Nonlinear Discrete-time Hazard Models for Entry into Marriage

Nonlinear Discrete-time Hazard Models for

Entry into Marriage

Heather Turner, Andy Batchelor, David Firth

Department of StatisticsUniversity of Warwick, UK

8th March 2010

Heather Turner, Andy Batchelor, David Firth University of Warwick

Nonlinear Discrete-time Hazard Models for Entry into Marriage

Motivating Application: The LII Survey

The Living in Ireland Surveys were conducted 1994-2001

For five 5-year cohorts of women born between 1950 and1975 we have the following data

I year of (first) marriageI year and month of birthI social classI highest level of education attainedI year highest level of education was attained



When do women get married?

We can use methods from survival analysis to model thetiming of marriage

Consider time starting from the legal age of marriage,then the survival time, T is the time until a personmarries

The time of marriage is recorded to the nearest year, sowe will use a discrete-time analysis



Discrete-time Hazard Models

For discrete-time the hazard of marriage occuring at timet is defined as

h(t) = P (T = t|T ≥ t)

We are interested in the shape of the hazard over the lifecourse and how the hazard is affected by covariates



Cox Proportional Odds Model

A popular choice is the proportional odds model proposedby Cox (JRSSB, 1972):

h(t|xit)

1− h(t|xit)=

h0(t)

1− h0texp x′

itβ

where h0(t) is the baseline hazard

Taking logs we obtain

logit(h(t|xit)) = logit(h0(t)) + x′itβ

= lt + x′itβ

I semi-parametric - makes no assumption about the shapeof the hazard function



Episode-splitting

A simple way to estimate the proportional odds model isto generate an event history for each observation

Pseudo observations are created at each time point fromtime 0 up to marriage or censoring - this is known asepisode-splitting

The parameters in the proportional odds model can thenbe estimated by fitting a logistic regression model to abinary indicator of marriage at each time point (married= 1, unmarried = 0)



Cox Proportional Odds Model

15 19 23 27 31 35 39 43

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Age (years)

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ge



Sidenote: interval-censored data

A similar model can be obtained by assuming that thedata are interval-censored observations of acontinuous-time proportional hazards model

The coefficients in the model

cloglog(h(t|xit)) = lt + x′itβ

are then the coefficients of the proportional hazards model

This relationship breaks down however if αt is replaced bya parametric function



Blossfeld and Huinink Model

Blossfeld and Huinink (Am. J. Sociol., 1991) propose thefollowing parametric baseline

logit(h0(t|ageit)) = l(ageit)

= c + βl log(ageit − 15) + βr log(45− ageit)

I describes the nature of the time dependenceI fixes the support of the hazard to be 15 to 45 years



BH Model

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Age (years)

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arria

ge



Effect of Endpoints

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Age (years)

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arria

ge

Hazard support

15−45 years12−75 years



Nonlinear Discrete-time Hazard Model

An obvious extension of the BH model is to treat theendpoints as parameters

l(ageit) = c + βl log(ageit − αl) + βr log(αr − ageit)

I nonlinear - need to extend available softwareI near-aliasing between parameters - need to

reparameterise



Developing the Nonlinear Model

First analyse using the BH model as a reference

Then analyse using the extended model and illustratenear-aliasing

Finally analyse using a re-parameterised nonlinear discretemodel

I compare to BH modelI refine model for the LII data



BH Models

The BH models can be fitted using the glm function in R.

Following the model building strategy of Blossfeld &Huinink (1991), we select

I a cohort factorI a time-varying indicator of educational status (in/out)

For the 1970-1974 cohort the conditional odds ofmarriage are 24% of those for the 1950-1954 cohort

For women in education the conditional odds of marriageare 11% of those for women not in education



Selected BH Model

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Age (years)

Pro

babi

lity

of M

arria

ge(1949,1954](1954,1959](1959,1964](1964,1969](1969,1974]

Deviance = 12073 Residual d.f. = 31001



Nonlinear Discrete-time Hazard Models

The nonlinear discrete-time hazard model is an example ofa generalised nonlinear model, which can be fitted usingthe gnm package in R (Turner and Firth, R News, 2007)

