Technical appendix
The number of deaths D reported for a given area and time period is assumed to be Poissondistributed, with mean ({mathbb{E}})(D) and variance ({mathbb{V}})(D) satisfying the equality ({mathbb{E}})(D) = λP = ({mathbb{V}})(D), where P denotes the population denominator [1]. The agespecific or crude death rate R, defined as the ratio D/P, is usually multiplied by 100,000 and reported as a rate per 100,000 population.
Poissongamma relationship
For a positive integer x ≤ P, it can be shown [2] that there exists a gamma random variable G such that ({mathbb{E}})(G) = x = ({mathbb{V}})(G) and
$$Pr left( {D ge xlambda } right) = Pr left( {G le lambda Px} right)$$
(A1)
Recall that if G is gammadistributed with shape parameter α > 0 and scale parameter β > 0, then its mean and variance are ({mathbb{E}})(G) = αβ and ({mathbb{V}})(G) = αβ^{2}. Conversely, the parameters are given by α = ({mathbb{E}})(G)^{2}/({mathbb{V}})(G) and β = ({mathbb{V}})(G)/({mathbb{E}})(G). Thus, with ({mathbb{E}})(G) = x = ({mathbb{V}})(G) in Eq. A1, the corresponding gamma distribution has α = x and β = 1.
For the rate R = D/P, with P = p, y = x/p, and v = x/p^{2}, Eq. A1 becomes
$$Pr (R ge ylambda ) = Pr (Z le lambda y,v)$$
(A2)
where Z = G/P is gammadistributed with mean y and variance v.
Gamma CI for agespecific and crude rates
When D = x is observed, the ratio y = x/p is an estimate of ({mathbb{E}})(R) = λ. An equaltailed 100(1 − a) percent CI [L(y), U(y)] for the parameter λ, e.g., with a = 0.05, is obtained as a solution to the following two equations:
$$Pr left[ {R ge ylambda = Lleft( y right)} right] = a/2$$
(A3a)
$$Pr left[ {R le ylambda = Uleft( y right)} right] = a/2$$
(A3b)
Eqs. A3a and A3b follow from looking upon L(y) as the largest λ for which Pr(R ≥ yλ) ≤ a/2 and U(y) as the smallest λ for which Pr(R ≤ yλ) ≤ a/2; see [22] and theorem 9.2.3.a in [23].
$$a/2 = Pr [R ge ylambda = L(y)] = Pr [Z le Lleft( y right)y,v]$$
where Z is gammadistributed with mean y and variance v, i.e., with parameters α = y^{2}/v = x and β = v/y = 1/p. Thus, the lower CI limit L(y) is obtained as the (a/2)quantile of the gamma(x, 1/p) distribution. For y = 0 = x, L(0) = 0 by convention.
Similarly, from Eqs. A2 and A3b,
$$1 – left( {a/2} right) = Pr [R > ylambda = U(y)] quadquadquadquad, = Pr [R ge y + 1/plambda = U(y)] quadquadquadquad, = Pr [Z^{prime} le U(y)y^{prime},v^{prime}]$$
where the second equality is due to x being a positive integer, so that D/p > x/p if and only if D/p ≥ (x + 1)/p, and Z′ is a gamma random variable with mean y′ = y + 1/p and variance v′ = v + 1/p^{2}. Because y′^{2}/v′ = x + 1 and v′/y′ = 1/p, the upper CI limit U(y) is obtained as the (1 − a/2)quantile of the gamma(x + 1, 1/p) distribution.
Approximate gamma CIs for ageadjusted rates
With n age groups, let D_{i} denote the number of deaths for group i. The D_{i} are assumed to be independent Poisson random variables, and the agespecific rates R_{i} are defined as the ratios D_{i}/P_{i}, with means ({mathbb{E}})(R_{i}) = λ_{i} and variances ({mathbb{V}})(R_{i}) = λ_{i}/p_{i}.
