Sullivan active life expectancy over three age bands with life expectancy and dependent years

Signature

e_x = (L_1 + L_2 + L_3) / l_x; ALE_x = ((1 - pi_1) * L_1 + (1 - pi_2) * L_2 + (1 - pi_3) * L_3) / l_x; DLE_x = e_x - ALE_x; s_A = ALE_x / e_x

Function

Sullivan active life expectancy from a period life table and dependency prevalence

Computational function

• Computational function for Sullivan active life expectancy with its standard error over any number of age bands

  Takes a period life table's survivors at the starting age and person-years by age band, the dependency prevalence measured in each band and the number of survey respondents behind each prevalence, and returns active life expectancy, life expectancy, expected dependent years, the active share, the standard error of active life expectancy and a normal 95% interval, for any number of bands. The point estimate is HE-FM-ALE-001. The variance is the standard estimator Imai and Soneji print as their equation 13, the sum over bands of pi_i (1 minus pi_i) L_i squared over N_i, divided by l_x squared; each band's term takes the form of the squared standard error of HE-FM-BINOM-004, with survey respondents in place of that record's patients, times the squared weight L_i over l_x (derived), so the bands are treated as independent binomial samples and the survey respondents N_i are inputs the formula does not hold. For the 1999 United States population with an activities of daily living definition, Imai and Soneji report 14.31 years at 65 with a 95% interval of 14.24 to 14.37 when the death rates are treated as known. The calculator line writes three bands out term by term; the loops over bands are in the Excel, R and Python code.

  Inputs and outputs: l_x is the life-table survivors at age x; L_1, L_2 and L_3 are the person-years in each band; pi_1, pi_2 and pi_3 are the dependent shares of the bands; N_1, N_2 and N_3 are the survey respondents in each band; w_1, w_2 and w_3 are the band weights L_i over l_x, in years per survivor; ALE_x is active life expectancy in years; V_ALE is its variance in years squared and SE_ALE its standard error in years; lo_ALE and hi_ALE are the 95% interval limits in years. In the code L, pi and N are vectors with one entry per band, and the outputs add le, dle and share.

  Assumption: The assumptions of HE-FM-ALE-001 hold (HE-AS-ALE-001 to HE-AS-ALE-004). The number dependent in each band follows an independent binomial process estimated by the sample fraction, as Imai and Soneji describe the standard approach, so the variance reflects survey sampling only; the life table is treated as known, which Imai and Soneji report is typical in the literature because the death and population counts behind it are large. Simple random sampling is assumed; under a complex design the prevalence is a weighted average and design effects are not captured by this formula.

  Worked example (Article baseline with illustrative survey sizes): With the article's baseline life table and prevalence and illustrative samples of 2,000, 1,500 and 800 respondents, active life expectancy is 14.475 years with a standard error of 0.1083 and a 95% interval of 14.2626 to 14.6874 years (computed here). The 85 and over band, with the smallest sample and a prevalence of one half, contributes 24% of the variance (derived). l_x = 100000; L_1 = 900000; L_2 = 650000; L_3 = 300000; pi_1 = 0.1; pi_2 = 0.25; pi_3 = 0.5; N_1 = 2000; N_2 = 1500; N_3 = 800; w_1 = 9; w_2 = 6.5; w_3 = 3; ALE_x = 14.475; V_ALE = 0.0117; SE_ALE = 0.1083; lo_ALE = 14.2626; hi_ALE = 14.6874

  Worked example (Same table with four times the respondents): With 8,000, 6,000 and 3,200 respondents the variance falls to a quarter and the standard error halves to 0.0542 years, an interval of 14.3688 to 14.5812 (computed here). l_x = 100000; L_1 = 900000; L_2 = 650000; L_3 = 300000; pi_1 = 0.1; pi_2 = 0.25; pi_3 = 0.5; N_1 = 8000; N_2 = 6000; N_3 = 3200; w_1 = 9; w_2 = 6.5; w_3 = 3; ALE_x = 14.475; V_ALE = 0.0029; SE_ALE = 0.0542; lo_ALE = 14.3688; hi_ALE = 14.5812

