizes a Taylor series approximation to determine the variance of a function involving multiple variables [9]. However, this approach may result in asymmetric and potentially implausible CIs, especially for rare outcomes [6,8]. The Greenland method approximates variance by separately analyzing the variability in exposure and outcome [10]. Nevertheless, this method heavily relies on specific distributional assumptions and may underperform in small or skewed samples [7]. In contrast, the Monte Carlo simulation uses a stochastic approach by resampling the probability distribution of input variables to generate a range of outcomes, though this process can be computationally demanding, especially in large-scale studies [8].

Although these methods each have their own advantages and limitations, comprehensive and systematic comparisons are lacking in the literature [4,6-8]. In particular, to estimate the 95% CI of the PAF, it is necessary to calculate the variance of the PAF estimator (V[PÂF]). This variance is influenced by several factors: the variance of the beta estimator ( β^), the prevalence of the exposure in the total population (Pe), the relative risk of the disease given the exposure (RR), and the total population (Tp) [3]. However, previous studies have not extensively identified the differential impacts of individual changes in these parameters on the 95% CI estimates for PAFs [3].

This study aimed to perform a comprehensive comparison and analysis of 3 commonly used methods (Delta, Greenland, and Monte Carlo) for estimating the 95% CIs of PAFs and to identify the individual impacts of the primary parameters involved.

METHODS

Notations and Definitions

We used the following notations and definitions in this study, as shown in Table 1.

Levin Formula

The Levin formula uses the Pe, whereas the Miettinen formula uses the prevalence of exposure among cases (Pc) [11]. Both methods use the RR. The Levin formula is easier to use because it requires less data. Typically, the Pe and the RR are easier to obtain. In contrast, the Miettinen formula requires knowledge of the prevalence of the exposure among those with the disease, necessitating more specific data. Therefore, the 95% CI is generally calculated based on the Levin formula, which is more commonly used for estimating the PAF [11].

Levin formula: PAF=Pe(RR-1)Pe(RR-1)+1; Miettinen formula: PAF=Pc(RR-1RR)

Delta, Greenland, and Monte Carlo Methods for the Estimation of 95% Confidence Intervals

The Delta method approximates V[PÂF] utilizing the Taylor series expansion [9]. V[PÂF] is estimated from the variances of the Pe estimator (V[ P^e]) and β^, as well as the RR and Pe. The 95% CIs for the PAF are determined by adding and subtracting 1.96 times the standard error (SE) of the PAF, which is calculated as the square root of V[PÂF]. This approach corresponds to the 2.5th percentiles and 97.5th percentiles of the standard normal distribution [12].

O=Pe1-Pe

V[P^e]=Pe(1-Pe)Tp

V[PA^F]=V[P^e](RR-1)2+V[β^](RR×Pe)2(Pe(RR-1)+1)4

95%CI of PAF=PAF±1.96×SEPAF=PAF±1.96×V[PA^F]

The Greenland method is an alternative approach to the Delta method for estimating the V[PÂF] [13]. This method takes into account the odds of Pe (O) and provides a more detailed estimation by considering the variability in the RR and the Pe. Specifically, V[PÂF] is calculated using the following formula:

V[PA^F]=[(V[P^e])((RR-1)2+V[β^](RR)2)+V[β^]O2RR2]+(1-PAF)2(1+O)2

The method incorporates the squared difference between the RR and 1.00, the square of [ β^], and their interactions with the O and Tp. Once V[PÂF] is calculated, the 95% CI for the PAF is obtained by transforming the PAF using the exponential function and the standard normal quantile (±1.96), adjusted for the V[PÂF], as shown below:

95%CI of PAF=1-(1-PAF)e(±1.96×V[PA^F])

In a Monte Carlo simulation, 10 000 random samples are generated for both RR and Pe based on the 4 primary parameters: [ β^], Pe, RR, and Tp [14]. RR estimation employs a log-normal distribution, while Pe estimation utilizes a binomial distribution. For each iteration, the simulation estimates the distribution of the PAF by calculating the individual PAF using the estimated RR and Pe. Point estimates from each simulation are used to determine the median PAF and its 95% CI, defined as the 2.5th percentiles and 97.5th percentiles of the PAF distribution, respectively (Supplemental Material 1).

