t-year death probability from a life table

Computes the t-year death probability $${}_t q_x = \Pr[T(x) \le t] = 1 - {}_t p_x$$ using an annual life table, allowing for fractional ages under standard actuarial assumptions.

Usage

t_qx(lt, x, t, frac, tidy = FALSE, check = TRUE, tol = 1e-10)

Arguments

lt: A lifetable object as produced by lifetable. Must contain columns x and lx. Columns qx or px are used if present.
x: Integer age(s) at which the interval starts.
t: Nonnegative numeric duration(s) in years (can be fractional).
frac: Fractional-age assumption: "UDD", "CF", "CML" (alias of CF), or "Balducci". If not specified and lt carries a frac attribute (set by lifetable), that value is used. Passed directly to t_px.
tidy: Logical. If TRUE, returns a tibble with columns x, t, frac, tqx.
check: Logical. If TRUE, performs validity checks.
tol: Numeric tolerance for integer checks on x.

Value

Numeric vector of ${}_t q_x$, or a tibble if tidy = TRUE.

Details

Uses t_px for input and horizon conventions; the identity is: $${}_t q_x = 1 - {}_t p_x.$$ Small fractional losses are formed directly before subtracting from one, preserving probabilities whose survival rounds to one.

The identity ${}_tq_x = {}_td_x / \ell_x$ (Finan, Section 22) holds for integer $t$, where ${}_td_x = \ell_x - \ell_{x+t}$ is the expected number of deaths between ages $x$ and $x+t$.

For fractional durations, the result depends on the chosen assumption (UDD, CF, or Balducci); see t_px for details and formulas (Finan, Section 24).

The deferred death probability ${}_{n|}q_x$ can be obtained as (Finan, Section 23.4): $${}_{n|}q_x = {}_np_x \cdot q_{x+n} = {}_{n+1}q_x - {}_nq_x.$$

Examples

x  <- 0:5
lx <- c(100000, 99500, 99000, 98200, 97000, 95000)
lt <- lifetable(x = x, lx = lx, omega = 5, close = TRUE)

# Integer death probability (Finan, Section 22)
t_qx(lt, x = 0, t = 3)  # (l0 - l3) / l0
#> [1] 0.018

# t = 0 always returns 0
t_qx(lt, x = 0, t = 0)
#> [1] 0

# Fractional age under UDD (Finan, Section 24.1)
t_qx(lt, x = 0, t = 2.5, frac = "UDD")
#> [1] 0.014

# Finan Example 22.2a: number of deaths between ages 2 and 5
# 3_d_2 = l_2 - l_5 = 98995 - 97468 = 1527
lt_22 <- lifetable(
  x = 0:5,
  lx = c(100000, 99499, 98995, 98489, 97980, 97468)
)
t_qx(lt_22, x = 2, t = 3) * lt_22$lx[lt_22$x == 2]  # 1527
#> [1] 1527

# Deferred death probability (Finan, Section 23.4):
# 2|1_q_0 = 2_p_0 * q_2 = 3_q_0 - 2_q_0
t_qx(lt, x = 0, t = 3) - t_qx(lt, x = 0, t = 2)
#> [1] 0.008

# Vectorized with tidy output
t_qx(lt, x = c(0, 1), t = c(1.5, 2.2), frac = "Balducci", tidy = TRUE)
#> # A tibble: 2 × 4
#>       x     t frac         tqx
#>   <int> <dbl> <chr>      <dbl>
#> 1     0   1.5 Balducci 0.00751
#> 2     1   2.2 Balducci 0.0155 

# Use in a tidy pipeline
if (requireNamespace("dplyr", quietly = TRUE)) {
  tibble::tibble(age = c(0, 1, 2), duration = c(3, 2.5, 1.7)) |>
    dplyr::mutate(
      surv  = t_px(lt, x = age, t = duration),
      death = t_qx(lt, x = age, t = duration)
    )
}
#> # A tibble: 3 × 4
#>     age duration  surv  death
#>   <dbl>    <dbl> <dbl>  <dbl>
#> 1     0      3   0.982 0.018 
#> 2     1      2.5 0.981 0.0191
#> 3     2      1.7 0.983 0.0166