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Combine Potential Impact Fractions and Population Attributable Fractions

totalpifpaf.Rd

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,
  var_p = NULL,
  var_beta = 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,
  var_p = NULL,
  var_beta = 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,
  var_p = NULL,
  var_beta = 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,
  var_p = NULL,
  var_beta = 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

var_p, var_beta

(Optional) Covariance matrices between the estimates of p (respectively beta) of different fractions. Row and column names are the identifiers given to the parameters as named lists in pif() or paf(); a vector-valued parameter repeats its identifier once per element. The diagonal blocks must equal the variances given to each fraction. Fractions that share an identifier are always correlated, so these matrices are only needed for correlations between different estimates (e.g. relative risks from the same meta-analysis).

conf_level

Confidence level for the confidence interval (default 0.95).

Link function such that the pif confidence intervals stays within the expected bounds.

(Optional). If link is a function then yhe inverse of link. For example if link is logit this should be inv_logit.

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-114269487842917] ──
#> 
#> PIF = 27.862% [95% CI: 3.823% to 78.962%]
#> standard_deviation(pif %) = 23.319
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 26.372% (sd %: 36.119) --- [deltapif-106199914454589]
#> • 29.294% (sd %: 29.772) --- [deltapif-256440833799741]
#> ────────────────────────────────────────────────────────────────────────────────

#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-0415406398921884] ──
#> 
#> PAF = 44.018% [95% CI: 15.026% to 77.759%]
#> standard_deviation(paf %) = 18.760
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 41.884% (sd %: 28.509) --- [deltapif-0558514421664496]
#> • 46.068% (sd %: 24.552) --- [deltapif-00751059001682063]
#> ────────────────────────────────────────────────────────────────────────────────

# 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-138291086057034] ──
#> 
#> PAF = 68.841% [95% CI: 62.782% to 74.901%]
#> standard_deviation(paf %) = 3.092
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 61.612% (sd %: 3.740) --- [deltapif-120376783938691]
#> • 18.832% (sd %: 1.529) --- [deltapif-143127077688999]
#> ────────────────────────────────────────────────────────────────────────────────

# Fractions that use the same relative risk are correlated. Declare it
# by giving `beta` the same identifier (name) in a list:
paf_women <- paf(0.32, beta = list("rr_meta" = log(1.3)), var_beta = 0.01,
                 var_p = 0.001, label = "Women")
paf_men   <- paf(0.27, beta = list("rr_meta" = log(1.3)), var_beta = 0.01,
                 var_p = 0.001, label = "Men")
paf_total(paf_men, paf_women, weights = c(0.49, 0.51))
#> 
#> ── Population Attributable Fraction: [deltapif-217757602827549] ──
#> 
#> PAF = 8.139% [95% CI: 1.465% to 14.361%]
#> standard_deviation(paf %) = 3.287
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 7.493% (sd %: 3.111) --- [Men]
#> • 8.759% (sd %: 3.552) --- [Women]
#> ────────────────────────────────────────────────────────────────────────────────

# Correlations between different estimates are given by identifier
paf_lead <- paf(0.2, beta = list("lead" = log(2.2)), var_beta = 0.04,
                var_p = 0.001, label = "Lead")
paf_rad  <- paf(0.1, beta = list("radiation" = log(1.2)), var_beta = 0.02,
                var_p = 0.0001, label = "Radiation")
cov_rr <- matrix(c(0.04, 0.01, 0.01, 0.02), ncol = 2,
                 dimnames = list(c("lead", "radiation"), c("lead", "radiation")))
paf_ensemble(paf_lead, paf_rad, var_beta = cov_rr)
#> 
#> ── Population Attributable Fraction: [deltapif-0876777326556075] ──
#> 
#> PAF = 20.936% [95% CI: 7.889% to 33.983%]
#> standard_deviation(paf %) = 6.657
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 19.355% (sd %: 6.233) --- [Lead]
#> • 1.961% (sd %: 1.642) --- [Radiation]
#> ────────────────────────────────────────────────────────────────────────────────

# 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-0654550251367236] ──
#> 
#> PAF = 75.731% [95% CI: 72.232% to 78.788%]
#> standard_deviation(paf %) = 1.668
#> ────────────────────────────────── Components: ─────────────────────────────────
#> • 76.180% (sd %: 2.271) --- [deltapif-0360278296382459]
#> • 75.299% (sd %: 2.435) --- [deltapif-147619690566501]
#> ────────────────────────────────────────────────────────────────────────────────