mathematical hazards models and model life tables formal
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Mathematical Hazards Models and Model Life TablesFormal Demography
Stanford Summer Short CourseJames Holland Jones, Instructor
August 12, 2005
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Outline
1. Mathematical Hazards Models
(a) Gompertz-Makeham(b) Siler(c) Heligman-Pollard
2. Relational Mortality Models
3. Model Life Tables
(a) Coale-Demeny(b) INDEPTH
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Basic Quantities in the Analysis of Mortality
Survival Function
S(x) = Pr(X > x)
for continuous X, strictly decreasing
Complement of the cumulative distribution function of F (x)
S(x) ≡ l(x) = 1− F (x), F (x) = Pr(X ≤ x)
Also the integral of the probability density function f(x):
S(x) = Pr(X > x) =∫ ∞
x
f(t)dt
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Thus, given a survival function, we can calculate the probability density function
f(x) ≡ d(x)dt = −dS(x)dx
and f(x)∆x is the approximate probability that a death will occur at time x
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More Basic Quantities: The Hazard Function
The Hazard is the demographic force of mortality (typically indicated µ(x))
It is defined as
h(x) ≡ µ(x) = lim∆x→0
Pr[x ≤ X < x + ∆x|X ≥ x]∆x
If X is a continuous random variable
h(x) =f(x)S(x)
= −d log[S(x)]
The cumulative hazard is
H(x) =∫ x
0
h(u)du = − log[S(x)]
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Thus, for continuous lifetimes
S(x) = exp[−H(x)] = exp[−
∫ x
0
h(u)du
]
This is exactly as it should be, as it matches our lifetable definitions:
l(x) = exp[−
∫ x
0
µ(t)dt
]
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Gompertz Mortality
Gompertz (1825) suggested that a “law of geometric progression pervades” inmortality after a certain age
Gompertz mortality can be represented as
µ(x) = αeβx
α is known as the baseline mortality, whereas β is the senescent component
Makeham (1860) extended the Gompertz model by adding a constant γ
Note that since the Gompertz model is for a mortality hazard, we can integrate itto give us the the survival function:
h(x) = αeβx, S(x) = exp[α
β
(1− eβx
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Also note that log-mortality is a linear function of age
log µ(x) = log(α) + βx
This suggests a regression approach my be useful
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Gompertz Fits So Well!
20 40 60 80 100
−8
−6
−4
−2
Age
log(
nMx)
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Or Does It?!
20 40 60 80 100
−8
−6
−4
−2
0
Age
log(
nMx)
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Other Mathematical Models of Mortality
Advantages:
• Compact, small number of parameters
• Highly interpretable
• Generalizable
• Good for comparative work (particularly interspecific comparisons)
Disadvantages:
• Hard to fit
• Numerical estimates frequently unstable (correlation between components)
• Almost certainly “wrong”
• What if there is a new source of mortality that is not covered by the model?
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Two Models
Despite their shortcomings, two models are of interest:
1. Siler 5-Component Competing Hazard2. Heligman-Pollard 8-Component Model
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Siler 5-Component Competing Hazard Model
Developed to facilitate interspecific comparisons
Model of mortality hazard, µ(x)
Competing hazard model
h(x) = ae−bx + c + defx
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Siler Fit to Coale-Demeny West 15
0 20 40 60 80 100
1e−
041e
−03
1e−
021e
−01
1e+
00
Age
Mor
talit
y H
azar
dinfantadultsenescenttotal
Siler 5−Parameter for West 15 Model Life Table
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Heligman-Pollard 8-Component Model
Heligman-Pollard: models mortality probability qx
qx
1− qx= A(x+B)C
+ D exp[−E
{log
x
F
}2]
+ GHx
where qx is the mortality probability at age x = 0, 1, 2, . . . , ω
The HP model is a beast to fit
Tremendous identifiability problems because the parameters are highly correlated
Dellaportas et al. (2001) suggest a Bayesian strategy to improve model fit andnumerical properties
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Heligman-Pollard Fit to UK 1995
0 20 40 60 80 100
5e−
062e
−05
1e−
045e
−04
2e−
03
Age
pi_x
infantaccidentsenescenttotal
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Relational Life Tables
Sometimes, we actually have some data, but not quite enough to make a full lifetable
Brass (1971) suggested a regression procedure in which observed mortality wasregressed onto a mortality standard, X(S)
Mortality probabilities, survivorships, etc. are distributed on [0, 1] and are notnormally distributed (and the errors associated with them are certainly not)
This makes regression tricky
The usual way to handle this is to use a transformation of the data known as a logit
for some 0 ≤ x ≤ 1,
Y = logit(x) = log(
x
1− x
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x =eY
1 + eY
Note that this transforms a variable that ranges over [0 1] to one that ranges over[−∞ ∞]
