Parametric Survival Models, Life Tables, and Their Relationships

Problem Preview: A population follows Gompertz's law with $B=0.0003$ and $c=1.07$. You need ${}_{10}{p}_{50}$. Half the candidates write the integral of $B{c}^{50+s}$ as $B{c}^{50}\cdot 10$, treating the exponential as constant. The correct integral is $B{c}^{50}(c^{10}-1)/\ln c$, giving 0.8813 instead of 0.9133.

HIGH-FREQUENCY: The relationships between the survival function, force of mortality, and density function appear on nearly every FAM exam.

The survival function $S_0(x)$ gives the probability that a newborn survives to at least age $x$.

If $X$ is the age-at-death random variable:

$$S_0(x)=\mathrm{Pr}(X>x)=1-F_0(x)$$

Properties: $S_0(0)=1$, $S_0(\omega )=0$, and $S_0(x)$ is non-increasing. Here $\omega$ is the limiting age.

$$f_0(x)=-\frac{d}{dx}{S}_0(x)=F_0^{\prime }(x)$$

HIGH-FREQUENCY: The force of mortality is the single most important function in survival analysis for this exam.

Common mistakes

• Confusing the survival function argument. $S_0(x)$ conditions on birth. ${}_t{p}_x$ conditions on survival to age $x$. Writing ${}_{10}{p}_{50}=S_0(60)$ is wrong, the correct formula is $S_0(60)/S_0(50)$. Trap: an answer choice that equals $S_0(60)$ alone.
• Forgetting the negative sign in the force of mortality. ${\mu }_x=-S_0^{\prime }(x)/S_0(x)$, not $S_0^{\prime }(x)/S_0(x)$. A negative force of mortality is always wrong.
• Misapplying the Gompertz integral. The integral of $B{c}^{x+s}$ from 0 to $t$ is $B{c}^x(c^t-1)/\ln c$, not $B{c}^x\cdot t$. Treating $c^{x+s}$ as constant gives a linear approximation. Trap: a value close to correct but systematically off for large $t$.

Bottom line

• Survival function $S_0(x)=\mathrm{Pr}(X>x)$, unconditional from birth; $S_0(0)=1$, $S_0(\omega )=0$
• Force of mortality ${\mu }_x=f_0(x)/S_0(x)=-\frac{d}{dx}\ln S_0(x)$, the master function that determines the entire model
• Recovery: $S_0(x)=\exp (-{\int }_0^x{\mu }_{t}dt)$, exponentiate the negative integral to get survival
• Conditional survival: ${}_t{p}_x=S_0(x+t)/S_0(x)=\exp (-{\int }_0^t{\mu }_{x+s}ds)$, conditional on reaching age $x$

Exam shortcut

When you see a Gompertz or Makeham problem, write the integral formula immediately: $B{c}^x(c^t-1)/\ln c$. Half the work on these problems is evaluating this integral. For life table problems, draw a quick timeline showing which $l$ values sit at which ages, this prevents dividing by the wrong denominator. Recovery rule: "Exponentiate the negative integral of the force to get survival." This one sentence connects $\mu$ to $S$.

The full lesson (about 3,270 words, 22 min read) adds 5 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 8a

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