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IAI Actuarial Core Principles · Risk Modelling and Survival Analysis

Graduation and Graduation Tests for CS2 Survival Models

Graduation smooths crude mortality estimates so they follow a sensible underlying pattern. You pick a method (parametric, standard table or spline), then test the result. Chi-squared and individual standardised deviations tests check closeness to the data. Signs and cumulative deviations tests check bias. Runs and serial correlation tests check clumping. Judge smoothness from third differences.

What this chapter covers

Crude mortality rates are calculated directly from observed deaths and exposure at each age. They are noisy because each age has limited data. Graduation replaces them with smoothed estimates that reflect the true, underlying pattern of mortality, which we expect to change gradually with age.

The chapter has two halves. First, you learn how to graduate: fit a parametric formula such as a Gompertz or Makeham type, adjust a standard table, or fit a spline. Second, you learn how to test the graduated rates against the data. The tests ask two separate questions. Does the graduation stay close enough to the crude rates? And is the graduation free from systematic bias and sufficiently smooth?

This chapter builds on the survival models and the estimation of mortality rates in the earlier part of the paper, including the exposed to risk, the central rate and the binomial and Poisson models for deaths. It also leans on CS1 ideas: hypothesis testing, the chi-squared distribution, the normal approximation and the sign and runs ideas from non-parametric testing. Written questions often ask you to carry out several tests on one data set and then give a conclusion.

Graduation tests are procedural, so marks are reliable if you practise. A typical written question gives a table of ages, deaths, exposures and graduated rates, then asks you to run a chi-squared test, a standardised deviations test, a signs test, a cumulative deviations test, a runs test or a serial correlation test. You may also be asked to comment on smoothness from the differences of the graduated rates. Each test has a fixed method, so you can score full working marks. The topics also suit short multiple-choice questions on what a test detects. Paper B is a computer-based exam, so the topic may be examined there too. Check the current syllabus and past papers.

Graduation and graduation tests: topics in the order to study them

  1. 1Crude vs Graduated Estimates of MortalityStart here to understand why crude rates are irregular and what graduation aims to achieve, including the balance between smoothness and adherence to data.
  2. 2Methods of GraduationLearn how graduated rates are produced before you learn how to judge them: parametric formulae, standard tables and splines, plus when each is suitable.
  3. 3Graduation Tests: Goodness of FitLearn the tests of closeness to the data next: the chi-squared test and the individual standardised deviations test. Both start from expected deaths. For the standardised deviations test, compare the deviations with N(0,1). About 5% should lie beyond ±2 and about a third beyond ±1. Check the number and proportion in each range.
  4. 4Serial Correlation and Smoothness TestsFinish with the tests for bias (signs test and cumulative deviations test) and for clumping of deviations (runs test and serial correlation test). Then learn to check smoothness by looking at whether the third differences of the graduated rates are small and progress regularly. This is a judgement check, not a formal test. Most of these tests use the standardised deviations you already know. The exceptions are the signs test, which uses only the signs of the deviations, and the cumulative deviations test, which uses the sum of (actual − expected) deaths divided by √(sum of expected deaths).

How to prepare Graduation and graduation tests

This chapter rewards method. Learn each test as a short recipe: hypothesis, statistic, distribution, degrees of freedom and conclusion. Then practise until you can run the full set quickly.

  1. Read how crude rates are formed and write down in your own words why they are rough and what graduation changes.
  2. List the methods of graduation with one line each on what is fitted and what is assumed. Note that fitted parameters reduce the degrees of freedom in the chi-squared test.
  3. Build a one-page sheet of the tests. For each, write what it detects, the statistic, its distribution under the null hypothesis and the usual decision rule.
  4. Work one full data set by hand: compute expected deaths, standardised deviations, then every test in turn. Use the same set to see how results differ across tests.
  5. Practise degrees of freedom. Count the age groups, subtract the number of fitted parameters, and check any grouping of small expected values.
  6. Paper B is computer-based, so also practise the same tests in R on a data set. Learn to read the output and write a conclusion in words.
  7. Do past written questions timed. Finish with a clear conclusion on whether the graduation is acceptable overall. Interpret any failure and discuss what it implies for the graduation.

Common mistakes in Graduation and graduation tests

  • Using the wrong degrees of freedom in the chi-squared test

    Fix: Write the count of groups, subtract the number of fitted parameters, and adjust for any grouping before looking up the critical value.

  • Treating a pass on the chi-squared test as proof the graduation is good

    Fix: Run every relevant test. Say which feature each test checks, and conclude only after seeing all results.

  • Mixing up what each test detects

    Fix: Keep a table in your notes: overall closeness, individual large deviations, bias, clumping, adjacent correlation and smoothness.

  • Computing expected deaths from the crude rates

    Fix: Expected deaths use the graduated rates times exposure. Actual deaths come from the data.

  • Giving a calculation with no conclusion

    Fix: Always compare with the critical value or p-value, state whether you reject the null hypothesis and say what that means for the graduation.

  • Ignoring the trade-off between smoothness and fit in discussion questions

    Fix: Prepare a short explanation of why too much smoothing can bias results and why too little keeps random noise.

Last-day revision: Graduation and graduation tests

  • Graduation smooths crude rates because true mortality is expected to change smoothly with age.
  • Two aims pull against each other: smoothness and adherence to the data.
  • Methods include parametric formulae, standard tables and splines.
  • Standardised deviation: z = (d − E·q) ÷ √(E·q(1−q)) for the binomial model, or z = (d − E·μ) ÷ √(E·μ) for the Poisson model. Here d is actual deaths and E·q or E·μ is expected deaths.
  • Individual standardised deviations test: compare the deviations with N(0,1). About 5% should lie beyond ±2 and about a third beyond ±1. Check the number and proportion in each range. Far more than expected suggests a poor fit.
  • Chi-squared test: sum the squared standardised deviations and compare with chi-squared with degrees of freedom reduced for fitted parameters.
  • Chi-squared test can miss bias and clustering of deviations of the same sign.
  • Signs test checks whether positive and negative deviations are balanced, which detects bias. It uses only the signs of the deviations, not their size.
  • Cumulative deviations test also checks for bias. It uses the sum of (actual − expected) deaths divided by √(sum of expected deaths), which is not a sum of standardised deviations.
  • Runs test checks for clumping of deviations of the same sign.
  • Serial correlation test checks whether adjacent deviations are correlated.
  • Smoothness is judged by checking that the third differences of the graduated rates are small and progress regularly. It is not a formal test.
  • Judge the graduation overall. Interpret any failure and discuss its implications.
  • Always state hypotheses, statistic, critical value and a conclusion in context.

Graduation and graduation tests practice questions

Graduation and graduation tests in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Graduation and graduation tests: frequently asked questions

What is the difference between crude and graduated mortality rates?

Crude rates come straight from observed deaths and exposure at each age, so they contain random variation. Graduated rates are smoothed estimates that reflect an assumed underlying pattern. The aim is to be close to the data while remaining smooth.

Why do we need more than one graduation test?

Each test checks a different feature. The chi-squared test looks at overall closeness, but other tests look at bias, clumping and adjacent correlation, and smoothness is checked separately from differences. A graduation can pass one test and fail another.

How do fitted parameters affect the chi-squared test?

Each parameter estimated from the data reduces the degrees of freedom by one. Grouping of cells with small expected values also changes the count. Use the correct degrees of freedom before comparing with the critical value.

Will this chapter appear in the computer-based paper?

It may. Paper B is a computer-based exam, but the content of each paper can change, so check the current syllabus and past papers. Practise both the hand method and the R method for expected deaths, standardised deviations and test statistics.