Graduation and graduation tests: formula sheet

Key formulas

Crude central rate

m̂x = θx ÷ Ex^c

θx = observed deaths aged x; Ex^c = central exposed to risk. This estimates the force of mortality μ, assumed constant over the year.

Crude initial rate

q̂x = θx ÷ Ex

Ex = initial exposed to risk. Use when exposure is measured at the start of the year of age.

Crude estimate as a random variable

crude rate = true rate + random error

Explains irregularity. Variance falls as exposure rises, so large groups give smoother crude rates.

Graduated estimates

graduated rate at x = fitted value from a smooth function of age

Written with a ˚ over the symbol, such as q̊x or μ̊x, to distinguish from crude estimates.

Gompertz law

μx = B c^x (equivalently ln μx = ln B + x ln c)

Two parameters. Same as GM(0,2) with B = e^β0 and c = e^β1.

Makeham law

μx = A + B c^x

Three parameters. Same as GM(1,2). The constant A covers age-independent mortality.

GM(r,s) family

μx = Σ (i = 0 to r−1) αi x^i + exp( Σ (j = 0 to s−1) βj x^j )

r counts the polynomial terms and s counts the exponent terms. Total parameters = r + s.

Age shift to a standard table

qx = q^s_{x+k} or μx = μ^s_{x+k}

One parameter k. If k > 0 your group has heavier mortality than the standard table.

Linear transformation of a standard table

qx = a + b q^s_{x+k} or μx = a + b μ^s_{x+k}

Parameters a, b and optionally k. With a = 0 and b > 1, mortality is heavier than the standard at every age.

Degrees of freedom after graduation

d.f. = number of age groups − number of parameters estimated

Used in the chi-square test. This applies to parametric and standard table graduations, where parameters are fitted to the data.

Standardised deviation

zx = (θx − Ex·μ̂x) ÷ √(Ex·μ̂x)

θx is actual deaths, Ex·μ̂x is expected deaths. Under the null hypothesis zx is approximately N(0,1).

Chi-squared statistic

χ² = Σ zx²

Compare with χ² on m − (number of fitted parameters) degrees of freedom, where m is the number of age groups. Reject if the statistic exceeds the upper critical value. It is a one-sided test.

Individual standardised deviations test

Expect about 5% of |zx| > 1.96 and about 68% within ±1

Check the count outside ±1.96 and the shape of the distribution. Look for large single deviations, and for deviations that are too large or too small overall.

Signs test

P = number of positive zx, out of m. Under H0, P ~ Binomial(m, ½)

Two-sided test for bias. Zero deviations are normally excluded from m. For large m, use the normal approximation.

Cumulative deviations test

Σ(θx − Ex·μ̂x) ÷ √(Σ Ex·μ̂x) ~ N(0,1) approximately

Two-sided test for overall bias. Cancellation of positive and negative deviations is the point, so it detects consistent bias that the chi-squared test can miss.

Grouping of signs test

Given n1 positive and n2 negative deviations, let G = number of positive runs. Use the null distribution of G given n1 and n2

Detects clumping. Too few groups means long runs of one sign. It is a one-sided test using tables, or a normal approximation for large n1 and n2.

Individual standardised deviation

z_x = (θ_x − E_x μ̂_x) ÷ √(E_x μ̂_x)

θ_x is actual deaths, E_x μ̂_x is expected deaths under the graduated rate. Under the null hypothesis z_x is approximately N(0,1).

Mean of deviations

z̄ = (1/m) Σ z_x

m is the number of ages. Sum over all m ages.

Lag-1 serial correlation coefficient

r₁ = [ (1/(m−1)) Σ_{x=1}^{m−1} (z_x − z̄)(z_{x+1} − z̄) ] ÷ [ (1/m) Σ_{x=1}^{m} (z_x − z̄)² ]

Numerator has m−1 terms, denominator has m terms. If the question gives a formula, use that version.

Test statistic

√m × r₁ ~ N(0,1) approximately under H₀

Large m is needed for the approximation. Reject independence if the value is too large and positive (one-tailed). Check the question for the form to use.

Third difference

Δ³μ_x = μ_{x+3} − 3μ_{x+2} + 3μ_{x+1} − μ_x

Built by repeated differencing: Δμ_x = μ_{x+1} − μ_x, then Δ² = Δ(Δμ), then Δ³.

Smoothness criterion

Δ³μ̂_x small relative to μ̂_x and changing progressively with x

A judgement test with no tabulated critical value.

Quick revision

• 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.

Common mistakes

• Saying crude rates are irregular because the data are wrong or contain errors. Fix: Say the irregularity comes from random variation in the number of deaths, even with perfect data. Small exposure makes it worse.
• Claiming graduated rates are always better because they are smoother. Fix: Explain that over-smoothing can introduce bias and hide real features. A good graduation balances smoothness and adherence.
• Using the standard table value at age x instead of x + k. Fix: Write the subscript x + k with the actual number, for example 60 + 2 = 62, before looking up the table.
• Getting the direction of the age shift wrong. Fix: If qx = q^s_{x+k} with k > 0, your group at age x has the mortality of the standard at an older age, so it is heavier.
• Using the wrong degrees of freedom for the chi-squared test Fix: Use m minus the number of parameters fitted. If rates are taken from a standard table with no fitting, use m.
• Treating the chi-squared test as proof the graduation is good Fix: State that it only checks overall size of deviations. It can miss bias, clumping and a few large outliers.
• Dividing both parts of r₁ by the same number Fix: Use m−1 for the lag-1 products and m for the squares, unless the question's formula says otherwise.
• Using a two-tailed critical value Fix: Clumping gives positive correlation, so use a one-tailed test: reject only if √m × r₁ is too large and positive, as stated in the question or your working.

Exam tips

• Always give the reason for irregularity as random sampling variation, and mention small exposure.
• When asked to discuss the trade-off, name both extremes and the consequence of each. Marks are for both sides.
• Show the crude rate formula and check whether the question uses central or initial exposure.
• Link the topic to testing. Mention that fit tests check adherence and smoothness tests check smoothness.
• For numerical comments, point to specific ages where the crude pattern is implausible.
• Know the formulas for Gompertz, Makeham and GM(r,s) exactly, including how r and s are defined. Questions often ask you to identify the GM(r,s) form of a law.
• In written answers, name the method and give one advantage and one disadvantage of it. Typical points are number of parameters, data needs, subjectivity and ease of testing.
• For standard table questions, show the age x + k row explicitly. This is where marks are gained and lost.