Let π_{i} denote the size of group i in the reference population, e.g., the projected year 2000 US population [24]. Let w_{i} denote the relative proportions for group i in the reference population: w_{i} = π_{i}/∑π_{j}. The ageadjusted death rate R′ is defined as
$$R^{prime} = sum w_{i} R_{i} = sum left( {w_{i} /P_{i} } right)D_{i}$$
Given the parameters λ_{i} and denominators P_{i} = p_{i}, the ageadjusted rate R′ has mean ({mathbb{E}})(R′) = λ′ = ∑w_{i} λ_{i} and variance ({mathbb{V}})(R′) = ∑w_{i}^{2} λ_{i}/p_{i}.
Fay–Feuer interval. Fay and Feuer [7] assume that Eq. A2 holds approximately for the ageadjusted rate R′, so that, for y = ∑(w_{i}/p_{i}) x_{i} and v = ∑(w_{i}/p_{i})^{2} x_{i},
$$Pr (R^{prime} ge ylambda^{prime}) approx Pr (Z le lambda^{prime}y,v)$$
(A4)
where Z is gammadistributed with ({mathbb{E}})(Z) = y and ({mathbb{V}})(Z) = v, i.e., with α = y^{2}/v and β = v/y. As for the crude rate R, an equaltailed 100(1 − a) percent CI for λ′ solves the equations:
$$Pr left[ {R^{prime} ge ylambda^{prime} = Lleft( y right)} right] = a/2$$
(A5a)
$$Pr left[ {R^{prime} le ylambda^{prime} = Uleft( y right)} right] = a/2$$
(A5b)
From Eqs. A4 and A5a, the lower limit L(y) can be resolved approximately from the lower tail probability of a gamma distribution with parameters α = y^{2}/v and β = v/y, again with the convention that L(0) = 0.
For the upper bound, note that a unit increment in the observed number of deaths x_{j} within group j results in the addition of the quantity w_{j}/p_{j} to the ageadjusted rate y = ∑(w_{i}/p_{i}) x_{i}. Because such a unit increment could be realized in any of the n groups,
$$Pr [R^{prime} > ylambda^{prime} = U(y)] ge Pr [R^{prime} ge y + kappa_{0} lambda^{prime} = U(y)]$$
where κ_{0} = max{w_{j}/p_{j}}. From Eq. A4, the righthand side in this last inequality is approximately equal to Pr[Z′ ≤ U(y)y′, v′], where Z′ is gammadistributed with mean y′ = y + κ_{0} and variance v′ = v + κ_{0}^{2}. Thus, an upper CI limit U(y) can be resolved from the upper tail probability of a gamma distribution with shape parameter α = y′^{2}/v′ and scale parameter β = v′/y′. Fay and Feuer [7] make the conjecture that the approximate gamma CI thus constructed remains conservative. Although this conjecture remains unproven, findings from the many simulation studies to date continue to support it, e.g., [4,5,6].
Tiwari modification. Tiwari et al. [8] developed a modification to the Fay–Feuer method described above by distributing an average increment 1/n uniformly across all age groups instead of a unit increment in a single age group:
$$y^{prime} = sum left( {w_{i} /p_{i} } right)left( {x_{i} + frac{1}{n}} right) = y + frac{1}{n}sum w_{i} /p_{i}$$
Thus, with κ_{1} = n^{−1} ∑w_{i}/p_{i} and κ_{2} = n^{−1} ∑(w_{i}/p_{i})^{2}, the gamma random variable Z′ above now has mean y′ = y + κ_{1} and variance v′ = v + κ_{2}. The Tiwari modification reduces the CI width relative to the Fay–Feuer method, because
$$Pr [R^{prime} ge y + kappa_{1} lambda^{prime} = U(y)] ge Pr [R^{prime} ge y + kappa_{0} lambda^{prime} = U(y)]$$
However, the resulting CI sometimes fails to retain the nominal coverage level, e.g., [4].
Fay–Kim modification. Fay and Kim [10] more recently developed a midp version of the Fay–Feuer CI. A modification of exact CIs from discrete data, midp CIs tradeoff guaranteed nominal coverage in all of the parameter space (which tends to result in overly wide CIs) for proximity to nominal coverage (and narrower CIs) for most parameter values.
