# Issues with incorrect computing of population attributable fraction (PAF) in a global perspective on coal-fired power plants and burden of lung cancer

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## Abstract

All observational studies are liable to confounding and Levin’s formula becomes useless in practice for unbiasedly estimating PAF. With respect to causal interpretation of PAF in public health setting, unbiased estimation of PAF requires several assumptions which are ignored in practice. We recommend using Miettinen PAF formula with careful consideration about possibility of bias in study design and analysis.

## Keywords

Population attributable fraction Confounding, ecological study## Abbreviation

- PAF
Population attributable fraction

To the Editor

_{it}= RR

_{0}

^{per capita coal capacity i(t-10)}. They calculated standardized attributable cases using

*PAF*

_{it}.

- 1.
There is a mistake in the calculation of PAF using proportion of males or females as p

_{e}. In the Levin’s formula, p_{e}is the proportion of the population exposed to the risk factor. The assumption of spatial homogeneity in exposure distribution in country level was implicit in the definition of risk factor i.e. per capital coal capacity; therefore p_{e}would be equal to 1. In this case Levin’s formula is equal to ((RR-1)/RR). - 2.
Levin’s formula is unbiased in the absence of confounding and effect modification [3, 4]. All observational studies are liable to confounding and formula 1 becomes useless in practice for unbiasedly estimating PAF [3]. The Meittinen formula (PAF = p

_{c}(RR_{adj}-1)/RR_{adj}) is appropriate for use in practice as it provide unbiased estimate of PAF with adjusted RR when confounding exists [3]. Opposed to the Levin’s formula, it requires information about the prevalence of exposure among the cases (p_{c}). As p_{e}is equal 1, implies p_{c}is equal one, the Levin and Meittinen formulas will give the same results in this special case. - 3.
One of the main assumptions underlying the PAF is no bias in the study design. In this observational, ecological study, data were collected for 83 countries and the unit of analysis was country. Thus, in an ecological study when the data were aggregated, the outcome measures are likely to be biased [5]. Ecological fallacy is another misinterpretation of ecological study results. Another issue is that RRs in PAF calculation were derived from sex-specific analysis, but all values of confounding variables except of smoking were not sex-specific. It is necessary to know that the ecological studies must use for generating hypothesis rather than deriving an adjusted association between risk factors and diseases.

- 4.
The authors used a longitudinal Poisson model to analyze the association of per capita coal capacity with incidence rate of lung cancer. In this model the exponentiated coefficients are incidence-rate ratio not risk ratio. However, when the disease is uncommon, odds ratio (OR), rate ratio and hazard ratio (HR) can be used instead of RR in Menttinen formula [3].

In sum, unbiased estimation of PAF requires several assumptions which are ignored in practice. We recommend using Miettinen PAF formula with careful consideration about possibility of bias in study design and analysis [3].

Yours Sincerely,

Dr. Ahmad Khosravi.

Dr. Mohammad Ali Mansournia.

## Notes

### Acknowledgements

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### Authors’ contributions

AKh wrote the paper and MMM revised the paper. Both authors approved the final version of the paper.

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## References

- 1.Lin CK, Lin RT, Chen T, Zigler C, Wei Y, Christiani DC. A global perspective on coal-fired power plants and burden of lung cancer. Environ Health. 2019;18(1):9. Epub 2019/01/30.CrossRefGoogle Scholar
- 2.Darrow LA. Commentary: errors in estimating adjusted attributable fractions. Epidemiology. 2014;25(6):917–8 Epub 2014/09/30.CrossRefGoogle Scholar
- 3.Mansournia MA, Altman DG. Population attributable fraction. BMJ. 2018;360:k757 Epub 2018/02/24.CrossRefGoogle Scholar
- 4.Darrow LA, Steenland NK. Confounding and bias in the attributable fraction. Epidemiology. 2011;22(1):53–8 Epub 2010/10/27.CrossRefGoogle Scholar
- 5.Sedgwick P. Ecological studies: advantages and disadvantages. BMJ. 2014;348:g2979 Epub 2014/08/19.CrossRefGoogle Scholar

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