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
Rates vs Probabilities
• 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
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]
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
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
Rates of Population Increase
r=CBR-CDR
R=r+NMR (met Migration rate)
Doubling time, Td~.69/r
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
Models of Mortality
• Summarizing variability in mortality by age:
– Gompertz model (Makeham extension)– Brass logit models– Coale and Demeny Models– United Nations Models
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
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]
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
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]
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]
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
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))