Components of the variance of a potential impact fraction

Splits the delta-method variance of a potential impact fraction (or population attributable fraction) into the part due to the uncertainty in the exposure prevalences p, the part due to the uncertainty in the relative-risk parameters beta, and, for totals and ensembles, the part due to the uncertainty in the weights.

Usage

variance_components(x, ...)

Arguments

x: A potential impact fraction created with pif(), paf(), pif_total(), pif_ensemble() or their paf counterparts.
...: Not used.

Value

A named numeric vector with entries p, beta and weights.

Details

As the prevalences and the relative risks are estimated independently, the variance of the fraction is the sum of a prevalence component and a relative-risk component: $$ \text{Var}\big(\widehat{\textrm{PIF}}\big) \approx \nabla_p \textrm{PIF}^{\top} \Sigma_p \nabla_p \textrm{PIF} + \nabla_\beta \textrm{PIF}^{\top} \Sigma_\beta \nabla_\beta \textrm{PIF}. $$ Comparing both components shows whether a more precise exposure survey or a more precise relative risk would most reduce the uncertainty of the fraction. For totals and ensembles the remaining variance (from var_weights and var_pif_weights) is reported as the weights component. Components are on the scale of the fraction (not of the link) and add up to variance().

Examples

my_pif <- pif(p = 0.085, p_cft = function(p) 0.85 * p, beta = log(1.59),
              var_p = 0.01^2, var_beta = 0.165^2, quiet = TRUE)
variance_components(my_pif)
#>            p         beta      weights 
#> 6.439931e-07 9.199774e-06 0.000000e+00 

# Share of the variance due to each source
variance_components(my_pif) / variance(my_pif)
#>         p      beta   weights 
#> 0.0654214 0.9345786 0.0000000