Data Simulation

A simulation was conducted in order to estimate the 95% CI of the PAF by systematically varying the hypothetical values of the 4 primary parameters (V[ β^], RR, Pe, and Tp). The simulation was performed across 3 methods: Greenland, Delta, and Monte Carlo, with the goal of encompassing a broad spectrum of epidemiological scenarios. The parameter ranges were as follows:

(1) Pe: Pe was generated across a wide spectrum, ranging from 0.01 to 0.99 (1-99%), to encompass the extensive range of prevalence rates encountered in epidemiological studies. This range can include diseases with a variety of characteristics, ranging from rare conditions to widespread health issues. The specific segmentation of Pe values was designed to provide a more detailed examination of the changes in the 95% CIs within the lower and higher extremes of prevalence, where the influence of prevalence on the estimation of the PAF is assumed to be more pronounced.

(2) RR: The RR systematically varied from 1.2 to 10.0, capturing a wide range of possible associations between exposures and outcomes. These variations allow the evaluation of the performance of the method under different association strengths, which can be adapted to a variety of epidemiological scenarios.

(3) Tp: Tp was varied from 100 to 10 million to simulate studies at different scales. This range considers the variability in study sizes, from small pilot studies to large-scale epidemiological studies. The variation in Tp allows us to examine the impact of the sample size on PAF estimates.

(4) V[ β^]: Instead of using the p-value of RR or V[ β^] directly to assess the variability or uncertainty of the RR, this simulation utilized the ratio of the upper to lower bound of the 95% CI for the RR (CIR). This choice stems from the common practice among researchers of employing the RR and its corresponding 95% CI as statistics for PAF calculation. The CIR offers a straightforward measure of the width of the CI and indicates the precision of the RR estimate. A wider CI (and thus, a higher CIR) signifies greater uncertainty in the RR, potentially impacting the reliability of the PAF estimate.

In addition to the primary parameters, supplementary auxiliary parameters necessary for computing the 95% CI of the PAF for each combination include the natural logarithm of the RR, denoted as β, and V[P^e]. We also derived O and the SE of the natural logarithm of RR. Furthermore, the Z-score was approximated, and the p-value was determined using the normal distribution approximation method. V[ β^] and V[PÂF] were computed by employing the Levin formula for PAF estimation [15].

Perturbation for Sensitivity Analysis

The absolute value of sensitivity | S(x, δ_x)| of the PAF to changes in the 4 parameters (V[ β^], Pe, RR, and Tp) were estimated using perturbation analysis [16]. The perturbation method involves applying a small perturbation (δ) to each parameter individually and assessing the resultant change in PAF [17]. This perturbation (δ) is applied to one parameter at a time, with all other parameters held constant at their baseline values, thus isolating the effects of each parameter according to the equation below:

S(x,δx)=△V[PA^F](x(1+δ))-△V[PA^F](x)x·δ

- △V[PÂF](x(1+δ)) represents the recalculated V[PÂF] after x has been adjusted by a small perturbation change δ.

- △V[PÂF](x) is the original V[PÂF] after adjusting x.

- x·δ represents the absolute change in x due to the proportional adjustment δ.

- x refers to parameters including V[ β^], Pe, RR, and Tp.

We introduced small perturbations (δ=0.01) to the parameters and recalculated V[PÂF] using both the Delta and Greenland methods to generate new V[PÂF] values that reflect the change (δ) made to the parameters. The Kruskal-Wallis test was applied to evaluate the differences in sensitivity values | S(x, δ_x)| among the 4 parameters after the perturbations, as well as to assess whether the median differences among the independent sensitivity samples obtained after perturbing each parameter were statistically significant. A significance level of 0.05 was set for all results, and a p-value of less than 0.05 was interpreted as indicating statistically significant differences in sensitivity | S(x, δ_x)| among the parameters.

