www.hist.umn.edu/~rmccaa calibrating paleodemography: fertility effects are so strong (and mortality...

www.hist.umn.edu/~rmccaa

Calibrating Paleodemography:Calibrating Paleodemography:fertility effects are so strong (and fertility effects are so strong (and

mortality so weak) that stable mortality so weak) that stable population analysis gives better population analysis gives better

results than quasi-stable or dynamic results than quasi-stable or dynamic methodsmethods

** * * * *

Robert McCaaRobert McCaaMinnesota Population Center Minnesota Population Center


“…“…can general magnitudes of fertility, can general magnitudes of fertility, mortality and growth be derived from a single mortality and growth be derived from a single

recorded age distribution alone?recorded age distribution alone?

Popoff and Judson, “Some Methods of Estimation Popoff and Judson, “Some Methods of Estimation for Statistically Underdeveloped Areas”, for Statistically Underdeveloped Areas”,

in in The Methods and Materials of DemographyThe Methods and Materials of Demography (Elsevier: 2004, 624):(Elsevier: 2004, 624):

““The answer is essentially negative. …The answer is essentially negative. …““Because past fertility is the dominant factor Because past fertility is the dominant factor

determining the shape of the age distribution, determining the shape of the age distribution,

… a rough estimate of the level of the … a rough estimate of the level of the birthrate birthrate

may be obtained by the examination of may be obtained by the examination of a single age structure.”a single age structure.”


What’s new??? What’s new??? --that is not already in “Paleodemography of the --that is not already in “Paleodemography of the

Americas” (Americas” (Backbone of HistoryBackbone of History, Cambridge, , Cambridge,

2002)?2002)? 1.1. Quasi-stable and dynamic models (simulated Quasi-stable and dynamic models (simulated

annealing optimization in Bonneuil, annealing optimization in Bonneuil,

forthcoming)forthcoming)

2.2. Graphical analysis using “faux” hazard Graphical analysis using “faux” hazard

rates, h(t), for both paleo and model rates, h(t), for both paleo and model

populationspopulations

3.3. Calibration of h(t) and age ratiosCalibration of h(t) and age ratios

4.4. When modeling plague epidemics, it is the When modeling plague epidemics, it is the

fertility that has the biggest impact on age fertility that has the biggest impact on age

structure (birth busts and booms following). structure (birth busts and booms following).


1.1. Why not quasi-stable or dynamic models?Why not quasi-stable or dynamic models?

• Quasi-stable (usually means varying mortality): Quasi-stable (usually means varying mortality):

it’s the fertility, stupid! The mortality signal is it’s the fertility, stupid! The mortality signal is

imperceptible except in extreme conditions.imperceptible except in extreme conditions.

• Dynamic models: Bonneuil’s simulated annealing Dynamic models: Bonneuil’s simulated annealing

optimization leads to the “closest path to a optimization leads to the “closest path to a

stable population”. The best! … but:stable population”. The best! … but:

– Results are heavily dependent on number of age groupsResults are heavily dependent on number of age groups

– Results range over the entire demographic experienceResults range over the entire demographic experience

– How would results vary if deposition period was in How would results vary if deposition period was in

centuries, rather than years?? Number of skeletons in centuries, rather than years?? Number of skeletons in

dozens instead of hundreds??dozens instead of hundreds??


1.1. Why not quasi-stable or dynamic models? Why not quasi-stable or dynamic models? (cont’d)(cont’d)

Dynamic models (Bonneuil, table 4), fertility:Dynamic models (Bonneuil, table 4), fertility:

Age groups Coale’s index Age groups Coale’s index iiff (with 95% confidence interval)(with 95% confidence interval)

33 0.44 [0.19, 0.52] 0.44 [0.19, 0.52]

44 0.43 [0.19, 0.49] 0.43 [0.19, 0.49]

55 0.51 [0.19, 0.52] 0.51 [0.19, 0.52]

66 0.47 [0.17, 0.49] 0.47 [0.17, 0.49]

77 0.39 [0.16, 0.42] 0.39 [0.16, 0.42]

88 0.39 [0.19, 0.42] 0.39 [0.19, 0.42]

99 0.34 [0.19, 0.42]0.34 [0.19, 0.42]

Range over much of human experience (iRange over much of human experience (iff = .16-.52) = .16-.52)


2. Graphical analysis using “faux” hazard rates2. Graphical analysis using “faux” hazard ratesDemographers know:Demographers know:

fertility fertility has the biggest impact on population age structure has the biggest impact on population age structure (and on the age distribution of deaths).(and on the age distribution of deaths).

Next figure shows fertility effects:Next figure shows fertility effects:

• Fertility varies from GRR = 2 to 6 (TFR=4-12!)Fertility varies from GRR = 2 to 6 (TFR=4-12!)

• Mortality is held constant (e0=20 years)Mortality is held constant (e0=20 years)

• Spread for adults is proportionally large.Spread for adults is proportionally large.


