Combine Potential Impact Fractions and Population Attributable Fractions

Combine potential impact fractions or the population attributable fractions to either generate the total fraction from the fractions of subpopulations (pif_total/paf_total) or the ensemble fraction of a population from different (independent) exposures.

Usage

paf_total(
  paf1,
  ...,
  weights,
  var_weights = 0,
  var_pif_weights = NULL,
  conf_level = 0.95,
  link = "log-complement",
  link_inv = NULL,
  link_deriv = NULL,
  quiet = FALSE,
  label = NULL
)

pif_total(
  pif1,
  ...,
  weights,
  var_weights = 0,
  var_pif_weights = NULL,
  conf_level = 0.95,
  link = "log-complement",
  link_inv = NULL,
  link_deriv = NULL,
  quiet = FALSE,
  label = NULL,
  is_paf = FALSE
)

paf_ensemble(
  paf1,
  ...,
  weights = NULL,
  var_weights = 0,
  var_pif_weights = NULL,
  link = "identity",
  link_inv = NULL,
  link_deriv = NULL,
  conf_level = 0.95,
  label = NULL,
  quiet = FALSE
)

pif_ensemble(
  pif1,
  ...,
  weights = NULL,
  var_weights = 0,
  var_pif_weights = NULL,
  link = "identity",
  link_inv = NULL,
  link_deriv = NULL,
  conf_level = 0.95,
  quiet = FALSE,
  label = NULL,
  is_paf = FALSE
)

Arguments

paf1: A population attributable fraction (class pif_class)
...: The remaining potential impact fractions or (respectively) population attributable fractions.
weights: A vector containing the proportion of the population for each of the categories (for each of the pifs given).
var_weights: link_covariance structure for the weights. Can be 0 (default) if the weights are not random, a vector if only the link_variances of the weights are available or a link_covariance matrix.
var_pif_weights: covariance matrix with row i and column j representing the covariance between the i-th potential impact fraction of the list and the j-th weight
conf_level: Confidence level for the confidence interval (default 0.95).
link: Link function such that the pif confidence intervals stays within the expected bounds.
link_inv: The inverse of link. For example if link is logit this should be inv_logit.
link_deriv: Derivative of the link function. The function tries to build it automatically from link using Deriv::Deriv().
quiet: Whether to show messages.
label: Character identifier for the impact fraction. This is for
pif1: A potential impact fraction (class pif_class)
is_paf: Whether the computed quantity is a population attributable fraction or not

Total potential impact fraction

Assuming the overall population can be subdivided into $N$ distinct subpopulations each of them with a different potential impact fraction (or population attributable fraction) we can estimate the total population attributable fraction or potential impact fraction of the whole population as:

$$ \text{PIF}_{\text{Total}} = \sum\limits_{i = 1}^{N} \pi_i \cdot \text{PIF}_i $$

where each $\text{PIF}_i$ corresponds to the potential impact fraction of the i-th subpopulation and $\pi_i$ correspond to the proportion of the total population occupied by $\text{PIF}_i$. The weights are such that $\sum_{i=1}^{N} \pi_i = 1$.

Ensemble potential impact fraction

If a population is exposed to $K$ different independent risk factors then the ensemble impact fraction of the combination of those factors can be written as:

$$ \text{PIF}_{\text{Ensemble}} = 1 - \prod\limits_{\ell = 1}^{K} \Big(1 - \pi_{\ell} \cdot \text{PIF}_{\ell}\Big) $$

where each $\text{PIF}_{\ell}$ corresponds to the potential impact fraction of the $\ell$-th risk factor for the same population.

Examples

#Potential impact fraction for women
pif_women <- pif(0.32, 0.1, 1.2, quiet = TRUE, var_p = 0.1)

#Potential impact fraction for men
pif_men <- pif(0.27, 0.1, 1.3, quiet = TRUE, var_p = 0.1)

#Population potential impact fraction with 49% men and 51% women
pif_total(pif_men, pif_women, weights = c(0.49, 0.51), link = "logit")
#> 
#> ── Potential Impact Fraction: [deltapif-127051311021181] ──
#> 
#> PIF = 27.862% [95% CI: 22.944% to 33.377%]
#> standard_deviation(pif %) = 23.319
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 26.372% (sd %: 36.119) --- [deltapif-0786362576309776]
#> • 29.294% (sd %: 29.772) --- [deltapif-0340637876031232]
#> ────────────────────────────────────────────────────────────────────────────────

#Population attributable  fraction for women
paf_women <- paf(0.32, 1.3, quiet = TRUE, var_p = 0.1)

#Population attributable  fraction for men
paf_men <- paf(0.27, 1.3, quiet = TRUE, var_p = 0.1)
paf_total(paf_men, paf_women, weights = c(0.49, 0.51), link = "logit")
#> 
#> ── Population Attributable Fraction: [deltapif-131763369782387] ──
#> 
#> PAF = 44.018% [95% CI: 38.601% to 49.581%]
#> standard_deviation(paf %) = 18.760
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 41.884% (sd %: 28.509) --- [deltapif-090281494922007]
#> • 46.068% (sd %: 24.552) --- [deltapif-16865047424422]
#> ────────────────────────────────────────────────────────────────────────────────

# Calculate the ensemble from lead and radiation exposure
paf_lead <- paf(0.2, 2.2, quiet = TRUE, var_p = 0.001)
paf_rad  <- paf(0.1, 1.2, quiet = TRUE, var_p = 0.0001)
pif_ensemble(paf_lead, paf_rad)
#> 
#> ── Population Attributable Fraction: [deltapif-0239959572121817] ──
#> 
#> PAF = 68.841% [95% CI: 64.670% to 73.013%]
#> standard_deviation(paf %) = 3.092
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 61.612% (sd %: 3.740) --- [deltapif-096812583702407]
#> • 18.832% (sd %: 1.529) --- [deltapif-0102603036376881]
#> ────────────────────────────────────────────────────────────────────────────────

# Totals and ensembles can be combined
pif_lead_women <- paf(0.27, 2.2, quiet = TRUE, var_p = 0.001)
pif_rad_women  <- paf(0.12, 1.2, quiet = TRUE, var_p = 0.001)
pif_women      <- pif_ensemble(pif_lead_women, pif_rad_women)
pif_lead_men   <- paf(0.30, 2.2, quiet = TRUE, var_p = 0.001)
pif_rad_men    <- paf(0.10, 1.2, quiet = TRUE, var_p = 0.001)
pif_men        <- pif_ensemble(pif_lead_men, pif_rad_men)

pif_total(pif_men, pif_women, weights = c(0.49, 0.51))
#> 
#> ── Population Attributable Fraction: [deltapif-106446420865157] ──
#> 
#> PAF = 75.731% [95% CI: 74.815% to 76.612%]
#> standard_deviation(paf %) = 1.668
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 76.180% (sd %: 2.271) --- [deltapif-0356124415722617]
#> • 75.299% (sd %: 2.435) --- [deltapif-0876777326556075]
#> ────────────────────────────────────────────────────────────────────────────────