For the midp interval, a solution to the following equations is sought:
$$Pr left[ {R^{prime} > ylambda^{prime} = Lleft( y right)} right] quadquadquadquad + left( {1/2} right) times Pr left[ {R^{prime} = ylambda^{prime} = Lleft( y right)} right] = a/2$$
(A7a)
$$Pr left[ {R^{prime} < ylambda^{prime} = Uleft( y right)} right] quadquadquadquad + left( {1/2} right) times Pr left[ {R^{prime} = ylambda^{prime} = Uleft( y right)} right] = a/2$$
(A7b)
Drawing B = b from a Bernoulli distribution with Pr(B = 1) = 1/2, Fay and Kim [10] define the midp version of the Fay–Feuer CI using the following gamma distribution:
$${text{gamma}}_{{text{midp}}} = b times {text{gamma}}left( {y^{2} /v,v/y} right) quadquadquadquadquadquad + left( {1 – b} right) times {text{gamma}}left( {y^{{prime}{2}} /v^{prime},v^{prime}/y^{prime}} right)$$
where y′ = y + κ_{0} and v′ = v + κ_{0}^{2} are as in the Fay–Feuer construction, above. Thus, the lower and upper limits are defined as the (a/2)th and (1−a/2)th quantiles of gamma _{midp}. The special case y = 0 is addressed using L(0) = 0 and U(0) defined as the (1−a)th quantile of the gamma(y^{2}/v, v/y) distribution. R syntax is provided to solve for L(y) and U(y) numerically [10].
Anderson–Rosenberg approximation. Anderson and Rosenberg [11] had introduced an approximation to the Fay–Feuer upper CI limit that alleviated the need to calculate κ_{0} = max{w_{j}/p_{j}}. Instead, the Poissongamma relationship in Eq. A1 is assumed to hold for an appropriately defined Poisson random variable D_{adj} corresponding to a crude rate that would have been equal to the ageadjusted rate R′, i.e., such that R′ = D_{adj}/P_{adj}. Therefore, a “standardized” gamma random variable G_{adj} is defined as Z/(v/y), where the gammadistributed Z has mean y and variance v. As a result, G_{adj} has mean and variance equal to y^{2}/v. Define x_{adj} = y^{2}/v and 1/p_{adj} = v/y. If x_{adj} was an integer, then there would exist a Poisson random variable D_{adj} with mean and variance equal to λ′ p_{adj} such that
$$Pr left( {D_{{{text{adj}}}} ge x_{{{text{adj}}}} lambda^{prime}} right) = Pr left( {G_{{{text{adj}}}} le lambda^{prime}p_{{{text{adj}}}} x_{{{text{adj}}}} } right)$$
(A6)
Because y^{2}/v will generally not be integer, x_{adj} is defined as the nearest integer instead (although this is not strictly necessary), and the equality in Eq. A6 is assumed to hold approximately. Either way, one proceeds as for the crude rate to derive CI limits L(y) and U(y) for λ′ as the (a/2)quantile of the gamma(x_{adj}, 1/p_{adj}) distribution and the (1 − a/2)quantile of the gamma(x_{adj} + 1, 1/p_{adj}), respectively.
Exact intervals. When there is a constant scalar c > 0 such that p_{i} = cπ_{i} for all i, the ageadjusted rate equals the overall crude rate, and the above CIs reduce to the exact gamma CI for λ = p^{−1} ∑λ_{i} p_{i} where p = ∑p_{i} and the total number of deaths D = ∑D_{i} follows a Poisson distribution with mean λp. In particular, when y = 0, v = 0 and x_{adj} is undefined. However, because the ageadjusted rate equals the crude rate in this case, the limits of all three approximate gamma CIs for the ageadjusted rate are defined to be those of the exact gamma CI for the crude rate, with p = ∑p_{i} and x = ∑x_{i} = 0. Thus, in this extreme case, L(0) = 0 and U(0) is the (1 − a/2)quantile of the gamma(1, 1/p) distribution.