  Excel: Name the band columns PersonYearsBandCF, DependentShareBandCF and SurveyRespondentsBandCF (one row per band, any number of rows) and the survivors cell SurvivorsAtStartAgeCF. A cell named SullivanALE holds =SUMPRODUCT(1-DependentShareBandCF,PersonYearsBandCF)/SurvivorsAtStartAgeCF, a cell named SullivanVariance holds =SUMPRODUCT(DependentShareBandCF*(1-DependentShareBandCF)*PersonYearsBandCF^2/SurveyRespondentsBandCF)/SurvivorsAtStartAgeCF^2, a cell named SullivanSE holds =SQRT(SullivanVariance), and cells named SullivanLower and SullivanUpper hold =SullivanALE-1.96*SullivanSE and =SullivanALE+1.96*SullivanSE. SUMPRODUCT evaluates the array products without array entry, so the formulas work in every Excel version. A blank respondents cell returns #DIV/0! in SullivanVariance.

  R: sullivan_ale <- function(lx, L, pi, N) { w <- L/lx; ale <- sum((1-pi)*w); le <- sum(w); v <- sum(w^2*pi*(1-pi)/N); se <- sqrt(v); list(ale = ale, le = le, dle = le-ale, share = ale/le, se = se, lower = ale-1.96*se, upper = ale+1.96*se) } Called as sullivan_ale(100000, c(900000, 650000, 300000), c(0.1, 0.25, 0.5), c(2000, 1500, 800)) it returns ale 14.475, le 18.5, dle 4.025, share 0.7824, se 0.1083 and limits 14.2626 and 14.6874. A band with N of 0 gives an infinite or undefined variance, so every N_i must be at least 1, and l_x must be positive.

  Python: def sullivan_ale(lx, L, pi, N): w = [x/lx for x in L]; ale = sum((1-p)*x for p, x in zip(pi, w)); le = sum(w); v = sum(x*x*p*(1-p)/n for x, p, n in zip(w, pi, N)); se = v**0.5; return ale, le, le-ale, ale/le, se, ale-1.96*se, ale+1.96*se Standard library only; sullivan_ale(100000, [900000, 650000, 300000], [0.1, 0.25, 0.5], [2000, 1500, 800]) returns 14.475, 18.5, 4.025, 0.7824, 0.1083, 14.2626 and 14.6874 (rounded). With every prevalence 0 it returns a standard error of 0 and active life expectancy equal to life expectancy; a respondent count of 0 or an empty list of bands raises ZeroDivisionError, so the inputs need at least one band, every N_i at least 1 and l_x above 0.

  Test (Standard error at the article baseline): With the inputs of the first worked example, SullivanSE is compared with the hand value 0.1083. Expected result TRUE. FALSE shows a variance that uses the weights unsquared, which gives 0.0464, and a doubled (0.2167) or halved (0.0542) cell. Excel check =ABS(SullivanSE-0.1083)<0.0001

  Test (Four times the respondents halve the standard error): With SurveyRespondentsBandCF set to 8,000, 6,000 and 3,200, SullivanSE is compared with 0.0542, half the baseline value. Expected result TRUE. FALSE shows the baseline respondents left in place (0.1083), and a doubled (0.1083) or halved (0.0271) cell. Excel check =ABS(SullivanSE-0.0542)<0.0001

  Test (Lower limit of the 95% interval at the article baseline): With the inputs of the first worked example, SullivanLower is compared with 14.2626. Expected result TRUE. FALSE shows a limit built from the variance in place of the standard error, 14.4520, and a doubled (28.5253) or halved (7.1313) cell. Excel check =ABS(SullivanLower-14.2626)<0.0001

  Common error (Weights left unsquared in the variance): Writing the variance as the sum of w_i pi_i (1 minus pi_i) over N_i gives 0.0022 in place of 0.0117 on the baseline, a standard error of 0.0464 in place of 0.1083 and an interval less than half as wide (derived). The weight enters the variance squared because each band's prevalence is multiplied by L_i over l_x.

