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t-year survival probability from a life table

Source: R/t_px.R
t_px.Rd

Computes the t-year survival probability $${}_t p_x = P[T(x) > t]$$ using an annual life table, allowing for fractional ages under standard actuarial assumptions.

Usage

t_px(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 survival 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.

tidy

Logical. If TRUE, returns a tibble with columns x, t, frac, tpx.

check

Logical. If TRUE, performs validity checks.

tol

Numeric tolerance for integer checks on x.

Value

Numeric vector of \({}_t p_x\), or a tibble if tidy = TRUE.

Details

The integer-year survival is obtained directly from the life table (Finan, Section 22): $${}_n p_x = \frac{\ell_{x+n}}{\ell_x}$$

For non-integer durations, let \(t = n + s\) with \(n = \lfloor t \rfloor\) and \(s \in [0,1)\). Then (Finan, Section 24): $${}_t p_x = {}_n p_x \times {}_s p_{x+n}$$

The fractional-year factor \({}_s p_y\) depends on the assumption:

  • UDD (Finan, Sec. 24.1): \({}_s p_y = 1 - s \times q_y\)

  • CF (Finan, Sec. 24.2): \({}_s p_y = (p_y)^s\)

  • Balducci (Finan, Sec. 24.3): \({}_s p_y = \frac{p_y}{1 - (1 - s) \times q_y}\)

Missing annual survival probabilities are derived directly from adjacent survivor counts, avoiding cancellation in \(1-q_y\). CF and Balducci retain logarithmic survival when that count ratio underflows. Supplied px values remain in use; input columns should describe a consistent annual life table.

If \(x + t > \omega\) (the terminal age), the function returns 0 by the established table-horizon convention, including open tables. This is truncation, not inference about survival beyond observed ages; extend the table before evaluating a longer horizon. Interval tables with nonannual widths from km_lifetable require annual conversion first.