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, p_cft_jacobian = 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. Either a numeric vector, for a fixed target that does not depend on the observed prevalence, or a function of p returning the counterfactual prevalence, for scenarios defined relative to the current exposure (e.g. function(p) 0.85 * p for a 15% reduction). In the latter case the uncertainty in p is also propagated through the counterfactual. See the section Fixed and relative counterfactuals.

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.

p_cft_jacobian

(Optional) Jacobian of the counterfactual prevalence p_cft with respect to p when the counterfactual is a function of the observed prevalence (for example a relative reduction p_cft = 0.85 * p). Entry [j, i] is the derivative of p_cft[j] with respect to p[i]. NULL (default) assumes p_cft is a fixed target.

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: $$ \dfrac{\partial \textrm{PIF}}{\partial p} = \dfrac{\mu^{\text{cft}}}{\big(\mu^{\text{obs}}\big)^2} \cdot \big( \text{RR}(\beta) - 1\big) $$ If the counterfactual is a function of the prevalence, \(p_{*} = g(p)\), the total derivative adds the term \(- J_g^{\top} \big( \text{RR}(\beta) - 1\big) / \mu^{\text{obs}}\) where \(J_g\) is the Jacobian of \(g\) evaluated at \(p\). The partial derivative of PIF with respect to beta is: $$ \dfrac{\partial \textrm{PIF}}{\partial \beta} = \Bigg( \dfrac{ \mu^{\text{cft}} \cdot p - \mu^{\text{obs}} \cdot p_{*} }{ \Big( \mu^{\text{obs}}\Big)^2 } \Bigg)\odot \text{RR}^{\prime}(\beta) $$ with \(\odot\) representing the Hadamard (elementwise) product.