Weighted Adjusted Fractions

Calculates the weighted adjusted potential impact fractions (or population attributable fractions). Each individual fraction $\widehat{\text{PIF}}_i$ is rescaled proportionally so that together they are consistent with the ensemble fraction.

Usage

weighted_adjusted_paf(
  paf1,
  ...,
  weights = NULL,
  var_weights = 0,
  var_pif_weights = NULL,
  var_p = NULL,
  var_beta = NULL,
  conf_level = 0.95,
  label_ensemble = NULL,
  label_sum = NULL,
  quiet = FALSE
)

weighted_adjusted_pif(
  pif1,
  ...,
  weights = NULL,
  var_weights = 0,
  var_pif_weights = NULL,
  var_p = NULL,
  var_beta = NULL,
  pif_total_link = "log-complement",
  pif_total_link_inv = NULL,
  pif_total_link_deriv = NULL,
  pif_ensemble_link = "identity",
  pif_ensemble_link_inv = NULL,
  pif_ensemble_link_deriv = NULL,
  conf_level = 0.95,
  label_ensemble = NULL,
  label_sum = NULL,
  quiet = FALSE
)

Arguments

paf1: A population attributable fraction (class pif_class). This is the first of the individual fractions to be adjusted.
...: The remaining potential impact fractions (class pif_class). All fractions must be of the same type (all PIF or all PAF).
weights: Weights for the ensemble (pif_ensemble). Passed directly to pif_ensemble() / paf_ensemble(). Defaults to NULL (equal weights of 1 for each fraction).
var_weights: Covariance structure for weights. Passed directly to pif_ensemble() / paf_ensemble(). Defaults to 0.
var_pif_weights: Covariance matrix between individual fractions and weights. Passed to pif_ensemble() / paf_ensemble(). Defaults to NULL.
var_p: Covariance matrix for the prevalence parameters p across all fractions. Passed to covariance() when computing cross-covariances. Defaults to NULL.
var_beta: Covariance matrix for the relative-risk parameters beta across all fractions. Passed to covariance() when computing cross-covariances. Defaults to NULL.
conf_level: Confidence level for the confidence interval (default 0.95).
label_ensemble: Character label for the internally constructed ensemble fraction. Defaults to NULL (auto-generated).
label_sum: Character label for the internally constructed sum (total) fraction. Defaults to NULL (auto-generated).
quiet: Whether to show messages.
pif1: A potential impact fraction (class pif_class). This is the first of the individual fractions to be adjusted.
pif_total_link: Link to pass to pif_total in the denominator of the adjustment
pif_total_link_inv: Inverse of the link to pass to pif_total in the denominator of the adjustment
pif_total_link_deriv: Derivative of the link to pass to pif_total in the denominator of the adjustment
pif_ensemble_link: Link to pass to pif_ensemble in the numerator of the adjustment
pif_ensemble_link_inv: Inverse of the link to pass to pif_ensemble in the numerator of the adjustment
pif_ensemble_link_deriv: Derivative of the link to pass to pif_ensemble in the numerator of the adjustment

Value

A named list of pif_class objects, one per input fraction, each being the weighted adjusted PAF (or PIF, respectively).

Formula

The weighted adjusted fraction for the $i$-th exposure is: $$ \widehat{\text{PIF}}_i^{\text{adj}} = \frac{\widehat{\text{PIF}}_i}{\sum_{j=1}^{n} \widehat{\text{PIF}}_j} \cdot \widehat{\text{PIF}}_{\text{Ensemble}} $$

Using the log-transform, the variance is: $$ \begin{aligned} \operatorname{Var}\!\Big[\ln \widehat{\text{PIF}}_i^{\text{adj}}\Big] &= \operatorname{Var}\!\left[\ln \widehat{\text{PIF}}_i\right] + \operatorname{Var}\!\left[\ln \textstyle\sum_j \widehat{\text{PIF}}_j\right] + \operatorname{Var}\!\left[\ln \widehat{\text{PIF}}_{\text{Ensemble}}\right] \\ &\quad + 2\Bigg[ \operatorname{Cov}\!\Big(\ln \widehat{\text{PIF}}_i, \ln \widehat{\text{PIF}}_{\text{Ensemble}}\Big) - \operatorname{Cov}\!\Big(\ln \widehat{\text{PIF}}_i, \ln \textstyle\sum_j \widehat{\text{PIF}}_j\Big) - \operatorname{Cov}\!\Big(\ln \widehat{\text{PIF}}_{\text{Ensemble}}, \ln \textstyle\sum_j \widehat{\text{PIF}}_j\Big) \Bigg] \end{aligned} $$

where each log-covariance is approximated via the delta method: $$ \operatorname{Cov}\!\big(\ln X, \ln Y\big) \approx \frac{\operatorname{Cov}(X, Y)}{X \cdot Y} $$

The covariances $\operatorname{Cov}(\widehat{\text{PIF}}_i, \cdot)$ are computed by the internal function cov_total_pif() applied to the individual fraction and the internally constructed sum/ensemble objects, which automatically propagates the full uncertainty structure already embedded in those objects.

Examples

paf_lead <- paf(0.2, 2.2, quiet = TRUE, var_p = 0.001, label = "Lead")
paf_rad  <- paf(0.1, 1.2, quiet = TRUE, var_p = 0.0001, label = "Radiation")

weighted_adjusted_paf(paf_lead, paf_rad)
#> $Lead
#> 
#> ── Population Attributable Fraction: [Lead_adj] ──
#> 
#> PAF = 52.726% [95% CI: 46.604% to 58.847%]
#> standard_deviation(paf %) = 3.123
#> 
#> $Radiation
#> 
#> ── Population Attributable Fraction: [Radiation_adj] ──
#> 
#> PAF = 16.116% [95% CI: 13.560% to 18.671%]
#> standard_deviation(paf %) = 1.304
#> 

pif_lead <- pif(0.2, p_cft = 0.1, beta = log(2.2), quiet = TRUE,
                var_p = 0.001, label = "Lead")
pif_rad  <- pif(0.1, p_cft = 0.05, beta = log(1.2), quiet = TRUE,
                var_p = 0.0001, label = "Radiation")

weighted_adjusted_pif(pif_lead, pif_rad)
#> $Lead
#> 
#> ── Potential Impact Fraction: [Lead_adj] ──
#> 
#> PIF = 9.591% [95% CI: 4.226% to 14.956%]
#> standard_deviation(pif %) = 2.737
#> 
#> $Radiation
#> 
#> ── Potential Impact Fraction: [Radiation_adj] ──
#> 
#> PIF = 0.972% [95% CI: 0.595% to 1.349%]
#> standard_deviation(pif %) = 0.192
#>

Usage

Arguments

Value

Formula

See also

Examples