% o

f d

ea

ths

e0 = 20 yearsage

01 5 15 25 35 45 550

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e20g2

e20g3

e20g4e20g5e20g6

% o

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e50g2

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e50g4e50g5e50g6

% o

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gross reproduction ratio = 3age

01 5 15 25 35 45 550

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e20g3e30g3e40g3e50g3

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2a. Fertility has big effects 2a. Fertility has big effects on age structure of deaths on age structure of deaths ee00 = 20, GRR = 2, 3, 4, 5, 6 = 20, GRR = 2, 3, 4, 5, 6


% o

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2b. Fertility offers a target 2b. Fertility offers a target for curve-fitting for curve-fitting

ee00 = 50, GRR = 2, 3, 4, 5, 6 = 50, GRR = 2, 3, 4, 5, 6


% o

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e0 = 20 yearsage

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2c. Mortality 2c. Mortality offers no target at alloffers no target at all

ee00 = 20, 30, 40, 50, GRR = 3 = 20, 30, 40, 50, GRR = 3


% o

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e0 = 20 yearsage

01 5 15 25 35 45 550

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e20g2

e20g3

e20g4e20g5e20g6

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01 5 15 25 35 45 550

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2d. Mortality effects on age 2d. Mortality effects on age structure are imperceptiblestructure are imperceptibleee00 = 20, 30, 40, 50, GRR = 4 = 20, 30, 40, 50, GRR = 4


e0 = 20 yearsGRR = 2.5, 2.9, 3.3, 3.7

prop

ortio

nal h

azar

d h(

t)

age

0 5 15 25 35 45 550

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w20g25w20g29w20g33w20g37

3a. Hazard rates h(t) 3a. Hazard rates h(t) ee00 = 20; GRR = 2.5, 2.9, 3.3, 3.7 = 20; GRR = 2.5, 2.9, 3.3, 3.7


e0 = 20 yearsGRR = 2.5, 2.9, 3.3, 3.7

prop

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w40g25w40g29w40g33w40g37

3b. Hazard rates h(t) e3b. Hazard rates h(t) e00 = 20 & 40; = 20 & 40; GRR = 2.5, 2.9, 3.3, 3.7GRR = 2.5, 2.9, 3.3, 3.7


Belleville h(t) fit to e0= 40 years and GRR = 2.5, 2.9, 3.3, 3.7

prop

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h(t)

95%ci

95%ci

3c. Fitting Belleville h(t) e3c. Fitting Belleville h(t) e00 = 20 & = 20 & 40; GRR = 2.5, 2.9, 3.3, 3.740; GRR = 2.5, 2.9, 3.3, 3.7


4. When modeling plague or 4. When modeling plague or other catastrophes, remember other catastrophes, remember

lagged effects and that fertility…lagged effects and that fertility…• ……has the biggest impact on age structure (birth has the biggest impact on age structure (birth

busts and booms, followed by echoes).busts and booms, followed by echoes).• Consider the 1630 plague of Parma (see Manfredi, Consider the 1630 plague of Parma (see Manfredi,

Iasio & Lucchetti, IJA, 2002) Iasio & Lucchetti, IJA, 2002) – Death rates Death rates

• increased 500% in 1630increased 500% in 1630• 1/2 of normal in 16311/2 of normal in 1631• 1/5 of normal in 16321/5 of normal in 1632• Normal in 1633; 1/2 of normal in 1634, etc.Normal in 1633; 1/2 of normal in 1634, etc.

– Birth rates:Birth rates:• Contracted in year 0 by 1/4Contracted in year 0 by 1/4• Returned to normal in year 1Returned to normal in year 1• Almost tripled pre-plague frequencies in year 2Almost tripled pre-plague frequencies in year 2• Doubled+ pre-plague in year 3Doubled+ pre-plague in year 3• Doubled in year 4Doubled in year 4• Increased 50% over normal in year 5Increased 50% over normal in year 5• Year 6 & 7 below normal; year 8 normal; 9 = double, year 10 Year 6 & 7 below normal; year 8 normal; 9 = double, year 10

= normal, etc.= normal, etc.• Smaller the population the greater the variance and Smaller the population the greater the variance and

the greater the effectsthe greater the effects


ConclusionsConclusions

1.1. Regardless of method, it is fertility that is Regardless of method, it is fertility that is being measured—mortality rarely leaves being measured—mortality rarely leaves a tracea trace

2.2. Therefore, quasi-stable and dynamic Therefore, quasi-stable and dynamic models that hold fertility constant and models that hold fertility constant and allow only mortality to vary, may be mis-allow only mortality to vary, may be mis-directed. directed.

3.3. Point estimates can be deceiving; graphs Point estimates can be deceiving; graphs may provide insight on how tenuous the may provide insight on how tenuous the findings are.findings are.

4.4. Complex models should be tested against Complex models should be tested against historical datasets, using a double-blindhistorical datasets, using a double-blind


Thank you.Thank you.

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[email protected]@umn.edu