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Crude Rates: measures of flows

Crude Rates: measures of flows. Main definition: No of events in (t,t+1)/Exposure in (t,t+1) (Births in year 1965)/(Midyear pop in 1965x1) (Births 1960-65)/{(Midyear pop 1960-65)*5} To remember: Events: counts from vital stats, surveys etc… Exposure: an abstraction or approximation.

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Crude Rates: measures of flows

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  1. Crude Rates: measures of flows • Main definition: • No of events in (t,t+1)/Exposure in (t,t+1) • (Births in year 1965)/(Midyear pop in 1965x1) • (Births 1960-65)/{(Midyear pop 1960-65)*5} • To remember: • Events: counts from vital stats, surveys etc… • Exposure: an abstraction or approximation

  2. Mortality rate at 4-5: Events/exposure Exposure=units are persons * unit of time Bounded by 0 and infinity Probability of dying between 4 and 5: Events/possible events Possible events=units are persons alive at 4 Bounded by 0 and 1 Rates vs Probabilities

  3. Nature of crude rates: CDR • CDR= Deaths/Pop= Dx /  Px • [(Dx/Px)*(Px/Pop)]=(Mx*Cx) • Weighted average of Mx, with age distribution, Cx, as weights • Would like to get measures reflecting Mx only • How does Mx look like? [Figure 1]

  4. Solutions to problems presented by CDR • Standardization: • SDR1=  Csx M1x where Csx is a ‘standard’ • SDR2=  Csx M2x • Comparison is between SDR1 and SDR2 • Life table: Mx----->S(x) [Figure 2] Life expectancy at birth, Eo Life expectancy at age x, Ex

  5. CBR • CBR=Births (t, t+1)/Exposure in (t,t+1)= • =Bx/Pop= Bx/ Px • =(Bx/Wx)*(Wx/35W15)*(35W15/ Wx)*(Wx/ Px) • =Fx * Rx*35C15 * w • A CBR depends on age and sex distributions • We only want to summarize Fx

  6. Rates of Population Increase r=CBR-CDR R=r+NMR (met Migration rate) Doubling time, Td~.69/r

  7. The age profile of Fx • Age specific fertility rates Fx have a universal shape [Figure 3] • Synthetic measures of fertility are (all summations are between 15 and 49): • TFR=  {15-49} Fx…….total fertility rate • GRR .45 * TFR…….gross reproduction rate • NRR .45 {15-49}Fx*S(x)..net reproduction rate

  8. Models of Mortality • Summarizing variability in mortality by age: • Gompertz model (Makeham extension) • Brass logit models • Coale and Demeny Models • United Nations Models

  9. Model of fertility I • What would “natural fertility” look like? • Reproduction span*(1/ Length Average Birth Interval) • Reproductive span=Menopause-Menarche~35*12=420 months • Birth Interval: • Conception time ~5 months • Pregnancy~9 • Post-partum fecundability~8 • Fetal losses~4 Expected births=420/26 = 16.2

  10. Models of fertility II • What is natural fertility, Nx? [Figure 4] • Variability in natural fertility…..K • Deviations from natural fertility..Vx and m • A model: • gx =K* Nx*exp(-Vx*m)….marital fertility • Fx= gx*Gx……………..…general fertility • [Figures 5 and 6]

  11. Popular standardized measures of fertility: the Princeton Study • If= births/women 15-49 • Ig=births to married women/ max births to married women  • Im=weighted number of married women 15-49/ weighted number of women 15-49 

  12. Historical Strategies: Iso-fertility curves • Disregarding illegitimate fertility we have: • If= Im*Ig • Two sets of factors operating on each measure • Location of societies in iso-fertility curves reveals societal strategies for reducing fertility [Figure 7]

  13. Age distributions, Cx • Cx’s reveal past history of mortality, fertility and migration [Figure 8 and 9] • Important result: when Fx and Mx are constant we generate a stable population with a unique r. If r=0 we say we have attained a stationary population[Figure 10]

  14. The mathematics of stable populations • N(x)=B(t-x)*S(x) • B(t-x)=Bo*exp(r*(t-x)) • N= N(x) • C(x)=N(x)/N • C(x)=CBR*S(x)*exp(-r*x)  • If r=0, C(x)=(1/Eo)*S(x) • a unique relation: NRR=exp(r*T) • T is the ‘length of a generation’~ Mean age of childbearing

  15. Important results • Age distributions are heavily affected by changes in fertility • They are less affected by changes in mortality • Momentum, M: • M ==(CBR*Eo / r * T) * (NRR-1)/NRR))

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