I parameters estimated by a modified IWLS algorithmI certain nonlinear terms inbuilt e.g. Mult, ExpI our terms cannot be expressed in terms of these

functions, so need to write custom "nonlin" function



Custom "nonlin" Function

LogExcess <- function(age, side = "left"){call <- sys.call()constraint <- ifelse(side == "left",

min(age) - 1e-5, max(age) + 1e-5)list(predictors = list(beta = ∼1, alpha = ∼1),

variables = list(substitute(age)),term = function(predLabels, varLabels) {

paste(predLabels[1], " * log("," -"[side == "right"], varLabels[1], " + "," -"[side == "left"], constraint," + exp(", predLabels[2], "))")

},call = as.expression(call))

}class(LogExcess) <- "nonlin"



Summary of Baseline ModelCall:

gnm(formula = marriages/lives ~ LogExcess(age, side = "left") +

LogExcess(age, side = "right"), family = binomial, data = fulldata,

weights = lives, start = c(-20, 3, 0, 3, 0))

Deviance Residuals:

Min 1Q Median 3Q Max

-0.8098 -0.4441 -0.3224 -0.1528 4.0483

Coefficients:

Estimate Std. Error z value Pr(>|z|)

(Intercept) -118.5395 201.6387 -0.588 0.55661

LogExcess(age, side = "left")beta 3.6928 1.1913 3.100 0.00194

LogExcess(age, side = "left")alpha -0.1432 0.8935 -0.160 0.87267

LogExcess(age, side = "right")beta 24.8623 38.5743 0.645 0.51923

LogExcess(age, side = "right")alpha 4.0247 1.7376 2.316 0.02054

Std. Error is NA where coefficient has been constrained or is unidentified

Residual deviance: 12553 on 31004 degrees of freedom

AIC: 12748

Number of iterations: 76



Parameter Correlations

c βl αl βr αr

c 1.00000βl -0.92563 1.00000αl -0.80861 0.95844 1.00000βr -0.99999 0.92688 0.80989 1.00000αr -0.99833 0.90319 0.77910 0.99808 1.00000



Example ’Recoil’ Plot

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Age

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10 20 30 40 50

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Age

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Age

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Is Near-aliasing a Problem?

Extended model can still be used as baseline hazard

logit(h(t|xit)) = l(ageit) + x′itβ

Near-aliasing will make models harder to fit - particularlywith several covariates

Not all parameters are interpretable



Re-parameterizing the Nonlinear Model

The nonlinear hazard model can be re-parameterized asfollows:

l(ageit) = γ − δ

{(ν − αl) log

(ν − αl

ageit − αl

)}+ δ

{(αr − ν) log

(αr − ν

αr − ageit

)}



Interpretation of Parameters

The parameters of the new parameterisation have a moreuseful interpretation than before:

Age (years)

Pro

babi

lity

of M

arria

ge

ααL νν ααR

expit((γγ))



New Parameter Correlations

γ ν δ αl αr

γ 1.00000ν 0.12956 1.00000δ 0.21943 -0.69849 1.00000

αl 0.27236 -0.42848 0.91425 1.00000αr 0.03231 -0.75428 0.93696 0.77910 1.00000

Table: Correlations between the estimated parameters of thereparameterized baseline model defined in Equation ??



Recoil Plots for Reparameterised Model

x

pred

ictC

urve

(x)

peak height (γ)−2.09 → −1.95

0.00

0.04

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0.12

x

pred

ictC

urve

(x)

peak location (ν)25.39 → 28

x

pred

ictC

urve

(x)

fall off (δ)0.34 → 0.15

x

pred

ictC

urve

(x)

left endpoint (αL)14.17 → 15.04

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x

pred

ictC

urve

(x)

right endpoint (αR)100.66 → 47.68

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10:50

rep(

0, 4

1)

●

Original ModelPerturbed ModelRe−fitted Model

Age

Pro

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of M

arria

ge



Analysis with the Reparameterised Model

We can now repeat the previous analysis using thenonlinear baseline hazard instead of the BH hazardfunction

I The model selection is qualitatively unchangedI The residual deviance is reduced by about 20 at the

expense of 2 d.f.I There is a lot of uncertainty about the right end-point -

in the final model it is estimated as 400 years with alarge standard error.



