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
polbrdn 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.
- expoprev
Proportion of the population exposed, in
[0, 1].- rrlo, rrup
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).
It has class c("rmbl_attrfrac", "rmbl_measure", "numeric"), so it
prints as a box with its inputs (kept in attribute "lakner");
arithmetic, [, c() and as.numeric() give plain
numbers.
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.
Examples
attrfrac(1.5, 0.4)
#> ╭── Population attributable fraction ─────────────────────────────────────────╮
#> │ Attributable fraction 16.67% │
#> │ Relative risk 1.5 │
#> │ Exposed 40.00% │
#> │ Formula Levin: p (RR - 1) / (1 + p (RR - 1)) │
#> │ │
#> │ About 16.7% of the cases in this population would not occur without the │
#> │ exposure, if the relative risk is causal. │
#> │ │
#> │ It holds only if the relative risk is causal and unconfounded, and │
#> │ only for a population exposed as this one is (Rothman et al. 2008, │
#> │ chapter 5). │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
attrfrac(2, 1) # everyone exposed: 1 - 1/RR
#> ╭── Population attributable fraction ─────────────────────────────────────────╮
#> │ Attributable fraction 50.00% │
#> │ Relative risk 2 │
#> │ Exposed 100.00% │
#> │ Formula Levin: p (RR - 1) / (1 + p (RR - 1)) │
#> │ │
#> │ About 50.0% of the cases in this population would not occur without the │
#> │ exposure, if the relative risk is causal. │
#> │ │
#> │ It holds only if the relative risk is causal and unconfounded, and │
#> │ only for a population exposed as this one is (Rothman et al. 2008, │
#> │ chapter 5). │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
# With the confidence limits of the relative risk.
paf <- attrfrac(1.08, 1, rrlo = 1.06, rrup = 1.09)
attr(paf, "ci")
#> lower upper
#> 0.05660377 0.08256881
# A protective exposure gives a negative fraction.
attrfrac(0.8, 0.5)
#> ╭── Population attributable fraction ─────────────────────────────────────────╮
#> │ Attributable fraction -11.11% │
#> │ Relative risk 0.8 │
#> │ Exposed 50.00% │
#> │ Formula Levin: p (RR - 1) / (1 + p (RR - 1)) │
#> │ │
#> │ The exposure is protective: without it there would be about 11.1% more │
#> │ cases. │
#> │ │
#> │ It holds only if the relative risk is causal and unconfounded, and │
#> │ only for a population exposed as this one is (Rothman et al. 2008, │
#> │ chapter 5). │
#> ╰─────────────────────────────────────────────────────────────────────────────╯
# A prevalence above 1 is refused.
try(attrfrac(1.5, 40))
#> Error : expoprev must be in [0,1]; got 40