  Common error (Reading the interval as covering all uncertainty): The interval reflects survey sampling only. Imai and Soneji's Table 2 shows that allowing for sampling variability in the death rates widens the interval at age 20 for the 1999 United States population, 54.40 to 54.53 years with the death rates known and 54.37 to 54.56 with them estimated; misclassification, a survey that leaves out care homes (HE-ER-ALE-004) and departures from stationarity are not in the standard error at all.

  Source: Imai K, Soneji S. On the estimation of disability-free life expectancy: Sullivan's method and its extension. Journal of the American Statistical Association. 2007;102(480):1199-1211. Section 3.1, equation 13, the standard variance estimator of Sullivan's estimator under an independent binomial process in each age interval, and Table 2, estimated disability-free life expectancy with 95% intervals with the death rates known and estimated; section 3.2, Proposition 2, that the standard variance estimator is consistent and approximately unbiased under the four stationarity assumptions.

  w_1 = L_1 / l_x; w_2 = L_2 / l_x; w_3 = L_3 / l_x; ALE_x = (1 - pi_1) * w_1 + (1 - pi_2) * w_2 + (1 - pi_3) * w_3; V_ALE = w_1^2 * pi_1 * (1 - pi_1) / N_1 + w_2^2 * pi_2 * (1 - pi_2) / N_2 + w_3^2 * pi_3 * (1 - pi_3) / N_3; SE_ALE = sqrt(V_ALE); lo_ALE = ALE_x - 1.96 * SE_ALE; hi_ALE = ALE_x + 1.96 * SE_ALE

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Implementations

• Excel

  Active life expectancy from named life-table and prevalence ranges

  Person-years for the bands are in a column range named PersonYearsByBand, the dependency prevalence of the same bands in a range of the same size named DependentShareByBand, and the survivors at the starting age in a cell named SurvivorsAtStartAge. The formula, in a cell named ActiveLifeExpectancy, works for any number of bands because SUMPRODUCT evaluates the array 1 minus the prevalence without array entry. A blank prevalence cell is treated as 0, so that band counts as wholly active, and a blank SurvivorsAtStartAge returns #DIV/0!.

  =SUMPRODUCT(1-DependentShareByBand,PersonYearsByBand)/SurvivorsAtStartAge

• Excel

  Life expectancy, dependent years and active share cells for active life expectancy

  A cell named LifeExpectancyAtStartAge holds the formula shown, the person-years over the survivors. A cell named DependentYearsExpected holds =LifeExpectancyAtStartAge-ActiveLifeExpectancy and a cell named ActiveShareOfRemainingLife holds =ActiveLifeExpectancy/LifeExpectancyAtStartAge. Multiplying DependentYearsExpected by a cost per dependent year gives the expected care cost of HE-FM-COST-001.

  =SUM(PersonYearsByBand)/SurvivorsAtStartAge

Assumptions

• Stationary mortality and dependency prevalence behind a period figure

  Sullivan's method rests on a period life table, so it assumes constant age-specific death rates, a constant birth rate and no net migration, and it adds that age-specific dependency prevalence is constant over time; Imai and Soneji prove that under these four assumptions the estimator is unbiased and consistent. The result describes the rates prevailing now and is not a forecast for people now aged x. Mathers and Robine found by simulation that it could not detect a sudden change in disability transition rates but gave a good estimate of the multistate value when prevalence changed smoothly over the longer term.

• Bands cover all remaining life on one life table

  l_x and every L_i come from the same life table on the same radix, the bands run without gaps from age x, and the last band is open-ended. Bands can have any widths; Imai and Soneji note that the result does not depend on the interval widths. If the last band stops at a fixed age, the result is a partial expectancy up to that age and not ALE_x.