Infinite Right End-point

It seems more appropriate to define the baseline hazard inwhich the right end-point tends to infinity:

l(ageit) = γ−δ

{(ν − αl) log

(ν − αl

ageit − αl

)− ageit − ν

}Re-fitting the final model with this baseline increases thedeviance by a negligible amount



Comparing Models

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Age (years)

Pro

babi

lity

of M

arria

ge

(1949,1954](1954,1959](1959,1964](1964,1969](1969,1974]


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Age (years)P

roba

bilit

y of

Mar

riage

(1949,1954](1954,1959](1959,1964](1964,1969](1969,1974]




Refining the Model

The model building strategy so far has been similar toBlossfeld and Huinink (1991) for comparison

Careful consideration of the fit of the model suggests thatimprovements can be made



Final Model with New Baseline

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Age (years)

Pro

babi

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of M

arria

ge(1949,1954](1954,1959](1959,1964](1964,1969](1969,1974]




Cohort Effect

We can investigate the cohort effect further by replacingthe cohort factor by a year-of-birth factor and plotting theresultant effects

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1955 1960 1965 1970

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5−

1.5

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50.

0

Year of Birth

Yea

r−of

−bi

rth

Effe

ct



Year-of-birth Effect

The plot suggests a more appropriate model

θ exp(λ(yrbi − 1950))

Replacing the year-of-birth factor with this nonlinear termreduces the deviance by 19 whilst gaining 2 d.f.



Checking the Fit

The new year-of-birth terms takes account of the effect ofthis factor on the magnitude of the hazard

To check for other effects on the hazard, we can groupthe data by year of age and cohort then plot thecorresponding observed and fitted proportions



Fit over Cohorts

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as.numeric(colnames(grp))

grpO

bs[i,

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(1949, 1954](5211)

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grpO

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(1955, 1959)(6283)

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grpO

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grpO

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(1965, 1969](6289)

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grpO

bs[i,

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(1969, 1974](6666)


grpO

bs[i,

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Age (years)

Pro

port

ion

mar

ried



Fit over Education Levels

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grpO

bs[i,

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No attainment/primary(2366)

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grpO

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Lower secondary(7900)

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Upper secondary(11507)

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grpO

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College(4829)

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grpO

bs[i,

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University(4407)


grpO

bs[i,

] ● Observed

Model 13 (common peak)Model 14 (separate peaks)

Age (years)

Pro

port

ion

mar

ried



Linear Dependence of Peak Location

Quantifying the education level by a dynamic measure ofyears in education ed, we incorporate a linear dependenceof peak location on ed :

l(xit) = γ − δ

{(ν0 + ν1edi − αl) log

(ν0 + ν1edi − αl

ageit − αl

)}+δ {ageit + ν0 + ν1edi}

This results in a non-proportional hazards model



Years Post-Education

Checking the fit against years post-education:

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−10 0 10 20 30

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Years post education

Pro

port

ion

mar

ried I lower rate of increase in

first 3 yearspost-education

I sharp change at 7 yearspost-education

I outlying points



Early Career Effect

The lower rate of increase during the first 3 yearspost-education may be explained by an early career effect

This can be incorporated in the model by including anappropriate indicator variable, significantly reducing thedeviance

The deviance does not significantly increase when the leftendpoint is constrained to 15 years



Effect of EducationPeak location varies from 20.78 years (primary education)to 26.89 years (university graduates)

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Age (years)

Pro

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of m

arria

geEducation level

PrimaryLower sec.Upper sec.PLCITUniversity



Effect of Year-of-birthPeak hazard varies from 0.17 (b. 1950) through 0.15 (b.1960) to 0.07 (b. 1970)

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Age (years)

Pro

babi

lity

of m

arria

geYear of Birth

195019601970



Summary

Estimating the support of the hazard function improves fit

Near-aliasing can occur in nonlinear models, but can beovercome by re-parameterisation

Our proposed model has more interpretable parameters,particularly location and magnitude of the maximumhazard

I can investigate effect of covariates on these features

The parametric form does impose some restrictions onthe shape of the hazard curve



References

A comprehensive manual is distributed with the packageat http://www.cran.r-project.org/package=gnm

A working paper on the marriage application is availableat www.warwick.ac.uk/go/crism/research/2007



http://www.cran.r-project.org/package=gnm

www.warwick.ac.uk/go/crism/research/2007

Acknowledgements

The data are from The Economic and Social ResearchInstitute Living in Ireland Survey Microdata File(©Economic and Social Research Institute).

We gratefully acknowledge Carmel Hannan forintroducing us to this application and providingbackground on the data.



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Nonlinear Discrete-time Hazard Models for Entry into Marriage

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