Gompertz Maximum Likelihood Estimation of Truncated Death … · 2020. 3. 25. · main ="2000...
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Gompertz Maximum Likelihood Estimation ofTruncated Death Distribuitons
Joshua R. Goldstein
November 8, 2019
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Big Picture
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Our challenge
We see only part of the picture (e.g., deaths aged 70 to 87).
No estimates of who died before or after
How can we estimate death rates without denominators?How can we estimate e(65) differences between groups?
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Our idea
We can combine
observed distribution of deaths (over limited range)our external knowledge of human mortality age-patterns
The hope is that this will produce good estimates of mortality rates, ofe(65), and of differences between groups
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Today’s agenda
Intro to Maximum Likelihood EstimationExample with simulated dataAttempt at validation with HMDPreliminary try at NUMIDENTLessions and directions
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Truncated Maximum Likelihood (in theory)
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Philosophy
For given data X , we can a likelihood associated with a particular value ofparameter θ.
We then choose the θ̂ to maximize this likelihood.
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A simple example
Likelihood for observation i withvalue xi :
Li = L(λ|xi) = fλ(xi)
Likelihood for all observations:
L =∏
iLi
Log-likelihood:
L =∑
ilog Li
If we observe x1 = 3 and x2 = 5, then
L(λ|x1) = λe−3λ
L(λ|x2|) = λe−5λ
L(λ|x1, x2) = λ2e( − 8λ)
L =∑
ilog Li = 2 log λ− 8λ
dLdλ = 2
λ− 8 = 0
So,λ̂MLE = 2/8 = 0.25
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We did this by hand, but can also do with the computerlambda.vec <- seq(.01, 1, .01)loglik.vec = 2 * log(lambda.vec) - 8 * lambda.vecplot(lambda.vec, loglik.vec)abline(v = lambda.vec[which.max(loglik.vec)])
0.0 0.2 0.4 0.6 0.8 1.0
−9
−8
−7
−6
−5
lambda.vec
logl
ik.v
ec
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For truncated distribution we observe only from a to b
We can define the conditional distribution
ftrunc = fθ(x)∫ ba fθ(x) dx
= fθ(x)Fθ(b)− Fθ(a)
with likelihood
L(θ|x) =∏ fθ(xi)
Fθ(b)− Fθ(a)
And then we maximize that.
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Validation with Simulated Gompertz
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Simulated Gompertz, without truncationsource("hmd_validation_functions.R")N = 2000set.seed(13)x <- rgompertz.M(N, b = 1/10, M = 75)Dx <- table(floor(x))plot(names(Dx), Dx, type = "h",
ylab = "Death Counts", xlab = "Age",main = "2000 Simulated Gompertz Deaths")
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2000 Simulated Gompertz Deaths
Age
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th C
ount
s
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MLE estimation
source("hmd_validation_functions.R")N = 1000set.seed(13)x <- rgompertz.M(N, b = 1/10, M = 75)Dx <- table(floor(x))fit <- counts.trunc.gomp.est(Dx= Dx, x.left = 0, x.right = 200,
b.start = 1/9, M.start = 80)(b.hat = exp(fit$par[1]))
## [1] 0.09572687
(M.hat = exp(fit$par[2]))
## [1] 75.01612
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How did we do?