R Shiny Application

We developed an application, entitled the “PAF 95% CI Explorer: Greenland, Delta, Monte Carlo Edition” (GDM-PAF CI Explorer), using Shiny within R, which incorporates a variety of functions and formulas. This tool allows individuals without proficiency in R programming to harness its well-developed functionality for easy analysis and visualization of their personal databases. The application is accessible online at https://sjunlee.shinyapps.io/GDMPAFCIExplorer/ (GitHub: https://github.com/s7unlee/GDM-PAF-CI-Explorer). All statistical analyses were conducted using R version 3.6.1 (R Foundation for Statistical Computing, Vienna, Austria).

Ethics Statement

This study was approved by the Institutional Review Broad of the Seoul National University Hospital (IRB No. C-1911-188-1084).

RESULTS

Despite an increase in the CIR with an RR of 1.2 (which resulted in a higher p-value), no discernible difference was observed in the 95% CIs derived from the 3 different methods (Figure 1). However, with an RR of 3.0, differences began to emerge among the 3 methods when the p-value of the RR was <0.001. Because the upper and lower bounds of the 95% CI in the Delta method are calculated at an equal distance from the point estimate (PAF), disparities arose in the Delta estimates compared to those of the Greenland and Monte Carlo methods as the p-value increased. When the p-value was <0.05, most cases with a Pe <0.9 exhibited uncertainty in the estimates from the Greenland method (Figure 1). For an RR of 5.0, the contrast between Delta and the other methods was evident, as the estimate of the Greenland method had already started to become uncertain with a p-value <0.001. As the RR increased, a trend for a decreasing p-value cut-off was observed, where the distinctions between the 3 methods began to appear.

With a similar p-value, increasing the RR resulted in different 95% CI estimates among the 3 methods, and the estimates diverged significantly with increasing RRs, particularly in the lower p-value range (Figure 2). Furthermore, as the p-value decreased, the RR cut-off, at which differences in the 95% CI estimates from the 3 methods began to emerge, also decreased (Figure 2).

Supplemental Material 2 illustrates the distribution of differences in | S(x, δ_x)| for the Delta and Greenland methods across the entire simulation dataset. The median | S(x, δ_x)| of V[ β^] exhibited the highest value, followed by Pe, RR, and Tp using both methods. In the Delta method, the median absolute values of sensitivity for V[ β^], Pe, RR, and Tp were 8.28E-02 (interquantile range [IQR], 3.47E-02 to 1.64E-01), 1.96E-02 (IQR, 5.81E-03 to 6.43E-02), 2.64E-03 (IQR, 6.96E-04 to 1.13E-02), and 1.07E-11 (IQR, 3.51E-14 to 3.15E-09), respectively (Supplemental Material 2). Based on the Greenland method, the median absolute values of sensitivity for V[ β^], Pe, RR, and Tp were 0.59 (IQR, 0.19 to 0.88), 0.28 (IQR, 0.10 to 0.61), 0.03 (IQR, 7.65E-03 to 0.10), and 7.51E-11 (IQR, 2.57E-13 to 2.25E-08), respectively (refer to Supplemental Material 2). The | S(x, δ_x)| values of V[ β^] and Tp remained constant across each parameter, while there was variability in V[ β^] of Pe and RR depending on the parameter values (Supplemental Material 2). The Kruskal-Wallis test yielded statistically significant differences between | S(x, δ_x)| for the 4 parameters (V[ β^], Pe, RR, and Tp) using both the Delta and Greenland methods.

Supplemental Material 3 presents the simulation results computed from combinations of Tp values of 1000 and 100 000; RRs of 1.2, 1.5, 3.0, 7.0, and 10.0; Pe values of 0.01, 0.05, 0.10, 0.50, and 0.90; and CIR values of 1.1, 1.3, 5.0, 10.0, 20.0, and 50.0. In the Delta method, the absolute value of sensitivity ranges for V[β^], Pe, RR, and Tp were 1.43E-04 to 0.60, 3.30E-06 to 1.22, 1.78E-07 to 0.82, and 3.89E-14 to 8.62E-07, respectively. For the Greenland method, these ranges were 1.44E-04 to 0.98, 1.74E-05 to 5.59, 1.83E-07 to 1.40, and 3.99E-14 to 4.46E-06, respectively.

