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Convert density/hazard to survival

convert_to_surv.Rd

Converts density or hazards from one of four input representations to survival probabilities at the same anchor time points (no interpolation).

Usage

convert_to_surv(
  x,
  times = NULL,
  input = "cont_haz",
  check = TRUE,
  integration = "trapezoid",
  clamp_surv = FALSE,
  eps = 1e-12
)

Arguments

x

(numeric() | matrix())
Input vector or matrix (rows = observations, columns = time points).

times

(numeric() | NULL)
Anchor time points. If NULL, extracted from names/colnames of x.

input

(character(1))
Input type. One of "disc_haz", "disc_dens", "cont_haz" or "cont_dens".

check

(logical(1))
If TRUE (default), run input validation checks. Disable only if you know the input is valid and want to skip checks for speed.

integration

(character(1))
Numerical integration rule for continuous inputs: "trapezoid" (default) uses the trapezoidal rule, while "riemann" is the left Riemann sum. Only used for "cont_dens" and "cont_haz".

clamp_surv

(logical(1))
If TRUE, clamp survival probabilities to [eps, 1] to avoid numerical issues.

eps

(numeric(1))
Small value used to clamp near-zero survival probabilities if clamp_surv = TRUE.

Value

A numeric vector or matrix of survival probabilities with the same dimensions as x.

Details

Let \(t_1,\dots,t_B\) denote the anchor time points, \(\Delta_j = t_j - t_{j-1}\), and \(S_j = S(t_j)\) the survival probabilities at the anchors. The conversion depends on the value of input as follows:

  • Discrete densities \(\tilde f_k\) ("disc_dens"): $$S_j = 1 - \sum_{k=1}^j \tilde f_k$$

  • Discrete hazards \(\tilde h_k\) ("disc_haz"): $$S_j = \prod_{k=1}^j (1 - \tilde h_k)$$

  • Continuous densities \(f_k\) ("cont_dens"):

    • Trapezoidal rule: $$S_j = 1 - \sum_{k=1}^j \frac{f_{k-1} + f_k}{2} \Delta_k$$ (with \(f_0 = f_1\))

    • Left Riemann sum: $$S_j = 1 - \sum_{k=1}^j f_k \Delta_k$$

  • Continuous hazards \(\lambda_k\) ("cont_haz"):

    • Trapezoidal rule: $$S_j = \exp\!\left(-\sum_{k=1}^j \frac{\lambda_{k-1} + \lambda_k}{2} \Delta_k\right)$$ (with \(\lambda_0 = \lambda_1\))

    • Left Riemann sum: $$S_j = \exp\!\left(-\sum_{k=1}^j \lambda_k \Delta_k\right)$$

For continuous inputs ("cont_dens" / "cont_haz"), numerical integration can be done either with the trapezoidal rule (integration = "trapezoid", default) or with a left Riemann sum (integration = "riemann"). Trapezoidal rule is more accurate (lower approximation error in the order of \(\Delta^2\) while the Riemann sum has an approximation error in the order of \(\Delta\)). At the first anchor both rules are identical, because no previous anchor value is available; therefore both use \(x_1 \Delta_1\).

Validation

If check = TRUE, we validate that the input is a proper discrete density/hazard matrix or vector using assert_prob(). For continuous hazards/densities, we only check that the input is a non-negative numeric matrix/vector.

Examples

# Continuous hazard => survival
haz_cont = c(0.02, 0.1, 0.2, 0.15)
times = c(0, 1, 2, 3)
convert_to_surv(haz_cont, times = times, input = "cont_haz")
#> [1] 1.0000000 0.9417645 0.8105842 0.6804506

# Discrete hazard => survival
haz_disc = c(0.1, 0.2, 0.15)
times = c(1, 2, 3)
convert_to_surv(haz_disc, times = times, input = "disc_haz")
#> [1] 0.900 0.720 0.612