Partial derivatives of PIF

Calculates the partial derivatives of a potential impact fraction with respect to the parameters p or beta.

Usage

deriv_pif_p(p, p_cft, rr, mu_obs = NULL, mu_cft = NULL)

deriv_pif_beta(p, p_cft, rr, rr_link_deriv_vals, mu_obs = NULL, mu_cft = NULL)

Arguments

p

Prevalence (proportion) of the exposed individuals for each of the N exposure levels.

p_cft

Counterfactual prevalence (proportion) of the exposed individuals for each of the N exposure levels.

rr

The relative risk for each of the exposure levels.

mu_obs

The average value of the relative risk in the observed population.

mu_cft

The average value of the counterfactual relative risk in the population.

rr_link_deriv_vals

The derivative of the relative risk function g with respect to the parameter beta evaluated at beta.

Value

The partial derivative (usually a vector)

Note

As p and beta are usually vectors these are vector-valued derivatives.

Formulas

The partial derivative of PIF with respect to p is: $$ \dfrac{\partial \textrm{PIF}}{\partial p} = \dfrac{\mu^{\text{cft}}}{\big(\mu^{\text{obs}}\big)^2} \cdot \big( \text{RR}(\beta) - 1) $$ The partial derivative of PIF with respect to beta is: $$ \dfrac{\partial \textrm{PIF}}{\partial \beta} = \Bigg(\dfrac{ \mu^{\text{obs}} \cdot p_{*} - \mu^{\text{cft}} \cdot p }{ \Big( \mu^{\text{obs}}\Big)^2 }\Bigg)\odot \text{RR}'(\beta)$$ with \(\odot\) representing the Hadamard (elementwise) product.