Supplemental Material 4 presents the p-values obtained based on the RR and CIR. When the RR was 1.2, the p-value was calculated as 0.02 with a CIR of 1.36. However, with the same CIR of 1.36, when the RR increased to 10, the p-value drastically decreased to <0.001. For example, as illustrated in Supplemental Material 5, when the RR was 1.55 with a CIR of 1.68, the p-value was <0.001, resulting in an RR of 1.55 (95% CI, 1.20 to 2.01). Conversely, when the RR was 10 with a p-value of <0.001, the result was an RR of 10.0 (95% CI, 2.43 to 41.20).

DISCUSSION

In this study, we compared 3 methods (Greenland, Delta, and Monte Carlo) to estimate the 95% CIs of PAFs. Our study aimed to comprehensively evaluate these methods, which have previously been used individually in the literature [1,4]. Additionally, we proposed a novel method for determining the most influential factor in estimating the 95% CI using sensitivity analysis based on perturbation. The purpose of this analysis was to ascertain which factor—Pe, RR, Tp, or V[β^] —has the greatest impact on 95% CI estimation. This is a novel approach that has not been commonly explored in previous studies.

The computation of the 95% CI for a PAF involves multiple factors, including Pe, Tp, V[β^], and RR. The sensitivity analysis found the most influential factor for 95% CI estimation was V[β^], followed by Pe, RR, and Tp. Since the calculation of the PAF depends on large populations (generally tens of thousands) [18], the number of participants had minimal influence on the 95% CI.

The Greenland, Delta, and Monte Carlo methods can yield divergent results under different circumstances, and no single method is unilaterally superior. Each method includes theoretical considerations that make it more effective in specific situations.

The Greenland method generally yields 95% CIs that are too broad for practical use [7]. However, its computational efficiency was evident in situations with very high prevalence (exceeding 0.9), where it outperformed the Monte Carlo method. This observation aligns with the findings of previous studies [7,19].

The Delta method relies on linear approximations, which may lead to deviations from actual values when exposure prevalence and RR values exceed certain thresholds. However, when using low or minimal RR values, low p-values, a low Pe, and a large Tp, all 3 methods produced comparable 95% CIs. In these conditions, where the CIs were consistent and narrow, the simplicity and ease of implementation of the Delta method become particularly advantageous. This suggests that, despite its limitations, the Delta method remains a useful tool when the CIs are relatively precise and stable [20].

Monte Carlo simulations, although computationally demanding [21], offer more accurate 95% CIs for PAFs [22]. The Monte Carlo method, based on random samples of the posterior region, may yield slightly different CIs with each application. However, this variability can be mitigated by employing a large number of simulations, albeit at the expense of increased computational runtime (Supplemental Material 6). In order to provide practical guidance for researchers in selecting the most appropriate method for estimating the 95% CIs of PAFs, we have summarized the key characteristics, advantages, limitations, and suitable conditions for each of the 3 methods in Supplemental Material 7. This table is designed to offer a concise comparison, enabling researchers to quickly identify the method that best fits their specific study conditions.

The association between effect size and statistical power explains why larger RR values typically result in smaller p-values, provided the sample size is sufficiently large. Larger effect sizes are easier to detect statistically, leading to more significant p-values, which in turn demonstrate stronger evidence supporting the presence of an effect as RR increases. However, to achieve the same p-value with smaller RR values, the 95% CI must be considerably narrower than that for larger RR values [23,24]. This indicates that the variability of the CIs increases as RR values become larger, which can affect the precision of 95% CI estimation. Particularly with exceptionally large RR values, non-linear effects can significantly increase. Under these circumstances, the Monte Carlo method, which is adept at handling uncertainty and variability, can provide more accurate results compared to the Greenland and Delta methods, which rely on linear functions.

