(a) What are the various indices of mortality measure? Explain the purpose and procedure for standardizing them. (20 marks) (b)…

(a) Indices of mortality and their standardisation

Indices of mortality:

• Crude death rate: CDR = D/P × 1000, where D = deaths in the year and P = mid-year population.
• Age-specific death rate: mₓ = Dₓ/Pₓ × 1000 (also by sex); cause-specific death rate: deaths from one cause per 100,000 population.
• Infant mortality rate: IMR = deaths under 1 year ÷ live births × 1000; the neonatal rate (under 28 days) and under-five mortality rate are defined likewise per 1000 live births.
• Maternal mortality ratio: maternal deaths per 100,000 live births.
• Life-table measures: qₓ and expectation of life eₓ⁰, which do not depend on the age structure.

Purpose of standardisation: the CDR reflects the age composition as well as the risk of dying, so a population with many old people has a high CDR even if its age-specific rates are low. Standardisation removes the effect of different age structures so that populations or years can be compared fairly.

Direct method (age-specific rates of the study population known and reliable):

1. Choose a standard population with age distribution Pₓˢ (for example the WHO World Standard Population or a census population).
2. Apply the study rates mₓ to it to get expected deaths ΣmₓPₓˢ.
3. SDR = ΣmₓPₓˢ/ΣPₓˢ.

Indirect method (study age-specific rates unknown or based on few deaths):

1. Take the standard age-specific rates mₓˢ and the study population's ages Pₓ.
2. Expected deaths E = ΣmₓˢPₓ; observed deaths D.
3. SMR = D/E (× 100); indirectly standardised rate = (D/E) × CDR of the standard population.

An SMR above 100 means higher mortality than the standard after allowing for age. Standardised rates are comparable only when the same standard population is used.

(b) Logistic curve and its properties

Let P(t) be the population and L its upper limit. Verhulst (1838) assumed growth proportional to P and to the unused fraction of capacity: dP/dt = rP(1 − P/L), r > 0. Since 1/[P(1 − P/L)] = 1/P + 1/(L − P), integration gives ln[P/(L − P)] = rt + c, so P/(L − P) = Ae^rt and P(t) = L/(1 + Be^−rt), B = 1/A. With P(0) = P₀, B = (L − P₀)/P₀. Write B = e^rβ, that is β = (1/r) ln[(L − P₀)/P₀]; then Be^−rt = e^r(β − t) and P(t) = L/(1 + e^r(β − t)). β > 0 corresponds to P₀ < L/2.

Three properties:

1. Asymptote: P → L as t → ∞, while P(0) = L/(1 + e^rβ) > 0; the relative growth rate (1/P)dP/dt = r(1 − P/L) falls linearly as P approaches L.
2. Inflection: P″ = Lr²e^r(β−t)(e^r(β−t) − 1)/(1 + e^r(β−t))³ is zero only at t = β, where P = L/2 and the growth rate dP/dt = rL/4 is greatest; the curve is convex before β and concave after.
3. Symmetry: P(β + s) + P(β − s) = L, so the curve is symmetric about (β, L/2); also ln[P/(L − P)] = r(t − β) is linear in t, which allows r and β to be fitted.

(c) TFR, GRR and NRR as measures of reproduction

All three are period measures built from age-specific fertility rates fₓ per woman at ages 15–49, but they count different things.

• TFR = Σfₓ (single-year ages; 5Σfₓ for five-year groups): the average number of children, sons and daughters together, a woman would bear at the current rates, with no allowance for mortality.
• GRR = Σfₓᶠ = TFR × proportion of female births (about 0.488 at 105 boys per 100 girls): the number of daughters per woman, still ignoring mortality, so GRR ≈ TFR/2.05.
• NRR = Σfₓᶠ(Lₓ/l₀): each daughter-birth is weighted by the probability that a girl survives to the mother's age x (female life table). It gives the daughters who will actually replace the mother.

Hence NRR ≤ GRR, NRR being GRR times an average survival factor. NRR = 1 means exact replacement (a stationary population in the long run), NRR > 1 growth and NRR < 1 eventual decline; GRR overstates replacement wherever girls die before childbearing. TFR alone does not measure replacement because it includes sons.

The replacement-level TFR is about 2.1: 1/0.488 = 2.05 births give one daughter, raised slightly for girls' mortality. India's TFR was 2.0 in NFHS-5 (2019-21), below replacement.