Scale Mortality Schedule to Target via Cumulative Hazard

Rescales an age-specific mortality schedule by matching the cumulative hazard to an empirical target (from Hoenig, Then et al., or a specified survival probability).

Usage

scale_mortality(M, age, M_target = NULL, tmax = NULL, p = 0.001)

Arguments

M: Numeric vector of instantaneous mortality rates.
age: Numeric vector of ages corresponding to M. Required for trapezoidal integration. Must be the same length as M and sorted in ascending order.
M_target: Target mean mortality. Can be a numeric scalar (fixed target), a function of tmax (e.g., function(tmax) 4.899 * tmax^(-0.916) for Then et al. 2015), or NULL to derive from survival probability p.
tmax: Maximum age (required if M_target is a function or NULL).
p: Probability of surviving to tmax. Used only if M_target = NULL. Default 0.001 (0.1% survival).

Value

Numeric vector of scaled mortality rates (same length as M).

Details

The scaling finds a proportional constant $c$ such that: $$M_{scaled}(t) = c \times M_{raw}(t)$$

The constant $c$ is determined by matching the cumulative hazard to the target. The cumulative hazard is computed via trapezoidal integration: $$H_{raw} = \int_0^{t_{max}} M_{raw}(a) \, da \approx \sum_i \frac{\Delta a_i}{2} \left[ M(a_i) + M(a_{i+1}) \right]$$

For the survival-probability target (M_target = NULL): $$c = \frac{-\ln(p)}{H_{raw}}$$ This ensures $S(t_{max}) = \exp\!\left(-c \cdot H_{raw}\right) = p$ exactly.

For a fixed mean-mortality target (numeric or function of tmax): $$c = \frac{\bar{M}_{target} \times t_{max}}{H_{raw}}$$ This interprets the target as the average mortality over the lifespan, so the cumulative hazard of the scaled schedule equals $\bar{M}_{target} \times t_{max}$.

This approach is preferred over arithmetic-mean-based scaling because it directly constrains the biologically relevant quantity (cumulative survival) rather than a proxy, and is invariant to the age grid spacing.

Examples

if (FALSE) { # \dontrun{
ages  <- seq(0.1, 30, length.out = 500)
M_raw <- M_chen_watanabe_L0(ages, Linf = 100, L0 = 25, k = 0.1,
                             two_phase = FALSE)

# Scale to survival probability
M_sp <- scale_mortality(M_raw, age = ages, tmax = 30, p = 0.01)

# Scale using Then et al. (2015) relationship
then_2015 <- function(tmax) 4.899 * tmax^(-0.916)
M_then <- scale_mortality(M_raw, age = ages, M_target = then_2015, tmax = 30)

# Scale to fixed target
M_fixed <- scale_mortality(M_raw, age = ages, M_target = 0.2, tmax = 30)
} # }