• Prevalence is the share of the band dependent at the time of the survey

  Each pi_i is the proportion of survey respondents in band i who are dependent when asked, a sample fraction, weighted where the survey design requires it. Imai and Soneji show that Sullivan's own 1971 estimator, the days a respondent spent disabled in the previous year divided by 365, measures something else and is not valid for the method. The survey covers the whole population of the band, including people in care homes and other institutions.

• Dependency treated as a single yes or no state

  Each surviving person in a band is either dependent in activities of daily living or not, under one definition held fixed across bands and over time. Independent years count fully and dependent years count zero, as in the other dichotomous health expectancies in the article; severity is not weighted, which is what separates the measure from HALE.

Worked examples

• Article baseline life table with dependency rising with age

  In the article's illustrative period life table, 100,000 people reach 65 and live 900,000, 650,000 and 300,000 person-years in the three bands, with dependency of 10%, 25% and 50%. Life expectancy at 65 is 18.5 years, 4.025 years are expected to be lived dependent, and active life expectancy is 14.475 years, 78.2% of remaining life, as in the article. All inputs are illustrative.

  l_x = 100000; L_1 = 900000; L_2 = 650000; L_3 = 300000; pi_1 = 0.1; pi_2 = 0.25; pi_3 = 0.5; e_x = 18.5; ALE_x = 14.475; DLE_x = 4.025; s_A = 0.7824

• Article Scenario A with prevention lowering dependency after 75

  A falls and reablement programme cuts dependency to 20% at 75 to 84 and 45% at 85 and over with mortality unchanged. Life expectancy stays at 18.5 years, dependent years fall to 3.55 and active life expectancy rises by 0.475 years to 14.95, an active share of 80.8%, as in the article. The inputs are illustrative.

  l_x = 100000; L_1 = 900000; L_2 = 650000; L_3 = 300000; pi_1 = 0.1; pi_2 = 0.2; pi_3 = 0.45; e_x = 18.5; ALE_x = 14.95; DLE_x = 3.55; s_A = 0.8081

• Article Scenario B with longer survival after 85

  Mortality improves after 85, so the last band holds 400,000 person-years in place of 300,000, with prevalence unchanged. Life expectancy rises to 19.5 years and active life expectancy to 14.975, but dependent years rise to 4.525 and the active share falls to 76.8%, as in the article. At an illustrative GBP 30,000 per dependent year, HE-FM-COST-001 gives expected care costs of GBP 135,750 per person against GBP 120,750 at baseline. The inputs are illustrative.

  l_x = 100000; L_1 = 900000; L_2 = 650000; L_3 = 400000; pi_1 = 0.1; pi_2 = 0.25; pi_3 = 0.5; e_x = 19.5; ALE_x = 14.975; DLE_x = 4.525; s_A = 0.7679

• Limiting case with no dependency in any band

  With every pi_i set to 0 the formula returns life expectancy, 18.5 years on the baseline life table, with no dependent years and an active share of 1. The case checks an implementation against the life-table identity.

  l_x = 100000; L_1 = 900000; L_2 = 650000; L_3 = 300000; pi_1 = 0; pi_2 = 0; pi_3 = 0; e_x = 18.5; ALE_x = 18.5; DLE_x = 0; s_A = 1

Common errors

• Weighting person-years by the dependency prevalence in place of its complement

  Multiplying each L_i by pi_i in place of 1 minus pi_i gives the expected years lived dependent, not the active years. On the article's baseline it returns 4.025 years in place of 14.475, and the error is hard to spot when prevalence is near one half in most bands.

• Applying an unweighted mean prevalence to life expectancy

  Averaging the band prevalences without weights and applying the mean to e_x ignores that the oldest band, where dependency is highest, holds the fewest person-years. On the baseline the mean of 0.2833 gives 13.2583 years, 1.2167 years below the Sullivan value of 14.475 (derived). The person-years of each band are the weights.