However, this study has several limitations. Although our sensitivity analyses considered 4 factors (Pe, RR, Tp, and V[β^]), real-world scenarios may involve additional factors not included in our analysis, potentially influencing PAF estimates. Specifically, the Monte Carlo method cannot be applied to sensitivity analysis of PAF. As the Monte Carlo method performs simulations based on the entire probability distribution of input parameters, it is less influenced by single values or small parameter changes (perturbations). Furthermore, all models were bound by inherent assumptions. For example, the Delta method assumes a linear relationship between the variables and a normal distribution of the estimated parameters, relying on a Taylor series approximation to estimate variance [9]. This can lead to asymmetric and potentially implausible CIs, particularly for rare outcomes or small sample sizes [6,8]. The Greenland method assumes specific distributional properties of exposure and outcome variables, which may not hold in skewed or small samples, potentially causing it to underperform in such scenarios [7]. Lastly, the Monte Carlo simulation method assumes that the resampled probability distributions accurately reflect the true underlying distributions of the input variables [8]. This method can be computationally intensive and may not be feasible for large-scale data or complex diseases. Thus, the Greenland, Delta, and Monte Carlo methods employed in this study could produce inaccurate results if their respective assumptions are violated.

Nonetheless, our study directly compared 3 methods for estimating the 95% CIs of PAFs and presented various epidemiological scenarios based on simulations. Furthermore, the sensitivity analysis expanded existing methods for estimating the PAF by systematically varying Pe, RR, Tp, and CIR, thereby quantitatively evaluating the dependency of the PAF on each parameter. Although calculations were only feasible using the Delta and Greenland methods, a consistent impact size of the parameters on the PAF was confirmed. Beyond merely comparing each method, we also developed a user-friendly R-shiny application, available on our webpage. This tool assists researchers, including clinicians who are new to this area, in choosing the most appropriate method for their specific research conditions.

In conclusion, the method for estimating the 95% CIs of PAFs should be carefully selected based on the unique circumstances of each study. Understanding the complexity of these methods is paramount for increasing the accuracy and reliability of PAF calculations for various diseases, including cancer. Our findings offer practical guidance for researchers conducting epidemiological studies on PAF calculations.

Supplemental Materials

Notes

Conflict of Interest

The authors have no conflicts of interest associated with the material presented in this paper.

Funding

This study was funded by the Korean Foundation for Cancer Research (grant No. CB-2017-A-2) and the National R&D Program for Cancer Control through the National Cancer Center (NCC) funded by the Ministry of Health & Welfare, Republic of Korea (HA21C0140).

Author Contributions

Conceptualization: Lee S, Moon S, Park SK. Data curation: Lee S, Moon S, Park SK. Formal analysis: Lee S, Moon S, Park SK. Funding acquisition: Park SK. Methodology: Lee S, Moon S. Project administration: Park SK. Visualization: Lee S. Writing – original draft: Lee S, Park SK. Writing – review & editing: Moon S, Kim K, Sung S, Hong Y, Lim W, Park SK.

Acknowledgements

Figure. 1.

95% CI for PAF based on Tp, RR, p-value, and CIR according to Delta, Greenland, and Monte Carlo methods (Tp=100 000). CI, confidence interval; PAF, population-attributable fraction; Tp, total population; RR, relative risk; CIR, ratio of upper-to-lower 95% CI of RR; Pe, prevalence of the risk factor in the population; AF, attributable fraction.

Figure. 2.

95% CI for PAF based on Tp, RR, p-value, and CIR according to Delta, Greenland, and Monte Carlo methods (Tp=10 000 000) based on the similar p-value range. CI, confidence interval; PAF, population-attributable fraction; Tp, total population; RR, relative risk of the disease given the exposure; CIR, ratio of upper-to-lower 95% CI of RR; Pe, prevalence of the risk factor in the population; AF, attributable fraction.

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Figure & Data

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