Anderson–Rosenberg CI as a modification of the Fay–Feuer CI. The Anderson–Rosenberg construction can be seen to follow that of the Fay–Feuer CI, with a gammadistributed Z′′ that has mean y′′ = y + κ and variance v′′ = v + κ^{2}, where κ = κ_{3} = 1/p_{adj} instead of κ = κ_{0} = max{w_{j}/p_{j}}. Indeed, with 1/p_{adj} = v/y and x_{adj} = y^{2}/v,
$$frac{{v^{primeprime} }}{{y^{primeprime} }} = frac{{v(y^{2} + v)/y^{2} }}{{(y^{2} + v)/y}} = frac{v}{y} = frac{1}{{p_{{{text{adj}}}} }}$$
$${text{and}}quad frac{{y^{{primeprime}{2}} }}{{v^{primeprime}}} = frac{{(y^{2} + v)^{2} /y^{2} }}{{v(y^{2} + v)/y^{2} }} = frac{{y^{2} + v}}{v} = x_{{{text{adj}}}} + 1$$
Furthermore, 1/p_{adj} can be expressed as follows:
$$begin{aligned}frac{1}{{p_{{{text{adj}}}} }} &= frac{v}{y} = sum left( {w_{i} /p_{i} } right)xi_{i} quad{text{with}}\xi_{i} &= frac{{(w_{i} /p_{i} )x_{i} }}{{sum (w_{j} /p_{j} )x_{j} }};{text{and}};sum xi_{i} = 1.end{aligned}$$
As a result,
$$y^{primeprime} = y + frac{1}{{p_{{{text{adj}}}} }} = sum (w_{i} /p_{i} )(x_{i} + xi_{i} )$$
and the Anderson–Rosenberg method is seen to result in incrementing the agespecific death counts from x_{i} to x_{i} + ξ_{i}, whereas in the Fay–Feuer method only the count x_{i*} for the age group i* for which w_{i*}/p_{i*} = max{w_{j}/p_{j}} is incremented—and in the Tiwari modification, the agespecific counts are incremented from x_{i} to x_{i} + ζ_{i}, where ζ_{i} = 1/n. Additionally,
$$kappa_{3} = frac{1}{{p_{{{text{adj}}}} }} = sum (w_{i} /p_{i} )xi_{i} le max { w_{i} /p_{i} } = kappa_{0}$$
since ∑ξ_{i} = 1. Thus, like the Tiwari modification, the Anderson–Rosenberg construction reduces the CI width relative to the Fay–Feuer method:
$$Pr [R^{prime} ge y + kappa_{3} lambda^{prime} = U(y)] ge Pr [R^{prime} ge y + kappa_{0} lambda^{prime} = U(y)]$$
Two questions emerge from the above derivations:

(1)
Under what circumstances does the Anderson–Rosenberg method result in a shorter CI than the Fay–Feuer method that retains nominal coverage?

(2)
Since both the Anderson–Rosenberg and Tiwari methods result in narrower CIs than the Fay–Feuer method, when is one preferable to the other?
To partially answer question 2, note that the Anderson–Rosenberg CI would be narrower than the Tiwari CI if (but not only if) κ_{3} ≤ κ_{1}, as that ensures
$$Pr [R^{prime} ge y + kappa_{3} lambda^{prime} = U(y)] ge Pr [R^{prime} ge y + kappa_{1} lambda^{prime} = U(y)]$$
By definition, the condition κ_{3} ≤ κ_{1} is realized when
$$sum (w_{i} /p_{i} )xi_{i} le frac{1}{n}sum (w_{i} /p_{i} )$$
which is equivalent to
$$frac{1}{n}sum (w_{i} /p_{i} )^{2} x_{i} le left{ {frac{1}{n}sum (w_{i} /p_{i} )x_{i} } right} left{ {frac{1}{n}sum (w_{i} /p_{i} )} right}$$
This last condition indicates that the slope of the line from the simple regression of the weightadjusted agespecific death rates w_{i} (x_{i}/p_{i}) = (w_{i}/p_{i}) x_{i} onto the weights w_{i}/p_{i} is negative or zero. This could be verified upfront for any set of ageadjustment weights (w_{1}, …, w_{n}) and population distribution (p_{1}, …, p_{n}), and it would be sufficient to ensure that the Anderson–Rosenberg CI will be narrower than the Tiwari CI. Of course, this leaves the issue of efficiency unresolved, as it would theoretically be possible for either the Anderson–Rosenberg or the Tiwari CIs to be so narrow as to fail to retain nominal coverage. The empirical simulations investigate situations where this may occur. In addition, these two CI methods are compared to the more recent Fay–Kim midp modification.
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