The share of all cases in a population that is due to an exposure: the
fraction that would not occur if the exposure were removed and nothing
else changed. It combines how strongly the exposure raises risk (the
relative risk) with how common it is (the prevalence); multiplied by the
number of cases it gives the attributable cases, as
pollution_burden does.
Arguments
- rr
Relative risk among the exposed, a single finite positive number. A relative risk below 1 gives a negative fraction (a protective exposure); 0, negative, infinite and missing values are errors.
- exposure_prevalence
Proportion of the population exposed, in
[0, 1].- rr_lower, rr_upper
Optional confidence limits of
rr; when both are given the fraction carries the limits that the same formula gives at them (Levin's formula is monotone in the relative risk, so the interval of the fraction is the image of the interval of the risk).
Value
A single number, the attributable fraction: in [0, 1) for
rr >= 1, negative for a protective exposure. When the limits
were given it carries an attribute ci, a named numeric vector
(lower, upper).
Details
Levin's formula (Levin 1953; Rothman, Greenland and Lash 2008, chapter 5): $$PAF = p (RR - 1) / (1 + p (RR - 1)),$$ with \(p\) the proportion exposed. With everyone exposed it is \(1 - 1/RR\). It assumes the relative risk is causal and unconfounded and that removing the exposure changes nothing else. A relative risk below 1 gives a negative fraction (the exposure protects).
When both confidence limits of the RR are given, the fraction's limits
are the formula applied to them: the formula increases with the RR, so
the interval of the fraction is the image of the interval of the risk.
Refusals: an rr or limit that is not a single finite positive
number (logical values included), a prevalence outside \([0, 1]\), only
one of the two limits, or limits that do not enclose rr.
References
Rothman, K. J., Greenland, S. and Lash, T. L. (2008). Modern Epidemiology, 3rd ed., chapter 5. Levin, M. L. (1953). The occurrence of lung cancer in man. Acta Unio Internationalis Contra Cancrum 9, 531-541.
See also
crf_pm25 for a relative risk from a
concentration, pollution_burden for the attributable
cases.
Examples
attributable_fraction(1.5, 0.4)
#> [1] 0.1666667
attributable_fraction(2, 1) # everyone exposed: 1 - 1/RR
#> [1] 0.5
# With the confidence limits of the relative risk.
paf <- attributable_fraction(1.08, 1, rr_lower = 1.06, rr_upper = 1.09)
attr(paf, "ci")
#> lower upper
#> 0.05660377 0.08256881
# A protective exposure gives a negative fraction.
attributable_fraction(0.8, 0.5)
#> [1] -0.1111111
# A prevalence above 1 is refused.
try(attributable_fraction(1.5, 40))
#> Error : exposure_prevalence must be in [0,1]; got 40