dprimarycensored: Compute the primary event censored PMF for delays

Description

This function computes the primary event censored probability mass function (PMF) for a given set of quantiles. It adjusts the PMF of the primary event distribution by accounting for the delay distribution and potential truncation at a maximum delay (D) and minimum delay (L). The function allows for custom primary event distributions and delay distributions.

Usage

dprimarycensored(
  x,
  pdist,
  pwindow = 1,
  swindow = 1,
  L = 0,
  D = Inf,
  dprimary = stats::dunif,
  dprimary_args = list(),
  log = FALSE,
  ...
)
dpcens(
  x,
  pdist,
  pwindow = 1,
  swindow = 1,
  L = 0,
  D = Inf,
  dprimary = stats::dunif,
  dprimary_args = list(),
  log = FALSE,
  ...
)

Value

Vector of primary event censored PMFs, normalized over [L, D] if truncation is applied

Arguments

x: Vector of quantiles
pdist: Distribution function (CDF). The package can identify base R distributions for potential analytical solutions. For non-base R functions, users can apply add_name_attribute() to yield properly tagged functions if they wish to leverage the analytical solutions.
pwindow: Primary event window
swindow: Secondary event window (default: 1)
L: Minimum delay (lower truncation point). If greater than 0, the distribution is left-truncated at L. This is useful for modelling generation intervals where day 0 is excluded, particularly when used in renewal models. Defaults to 0 (no left truncation).
D: Maximum delay (upper truncation point). If finite, the distribution is truncated at D. If set to Inf, no upper truncation is applied. Defaults to Inf.
dprimary: Function to generate the probability density function (PDF) of primary event times. This function should take a value x and a pwindow parameter, and return a probability density. It should be normalized to integrate to 1 over [0, pwindow]. Defaults to a uniform distribution over [0, pwindow]. Users can provide custom functions or use helper functions like dexpgrowth for an exponential growth distribution. See pcd_primary_distributions() for examples. The package can identify base R distributions for potential analytical solutions. For non-base R functions, users can apply add_name_attribute() to yield properly tagged functions if they wish to leverage analytical solutions.
dprimary_args: List of additional arguments to be passed to dprimary. For example, when using dexpgrowth, you would pass list(min = 0, max = pwindow, r = 0.2) to set the minimum, maximum, and rate parameters
log: Logical; if TRUE, probabilities p are given as log(p)
...: Additional arguments to be passed to the distribution function

Details

The primary event censored PMF is computed by taking the difference of the primary event censored cumulative distribution function (CDF) at two points, $d + \text{swindow}$ and $d$. The primary event censored PMF, $f_{\text{cens}}(d)$, is given by: $$ f_{\text{cens}}(d) = F_{\text{cens}}(d + \text{swindow}) - F_{\text{cens}}(d) $$ where $F_{\text{cens}}$ is the primary event censored CDF.

The function first computes the CDFs for all unique points (including both $d$ and $d + \text{swindow}$) using pprimarycensored(). It then creates a lookup table for these CDFs to efficiently calculate the PMF for each input value. For delays less than L, the function returns 0.

If truncation is applied (finite D or L > 0), the PMF is normalized to ensure it sums to 1 over the range [L, D\). This normalization uses: $$ f_{\text{cens,norm}}(d) = \frac{f_{\text{cens}}(d)}{ F_{\text{cens}}(D) - F_{\text{cens}}(L)} $$ where $f_{\text{cens,norm}}(d)$ is the normalized PMF. For the explanation and mathematical details of the CDF, refer to the documentation of pprimarycensored().

Examples

Run this code

# Example: Weibull distribution with uniform primary events
dprimarycensored(c(0.1, 0.5, 1), pweibull, shape = 1.5, scale = 2.0)

# Example: Weibull distribution with exponential growth primary events
dprimarycensored(
  c(0.1, 0.5, 1), pweibull,
  dprimary = dexpgrowth,
  dprimary_args = list(r = 0.2), shape = 1.5, scale = 2.0
)

# Example: Left-truncated distribution (e.g., for generation intervals)
dprimarycensored(1:9, pweibull, L = 1, D = 10, shape = 1.5, scale = 2.0)