• Reading the active share without the years of dependence

  A rising active life expectancy can come with more years of dependence. In the article's Scenario B active life expectancy rises by 0.5 years, yet the active share falls to 76.8% against 78.2% at baseline and dependent years rise by 0.5, which adds GBP 15,000 of expected care cost per person at GBP 30,000 a year (HE-FM-COST-001). Both levels, ALE_x and DLE_x, are reported with s_A.

• Prevalence from a household survey that leaves out care homes

  A survey of people living at home, like the non-institutionalised sample of Katz and colleagues, misses many of the most dependent people. If the measured prevalence at 85 and over were 0.40 in place of a whole-population 0.50, active life expectancy on the baseline would be 14.775 years, 0.3 years too high (illustrative, derived).

• Leaving a prevalence cell blank

  In Excel a blank cell in DependentShareByBand counts as 0, so that band is read as wholly active and no error appears. A blank prevalence for 85 and over on the baseline raises ActiveLifeExpectancy to 15.975 years against 14.475 (derived). A band with no survey estimate needs a value or an explicit flag before the result is used.

Sources

• Imai and Soneji on Sullivan's estimator and life expectancy from a period life table

  Imai K, Soneji S. On the estimation of disability-free life expectancy: Sullivan's method and its extension. Journal of the American Statistical Association. 2007;102(480):1199-1211. doi:10.1198/016214507000000040 (published version read in full). Section 2.2, equation 6 for the person-years in an interval and equation 7, life expectancy at age x as the person-years of the following intervals divided by l_x. Section 3.1, Sullivan's Method, equation 11, Sullivan's estimator of disability-free life expectancy as the sum of 1 minus the band prevalence times the person-years, divided by l_x, with equation 12 for the prevalence as the sample fraction of disabled respondents in the band. Section 3.2, Proposition 1, unbiasedness and consistency under the three stationarity assumptions of the period life table and constant age-specific disability prevalence, not depending on the interval widths. Section 3.3, Sullivan's own estimator of prevalence from days spent disabled in the previous year, equation 15, called incorrect.

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• Katz and colleagues on active life expectancy from life tables

  Katz S, Branch LG, Branson MH, Papsidero JA, Beck JC, Greer DS. Active life expectancy. New England Journal of Medicine. 1983;309(20):1218-1224. doi:10.1056/NEJM198311173092005. The abstract describes using life-table techniques to analyse the expected remaining years of functional well-being, in terms of the activities of daily living, for noninstitutionalized elderly people in Massachusetts in 1974, and names these expected years active life expectancy.

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• Kingston and colleagues on years with and without dependency by Sullivan's method

  Kingston A, Comas-Herrera A, Jagger C. Forecasting the care needs of the older population in England over the next 20 years: estimates from the Population Ageing and Care Simulation (PACSim) modelling study. Lancet Public Health. 2018;3(9):e447-e455. doi:10.1016/S2468-2667(18)30118-X (full text read). Methods, subsection Model outputs, where years with and without dependency at age 65 were calculated for each year by Sullivan's method, applying the age-sex-specific prevalence of dependency to the age-sex-specific life-table population generated from the survival probabilities. The Introduction states that care needs are most often assessed by needing help to undertake one or more basic activities of daily living.

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• Mathers and Robine on Sullivan's method when disability rates change

  Mathers CD, Robine JM. How good is Sullivan's method for monitoring changes in population health expectancies? Journal of Epidemiology and Community Health. 1997;51(1):80-86. doi:10.1136/jech.51.1.80. Abstract, which describes a simulation on abridged cohort life tables and states in its main results that the Sullivan method could not detect a sudden change in disability transition rates but provides a good estimate of the true multistate value if there are smooth and relatively regular changes in disability prevalence over the longer term.

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Canonical Identity