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,
  weights_sum_to_1 = TRUE
)

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: (Optional). If link is a function then yhe 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
weights_sum_to_1: Boolean flag indicating if the weights sum to 1 (normalized weights) or if they are not (unnormalized).

Value

A pif_class object containing the estimate for the total fraction or the ensemble fraction that aggregates several fractions (either potential impact or population attributable).

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-264893202856241] ──
#> 
#> PIF = 27.862% [95% CI: 22.944% to 33.377%]
#> standard_deviation(pif %) = 23.319
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 26.372% (sd %: 36.119) --- [deltapif-206365445053881]
#> • 29.294% (sd %: 29.772) --- [deltapif-063317817552485]
#> ────────────────────────────────────────────────────────────────────────────────

#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-0675244698081724] ──
#> 
#> PAF = 44.018% [95% CI: 38.601% to 49.581%]
#> standard_deviation(paf %) = 18.760
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 41.884% (sd %: 28.509) --- [deltapif-0845195738551864]
#> • 46.068% (sd %: 24.552) --- [deltapif-063953477020697]
#> ────────────────────────────────────────────────────────────────────────────────

# 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-210215247889527] ──
#> 
#> PAF = 68.841% [95% CI: 64.670% to 73.013%]
#> standard_deviation(paf %) = 3.092
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 61.612% (sd %: 3.740) --- [deltapif-021223603483794]
#> • 18.832% (sd %: 1.529) --- [deltapif-0906340178564226]
#> ────────────────────────────────────────────────────────────────────────────────

# 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-0876777326556075] ──
#> 
#> PAF = 75.731% [95% CI: 74.815% to 76.612%]
#> standard_deviation(paf %) = 1.668
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 76.180% (sd %: 2.271) --- [deltapif-141602391449351]
#> • 75.299% (sd %: 2.435) --- [deltapif-0480725266268846]
#> ────────────────────────────────────────────────────────────────────────────────