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

Mortality indices and standardization. Mortality measurement requires crude, age-specific and cause-specific rates because age structure affects comparisons. The crude death rate (CDR) is total deaths in a year divided by mid-year population, usually ×1000. Age-specific death rate (ASDR) for age group x is deaths in that group divided by mid-year population in that group ×1000. Infant mortality rate (IMR) is deaths under one year per 1,000 live births. Maternal mortality ratio (MMR) is maternal deaths per 100,000 live births. Under-five mortality rate (U5MR) is deaths before age five per 1,000 live births. They measure overall burden, age pattern, child survival and pregnancy-related risk. Standardization is needed because a population with more old people will have a higher CDR even if age-specific risks are lower. It removes the effect of different age distributions, so changes in the standardized rate reflect changes in age-specific mortality rather than demographic ageing. Direct standardization applies study ASDRs to standard population weights: expected deaths = standard group population × study ASDR (with per-1000 factor); total expected deaths divided by total standard population gives the directly standardized rate. Indirect standardization is used when study ASDRs are unstable: apply standard ASDRs to the study population to obtain expected deaths; SMR = observed deaths / expected deaths, often ×100. An SMR above 100 indicates higher mortality than standard. The indirectly standardized rate can be obtained as SMR × standard CDR, but the SMR itself is not multiplied by the standard CDR. In India, Sample Registration System data provide ASDRs, from which age-standardized mortality estimates are computed for interstate and temporal comparison.

Logistic curve. Let P(t) be population and L carrying capacity. The logistic growth model is dP/dt = rP(1-P/L), r>0. The model assumes density-dependent growth: at low P, growth is nearly exponential; near L, the limiting term (1-P/L) reduces growth. Separating variables, dP/[P(1-P/L)] = r dt. Since 1/[P(1-P/L)] = 1/P + 1/(L-P), integration gives ln(P/(L-P)) = rt + C. With P(0)=P0, C=ln(P0/(L-P0)). Hence P/(L-P) = (P0/(L-P0)) e^rt. Solving for P, for t>0, P(t)=L/[1+((L-P0)/P0)e^-rt]. Put β=(1/r)ln[(L-P0)/P0]; then ((L-P0)/P0)e^-rt=e^r(β-t), giving P(t)=L/[1+e^r(β-t)]. The stated condition β>0 requires P0<L/2; if P0=L/2, β=0, and if P0>L/2, β<0. Three properties: first, the curve is sigmoidal, with maximum growth at the inflection point P=L/2 and t=β; second, L is the upper asymptote, so growth slows as P approaches carrying capacity; third, r controls steepness, larger r producing faster approach to L, and it is symmetric about the inflection point, P(β+s)+P(β-s)=L.

Reproduction measures. TFR, GRR and NRR differ in sex specificity and mortality adjustment. TFR sums age-specific fertility rates; for 5-year groups, TFR=5ΣASFRₓ, or 5Σ(ASFRₓ/1000) if per 1,000 women. It measures total daughters and sons a woman would have, without mortality adjustment. GRR is TFR × proportion female births, i.e. the sum of female-specific ASFRs; it counts only female births and ignores survival. It overstates replacement if child mortality is high. NRR weights each female ASFR by the probability of surviving to that childbearing age, NRR=5Σ(lₓ/l₀)(female ASFRₓ/1000) for 5-year groups, where lₓ/l₀ is survival from birth to age x. Thus NRR estimates the average number of daughters who would survive to reproductive age per woman in a hypothetical generation. All three are period measures based on current ASFRs, with NRR additionally using current survival probabilities; NRR is the most complete for projection because it incorporates survival, while TFR is most commonly reported. NRR=1 indicates exact replacement; in India, TFR≈2.1 is commonly used as the replacement level, though it is higher where child mortality is substantial. Thus, demographic measurement moves from crude rates to age-, sex- and survival-adjusted indicators for better comparison and projection.