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2000 Simulated Gompertz Deaths, with MLE fit
Age
Dea
th C
ount
s
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Now artificially truncate to ages 65-90
l <- 65h <- 90x.trunc <- x[x > l & x < h]Dx <- table(floor(x.trunc))fit <- counts.trunc.gomp.est(Dx= Dx, x.left = l, x.right = h,
b.start = 1/9, M.start = 80)(b.hat = exp(fit$par[1]))
## [1] 0.1083535
(M.hat = exp(fit$par[2]))
## [1] 75.42279
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Plot the fit
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Age
Dea
ths
Simulated Truncated Gompertz, with MLE fit
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Other ingredients
A modelHere we pick Gompertz, with parameters β and M.Some code to implement optimization routineValidatationTest on simulated Gompertz deaths, to see if we estimate right valuesTest on HMD to see if it works with real dataApplicationNUMIDENT(Weighted Censoc)
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Major Assumptions
Gompertz model is appropriateUniform coverage across ages to preserve the cohort distribution ofdeaths
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CodeGompertz functions
source("hmd_validation_functions.R")
##fit <- counts.trunc.gomp.est(Dx= Dx, x.left = 0, x.right = 200,
b.start = 1/9, M.start = 80)(b.hat = exp(fit$par[1]))
## [1] 0.1716812
(M.hat = exp(fit$par[2]))
## [1] 78.73153
Optimization
## wrapper function that calls optim()counts.trunc.gomp.est <- function(Dx, x.left, x.right, b.start,M.start){...}
## our negative log-lilelihoodd.counts.gomp.negLL <- function(par, Dx, x.left, x.right){...}
## usagefit <- counts.trunc.gomp.est(Dx = my.Dx, x.left = 70, x.right = 87,b.start = 1/9, M.start = 80)
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Without Truncation
N = 1000set.seed(13)x <- rgompertz.M(N, b = 1/10, M = 75)Dx <- table(floor(x))fit <- counts.trunc.gomp.est(Dx= Dx, x.left = 0, x.right = 200,
b.start = 1/9, M.start = 80)(b.hat = exp(fit$par[1]))
## [1] 0.09572687
(M.hat = exp(fit$par[2]))
## [1] 75.01612
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Plot the fit
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names(Dx)
Dx
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Validating Method with HMD
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Our approach
Use HMD death counts for the year and ages we have NUMIDENTdeaths
(1988 to 2005, Ages 65 and over)
Fit Gompertz and if we matchMortality ratese(65)
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Import and prepare HMD datalibrary(data.table)## read in and define age and cohortdt.mx <- fread("~/Documents/hmd/hmd_statistics/death_rates/Mx_1x1/USA.Mx_1x1.txt")dt.mx[ , x := as.numeric(Age)]
## Warning in eval(jsub, SDenv, parent.frame()): NAs introduced by coercion
dt.mx[Age == "110+" , x := 110]dt.mx[, cohort := Year - x]dt <- fread("~/Documents/hmd/hmd_statistics/deaths/Deaths_1x1/USA.Deaths_1x1.txt")dt[ , x := as.numeric(Age)]
## Warning in eval(jsub, SDenv, parent.frame()): NAs introduced by coercion
dt[Age == "110+" , x := 110]dt[, cohort := Year - x]#### make array, age x cohort x sex.dt.long <- melt(dt, measure.vars = c("Male", "Female"), variable.name = "sex", value.name = "Dx")## make arraymy.array <- dt.long[cohort %in% my.cohorts, xtabs(Dx ~ x + cohort + sex)]dimnames(my.array)[[3]] <- c("m", "f") ## for backwards compatibility
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Fit HMD## [1] 1900## [1] 1901## [1] 1902## [1] 1903## [1] 1904## [1] 1905## [1] 1906## [1] 1907## [1] 1908## [1] 1909## [1] 1910## [1] 1911## [1] 1912## [1] 1913## [1] 1914## [1] 1915## [1] 1916## [1] 1917## [1] 1918## [1] 1919## [1] 1920## [1] 1921## [1] 1922## [1] 1923
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Plot parameter values
1900 1905 1910 1915 1920
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my.cohorts
b.ve
c
1900 1905 1910 1915 1920
7585
95
my.cohorts
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ec
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Look at cohort of 1920, pretending we observe only ages68 to 85
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names(Dx)
Dx
HMD females cohort of 1920, observed and fit
Very high M is not crazy: Period life table for 2016 has mode at age 88.Joshua R. Goldstein Gompertz Maximum Likelihood Estimation of Truncated Death DistribuitonsNovember 8, 2019 27 / 32
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e(65)
1900 1905 1910 1915 1920
1012
1416
1820
Male cohort e(65) estimates from truncated Gompertz MLE
cohort
e(65
)
For reference, period e(65, 2016) = 18.36.Joshua R. Goldstein Gompertz Maximum Likelihood Estimation of Truncated Death DistribuitonsNovember 8, 2019 28 / 32
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Mortality rates
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Selected Mortality Rates
1895 1900 1905 1910 1915 1920 1925
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MLE fits (dashed) vs HMD observations (solid) Males
cohort
Mac
(lo
g−sc
ale)
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Overall fit is remarkably good.But shouldn’t rely on for unobserved ages (see old age decline in upperright)Joshua R. Goldstein Gompertz Maximum Likelihood Estimation of Truncated Death DistribuitonsNovember 8, 2019 30 / 32
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Period perspective
1990 1995 2000 2005
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Year
Mxt
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ale)
MLE estimated hazards (dashed) vs HMD (solid), period perspective by age
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Trying out the method with NUMIDENT
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