Demography, Actuarial Science & Population Health
Chiang Life-Table q_x & Sullivan DFLE Formula
Convert observed age-group mortality rates n_m_x into exact interval death probabilities n_q_x = (n*n_m_x)/(1 + n*(1 - n_a_x)*n_m_x) via Chiang (1968) and compute Sullivan healthy person-years.
Convert central death rates n_m_x into exact interval probabilities n_q_x without >100% overflow!
Vital-statistics registries report central mortality rates n_m_x = Deaths / MidYear_PersonYears across n-year age groups (usually n = 5 years), NOT life-table interval death probabilities n_q_x! Multiplying naively as n * n_m_x overstates mortality because decedents stop contributing person-years mid-interval (leaving fraction n_a_x in (0, 1) of the n-year interval lived before death)! Chin Long Chiang (1968) derived the exact demographic conversion n_q_x = (n * n_m_x) / (1 + n * (1 - n_a_x) * n_m_x), yielding Cohort Deaths n_d_x = l_x * n_q_x, Interval Person-Years n_L_x = n * (l_x - n_d_x) + n * n_a_x * n_d_x (= n_d_x / n_m_x!), and Daniel Sullivan’s (1971) Disability-Free Person-Years n_DF_x = n_L_x * (1 - pi_x) at disability prevalence pi_x!
Verify exact integers on Interval_Width_n = 5 yr (B2), Central_Death_Rate_nmx = 0.012 (B3), Fraction_Lived_nax = 0.50 (B4), Entering_Survivors_lx = 103,000 (B5), and Disability_Prevalence_pix = 0.15 (B6 = 15%): Chiang Probability is n_q_x = (5 * 0.012) / (1 + 2.5 * 0.012) = 0.06 / 1.03 = 0.058252 (vs naive 0.060000!), Exact Interval Deaths n_d_x = 103,000 * (0.06 / 1.03) = 6,000, Interval Person-Years n_L_x = 5 * 97,000 + 2.5 * 6,000 = 500,000 (confirming 6,000 / 500,000 = 0.012!), and Sullivan Healthy Person-Years = 500,000 * 0.85 = 425,000!
| Abridged Life-Table Metric (n=5, m=0.012, a=0.50, l_x=103k, pi=15%) | Formula | Verified Output |
|---|---|---|
| Chiang Interval Death Prob (n_q_x) | (n * m) / (1 + n * (1 - a) * m) | 0.058252 (vs Naive 0.060000) |
| Cohort Deaths in Interval (n_d_x) | l_x * n_q_x | 6,000.00 deaths |
| Interval Person-Years Lived (n_L_x) | n*(l_x - d_x) + n*a*d_x | 500,000.00 person-years |
| Sullivan Disability-Free Years (n_DF_x) | n_L_x * (1 - pi_x) | 425,000.00 healthy years |
Chiang (1968) Exact Interval Mortality Probability n_q_x Formula
Validate n > 0, m >= 0, 0 < a < 1, l_x > 0, and 0 <= pi <= 1 (clamping n_q_x at 1 in extreme open-age cases).
=LET(n_wd, B2, m_rt, B3, a_fr, B4, IF(OR(n_wd <= 0, m_rt < 0, a_fr <= 0, a_fr >= 1), "Invalid Life-Table Input", ROUND(MIN(1, (n_wd * m_rt) / (1 + n_wd * (1 - a_fr) * m_rt)), 6)))Sullivan (1971) Disability-Free Person-Years n_DF_x Formula
Evaluate n_d_x with unrounded Chiang probability inside LET so demographic identities n_d_x / n_L_x = n_m_x hold to machine precision!
=LET(n_wd, B2, m_rt, B3, a_fr, B4, l_in, B5, pi_d, B6, IF(OR(n_wd <= 0, m_rt < 0, a_fr <= 0, a_fr >= 1, l_in <= 0, pi_d < 0, pi_d > 1), "Invalid Life-Table Input", LET(q_ch, MIN(1, (n_wd * m_rt) / (1 + n_wd * (1 - a_fr) * m_rt)), d_iv, l_in * q_ch, l_py, n_wd * (l_in - d_iv) + n_wd * a_fr * d_iv, ROUND(l_py * (1 - pi_d), 2))))Try the example
Use the original values or change them to test your understanding.