Concepts of Survival Models for CS2 Survival Analysis

What this chapter covers

Concepts of survival models: topics in the order to study them

1. 1Future Lifetime Random VariableEverything else is defined from T_x, its distribution function and survival function, so start here.
2. 2Force of Mortality and Hazard RateIt gives the second way to describe T_x and lets you convert between S, f and μ before using any tables.
3. 3Life Table Functions and NotationOnce the continuous model is clear, you can read l_x, d_x, q_x and p_x as its discrete form.
4. 4Expected Future Lifetime and Life ExpectancyIt needs the survival function and life table notation, and it practises integrating or summing survival probabilities.
5. 5Mortality Laws and Fractional Age AssumptionsYou apply the earlier results to specific laws and to non-integer ages, which needs μ, S and the table together.
6. 6Select, Ultimate and Aggregate MortalityIt extends life tables with duration since selection, so it comes last when basic notation is automatic.

How to prepare Concepts of survival models

1. Write the definitions of T_x, K_x, S_x(t), F_x(t), f_x(t) and μ_x in your own words and notation, with one line on what each means.
2. Practise converting between S, f and μ in both directions. Derive tμ_x relationships, such as S_x(t) = exp(−∫₀ᵗ μ_{x+s} ds), until it is routine.
3. Do life table questions with a small table. Compute tp_x, tq_x and deferred probabilities t|uq_x from l_x, and check every answer is between 0 and 1.
4. Practise expected lifetime: e_x as an integral of tp_x, and the curtate expectation as a sum, and compare the two.
5. For each mortality law and fractional age assumption, note its formulas and when each is used, then solve one numerical question for each.
6. Solve select table questions by reading the correct row for age at selection and duration, and note where the ultimate part begins.
7. Finish with timed past-paper style questions, mixing multiple-choice and written, and repeat any where you lost marks on notation or assumptions.

Common mistakes in Concepts of survival models

• Mixing up tp_x, tq_x and deferred probabilities such as t|uq_x.

  Fix: Translate each symbol into words before calculating, for example: survive t years, then die within the next u years.

• Using μ_x as if it were a probability.

  Fix: Remember μ_x is a rate and can exceed 1 in theory. Convert to probabilities only through the exponential formula.

• Applying the wrong fractional age assumption or mixing formulas between assumptions.

  Fix: Write the assumption at the top of your working, then use only its formulas. Check that the resulting probabilities are sensible.

• Confusing complete expectation of life with curtate expectation.

  Fix: Use an integral for e°_x and a sum for e_x. Remember that e°_x is roughly e_x + 0.5 only under suitable assumptions and not exactly.

• Reading the wrong row or column in a select table.

  Fix: Identify [x] and duration r first, then find x + r. Mark where the select period ends and the ultimate rates begin.

• Leaving out assumptions and units in written answers and computer output.

  Fix: State the model, notation, formula, working and result in that order, and say what the answer means in words.

Last-day revision: Concepts of survival models

• ✓T_x is the future lifetime of a life aged x; S_x(t) = P(T_x > t) = tp_x.
• ✓μ_x = f_x(0) ÷ S_x(0), and in general μ_{x+t} = f_x(t) ÷ S_x(t).
• ✓tp_x = exp(−∫₀ᵗ μ_{x+s} ds).
• ✓f_x(t) = tp_x × μ_{x+t}.
• ✓From a life table: tp_x for whole t = l_{x+t} ÷ l_x, and q_x = d_x ÷ l_x with d_x = l_x − l_{x+1}.
• ✓Deferred probability: t|uq_x = tp_x − t+up_x.
• ✓Complete expectation e°_x = ∫₀^∞ tp_x dt; curtate e_x = Σ (k from 1) kp_x.
• ✓Under uniform distribution of deaths, tq_x = t × q_x for 0 ≤ t ≤ 1, and μ_{x+t} = q_x ÷ (1 − t q_x).
• ✓Under constant force of mortality over the year, tp_x = (p_x)^t.
• ✓Gompertz: μ_x = Bc^x. Makeham: μ_x = A + Bc^x.
• ✓Select mortality: use q_[x]+r for a life selected at age x, with duration r; after the select period it matches ultimate rates.
• ✓State your assumptions clearly in written answers, especially for fractional ages.

Concepts of survival models practice questions

• For a life aged 40 in a select-and-ultimate table with a select period of 2 years, which statement about the notation is correct?
• Given q_40 = 0.02 and a uniform distribution of deaths (UDD) between integer ages, what is the force of mortality at exact age 40.5, to 4 de…
• A 1-year select period applies. Select rates are q[50] = 0.004 and q[51] = 0.0045, and ultimate rates are q51 = 0.0055 and q52 = 0.0065. For…
• Which feature of a select table with select period 3 years distinguishes the rates q[x], q[x-1]+1 and q[x-2]+2 for lives all aged x?
• Let K_x = floor(T_x) be the curtate future lifetime of a life aged x. Which statement is correct?
• In standard survival-model notation, for a life aged x with future lifetime random variable T_x, which expression correctly gives the force …
• A lifetime has force of mortality μ(t) = 0.01t for t ≥ 0, with t in years. What is the probability that a new life dies between times 5 and …
• For a life aged x, the force of mortality is mu_(x+t) = -d/dt ln(tp_x). Which expression gives the probability density function of T_x at ti…