Calibrating Paleodemography:
fertility effects are so strong (and
mortality so weak) that stable population
analysis gives better results than quasistable or dynamic methods
***
Robert McCaa
Minnesota Population Center
www.hist.umn.edu/~rmccaa
Popoff and Judson, “Some Methods of Estimation for
Statistically Underdeveloped Areas”,
in The Methods and Materials of Demography
(Elsevier: 2004, 624):
“…can general magnitudes of fertility, mortality and
growth be derived from a single recorded age
distribution alone?
“The answer is essentially negative. …
“Because past fertility is the dominant factor
determining the shape of the age distribution,
… a rough estimate of the level of the birthrate
may be obtained by the examination of
a single age structure.”
www.hist.umn.edu/~rmccaa
What’s new???
--that is not already in “Paleodemography of the
Americas” (Backbone of History, Cambridge, 2002)?
1. Quasi-stable and dynamic models (simulated
annealing optimization in Bonneuil,
forthcoming)
2. Graphical analysis using “faux” hazard rates,
h(t), for both paleo and model populations
3. Calibration of h(t) and age ratios
4. When modeling plague epidemics, it is the
fertility that has the biggest impact on age
structure (birth busts and booms following).
www.hist.umn.edu/~rmccaa
1.
•
Why not quasi-stable or dynamic models?
Quasi-stable (usually means varying mortality):
it’s the fertility, stupid! The mortality signal is
imperceptible except in extreme conditions.
•
Dynamic models: Bonneuil’s simulated annealing
optimization leads to the “closest path to a
stable population”. The best! … but:
–
Results are heavily dependent on number of age groups
–
Results range over the entire demographic experience
–
How would results vary if deposition period was in
centuries, rather than years?? Number of skeletons in
dozens instead of hundreds??
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1.
Why not quasi-stable or dynamic models? (cont’d)
Dynamic models (Bonneuil, table 4), fertility:
Age groups Coale’s index if (with 95% confidence interval)
3
0.44 [0.19, 0.52]
4
0.43 [0.19, 0.49]
5
0.51 [0.19, 0.52]
6
0.47 [0.17, 0.49]
7
0.39 [0.16, 0.42]
8
0.39 [0.19, 0.42]
9
0.34 [0.19, 0.42]
Range over much of human experience (if = .16-.52)
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2. Graphical analysis using “faux” hazard rates
Demographers know:
fertility has the biggest impact on population age structure
(and on the age distribution of deaths).
Next figure shows fertility effects:
• Fertility varies from GRR = 2 to 6 (TFR=4-12!)
• Mortality is held constant (e0=20 years)
• Spread for adults is proportionally large.
www.hist.umn.edu/~rmccaa
e20g2
% of deaths
5
4
% of deaths
6
2a. Fertility has big effects on
age structure of deaths
e0 = 20, GRR = 2, 3, 4, 5, 6
e20g3
3
e20g4
e20g5
e20g6
2
1
0
01
5
15
25
age
e0 = 20 years
www.hist.umn.edu/~rmccaa
35
45
55
6
2b. Fertility offers a target for
curve-fitting
e0 = 50, GRR = 2, 3, 4, 5, 6
% of deaths
5
4
e50g2
3
e50g3
2
e50g4
e50g5
e50g6
1
0
01
5
15
25
age
e0 = 50 years
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35
45
55
2c.
e0 = Mortality
20 years
offers no target at all
e0 = 20, 30, 40, 50, GRR = 3
age
6
4
% of deaths
% of deaths
5
e20g3
e30g3
e40g3
e50g3
3
2
1
0
01
5
15
25
age
35
gross reproduction ratio = 3
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45
55
55
2d. Mortality
on age
e0 = 50effects
years
structure are imperceptible
e0 = 20, 30, 40, 50, GRR = 4
age
6
% of deaths
5
4
3
e20g4
e30g4
e50g4
e40g4
2
1
0
01
5
15
25
age
35
gross reproduction ratio = 4
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45
55
3a. Hazard rates h(t)
e0 = 20; GRR = 2.5, 2.9, 3.3, 3.7
proportional hazard h(t)
e0 = 20 years
GRR = 2.5, 2.9, 3.3, 3.7
14
12
10
8
6
w20g37
w20g33
w20g29
w20g25
4
2
0
0
5
15
25
age
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35
45
55
3b. Hazard rates h(t) e0 = 20 & 40;
GRR = 2.5, 2.9, 3.3, 3.7
proportional hazard h(t)
e0 = 20 years
GRR = 2.5, 2.9, 3.3, 3.7
14
12
10
8
6
w20g37
w20g33
w20g29
w20g25
w40g37
w40g33
w40g29
w40g25
4
2
0
0
5
15
25
age
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35
45
55
3c. Fitting Belleville h(t) e0 = 20 & 40;
GRR = 2.5, 2.9, 3.3, 3.7
proportional hazard h(t)
Belleville h(t) fit to e0= 40 years and
GRR = 2.5, 2.9, 3.3, 3.7
14
12
95%ci
10
h(t)
8
6
95%ci
w40g37
w40g33
w40g29
w40g25
4
2
0
0
5
15
25
age
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35
45
55
4. When modeling plague or other
catastrophes, remember lagged
effects and that fertility…
•
•
–
–
•
…has the biggest impact on age structure (birth busts and
booms, followed by echoes).
Consider the 1630 plague of Parma (see Manfredi, Iasio &
Lucchetti, IJA, 2002)
Death rates
•
•
•
•
increased 500% in 1630
1/2 of normal in 1631
1/5 of normal in 1632
Normal in 1633; 1/2 of normal in 1634, etc.
•
•
•
•
•
•
•
Contracted in year 0 by 1/4
Returned to normal in year 1
Almost tripled pre-plague frequencies in year 2
Doubled+ pre-plague in year 3
Doubled in year 4
Increased 50% over normal in year 5
Year 6 & 7 below normal; year 8 normal; 9 = double, year 10 =
normal, etc.
Birth rates:
Smaller the population the greater the variance and the
greater the effects
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Conclusions
1. Regardless of method, it is fertility that is
being measured—mortality rarely leaves a
trace
2. Therefore, quasi-stable and dynamic models
that hold fertility constant and allow only
mortality to vary, may be mis-directed.
3. Point estimates can be deceiving; graphs may
provide insight on how tenuous the findings
are.
4. Complex models should be tested against
historical datasets, using a double-blind
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Thank you.
****
rmccaa@umn.edu
www.hist.umn.edu/~rmccaa