We now perform the following regression
Y = α + βX(S)
α is interpreted as the “level” of mortality
β is the “shape” of mortality
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Be Careful! The notational usage in Preston et al. (2001) on this topic is notparticularly standard
. qx is not the standard 5-year mortality probability at age x
. qx is really 1− lx
. Perhaps a better notation would be xq0
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Effect of Changing α
0 20 40 60 80
0.2
0.4
0.6
0.8
1.0
Age
lx
standardalpha<0alpha>0
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Effect of Changing β
0 20 40 60 80
0.2
0.4
0.6
0.8
1.0
Age
lx
standardbeta>1beta<1
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Model Life Tables
There are a variety of model life table systems
1. UN (1958, 1982)2. Coale & Demeny (1966, 1983)3. INDEPTH4. Weiss, 1973 (Obscure anthropological model life tables)
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Model Life Table Construction
Model life table construction involves three general steps
• Gather a lot of high quality life tables
• Use Multivariate Analysis Techniques (e.g., PCA) to find clusters
• Combine them to provide a standard
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Coale-Demeny Regional Model Life Tables
The Coale & Demeny is broken down into 4 “Regional” Models: North, South,East, and West
North Sweden pre-1920, Norway, Iceland: low infant and post-50 mortalitySouth Spain, Portugal, Southern Italy: high infant and post-65 mortalityEast Austria, Germany, Northern Italy, Hungary, Poland: mortality rates generally
high, particularly after 50West Everything else: Most frequent anthropological application
They are indexed (typically) by life expectancy at birth or age 10
Use of model life tables follows a simple recipe:
1. Use whatever information available to estimate life expectancy2. Use whatever information available to determine the mortality pattern (or use
West model life table)3. Use the table that fits your assumptions
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Model Life Tables Construction
Component mortality models (e.g., Coale-Demeny)
m = Ca
where m is a vector of logit(nqx) values for x ∈ 0 . . . k
C is a k× l +1 matrix of loadings of the k ages on the first l components of a PCA(with a leading column of ones)
a is an l + 1× 1 vector of coefficients
a estimated by regressing empirically-derived values of logit(nqx) on C
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AIDS Decremented Model Life Tables: INDEPTH Model 1 & 5
Two model life table families are relevant:
• Model 1: Primarily western Africa (similar to Coale-Demeny North)
• Model 5: Primarily eastern and southern Africa (unique)
AIDS-decremented life tables constructed by adding excess mortality in thecharacteristic AIDS pattern to the model 1 life table
Regress the difference in m between model 1 and model 5 life tables with a common◦e0
Let d denote the coefficients of this regression
The AIDS decremented model life table is generated by:
m = C(a + αd)
where α is a scale parameter determining the extent of mortality excess
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INDEPTH AIDS-Decremented Model Life Tables
0 20 40 60 80
0.2
0.4
0.6
0.8
1.0
Age
Sur
vivo
rshi
p
0 20 40 60 80
0.2
0.4
0.6
0.8
1.0
Age
Sur
vivo
rshi
p
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AIDS Creates More Orphans
0 5 10 15
0.0
0.1
0.2
0.3
0.4
Age
Con
ditio
nal P
roba
bilit
y of
Orp
hanh
ood
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It’s the Pattern of Mortality that Matters
0 5 10 15
0.0
0.2
0.4
Age
Pr(
Orp
han)
5 Year Decrement
0 5 10 15
0.0
0.2
0.4
Age
Pr(
Orp
han)
10 Year Decrement
0 5 10 15
0.0
0.2
0.4
Age
Pr(
Orp
han)
15 Year Decrement
0 5 10 15
0.0
0.2
0.4
Age
Pr(
Orp
han)
20 Year Decrement
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Howell Used Coale-Demeny Model Life Tables
0 20 40 60 80
0.0
0.2
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1.0
Age
Sur
vivo
rshi
p
!Kung Life Table Used Coale−Demeny
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Hill & Hurtado Did Not
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0.0
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Age
Sur
vivo
rshi
p
Ache Life Table Did Not Use Coale−Demeny
Ache!Kung
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0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0
Age
Sur
vivo
rshi
p
Ache!KungYanomamoGombe
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0 20 40 60 80
0.0
0.4
0.8
Age
Sur
vivo
rshi
p
15 20 25 30 35 40 45
0.0
0.4
0.8
1.2
Age
AS
FR
0.2 0.4 0.6 0.8 1.0
0.05
0.10
0.15
Relative Age
dx
Proportion in the Chimp Life Table
Pro
port
ion
in H
uman
Life
Tab
le
0.5
1.5
2.5
0.0 0.2 0.4 0.6 0.8 1.0
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0 20 40 60 80
0.0
0.4
0.8
Age
Sur
vivo
rshi
p
10 20 30 40
0.00
0.10
0.20
0.30
Age
Fer
tility
0 10 20 30 40
0.00
0.10
0.20
0.30
Age
Sen
sitiv
ity
0 10 20 30 40 50
0.00
0.05
0.10
0.15
Age
Sen
sitiv
ity
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