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Extending the discrete-time hazard model ALDA, Chapter Twelve

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Extending the discrete-time hazard model ALDA, Chapter Twelve Some departure from the norm will occur as time grows more open about it John Ashbery – PowerPoint PPT presentation

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Title: Extending the discrete-time hazard model ALDA, Chapter Twelve


1
Extending the discrete-time hazard model ALDA,
Chapter Twelve
Some departure from the norm will occur as time
grows more open about it John Ashbery
Judith D. Singer John B. Willett Harvard
Graduate School of Education
2
Chapter 12 Extending the discrete-time hazard
model
  • Alternative specifications for TIME in the
    discrete-time hazard model (12.1)must we always
    use the TIME indicators or might a more
    parsimonious representation for TIME be nearly as
    good?
  • Including time-varying predictors (12.3)as in
    growth modeling, the use of the person-period
    data set makes them easy to include (although be
    careful with interpretations)
  • Evaluating the assumptions of the discrete-time
    hazard modellike all statistical models, these
    invoke important assumptions that should be
    examined (and if necessary relaxed)
  • Linear additivity assumption (12.4)must all
    predictors operate only as main effects or can
    there be interactions?
  • Proportionality assumption (12.5)must the
    effects of all predictors be constant over time?

3
Pros and cons of the dummy specification for the
main effect of TIME?
  • Three reasons for considering an alternative
    specification
  • Your study involves many discrete time periods
    (because data collection is long or time is less
    coarsely discretized)
  • Hazard is expected to be near 0 in some time
    periods (causing convergence problems)
  • Some time periods have small risk sets (because
    either the initial sample is small or hazard and
    censoring dramatically diminish the risk set over
    time)

(ALDA, Section 12.1, pp 408-409)
4
An ordered set of smooth polynomial
representations for TIMENot necessarily the
best, but practically speaking a very good place
to start
(ALDA, Section 12.1.1, pp 409-412)
5
Illustrative example Time to tenure in colleges
and universities
Data source Beth Gamse and Dylan Conger (1997)
Abt Associates Report
  • Sample 260 faculty members (who had received a
    National Academy of Education/Spencer Foundation
    Post-Doctoral Fellowship)
  • Research design
  • Each was tracked for up to 9 years after taking
    his/her first academic job
  • By the end of data collection, n166 (63.8) had
    received tenure the other 36.2 were censored
    (because they might eventually receive tenure
    somewhere).
  • For simplicity, we wont include any substantive
    predictors (although the study itself obviously
    did)

(ALDA, Section 12.1.1 p 412)
6
Examining alternative polynomial specification
for TIMEDeviance statistics and fitted logit
hazard functions
The quadratic looks reasonably good, but can we
test whether its good enough?
(ALDA, Section 12.1.1, pp 412-419)
7
Testing alternative polynomial specification for
TIMEComparing deviance statistics (and AIC and
BIC statistics) across nested models
Two comparisons always worth making
(ALDA, Section 12.1.1, pp 412-419)
8
Including time-varying predictors Age of onset
of psychiatric disorder
Data source Blair Wheaton and colleagues (1997)
Stress adversity across the life course
  • Sample 1,393 adults ages 17 to 57 (drawn
    randomly through a phone survey in metropolitan
    Toronto)
  • Research design
  • Each was ask whether and, if so, at what age (in
    years) he or she had first experienced a
    depressive episode
  • n387 (27.8) reported a first onset between ages
    4 and 39
  • Time-varying question predictor PD, first
    parental divorce
  • n145 (10.4) had experienced a parental divorce
    while still at risk of first depression onset
  • PD is time-varying, indicating whether the
    parents of individual i divorced during, or
    before, time period j.
  • PDij0 in periods before the divorce
  • PDij1 in periods coincident with or subsequent
    to the divorce
  • Additional time-invariant predictors
  • FEMALE which well use now
  • NSIBS (total number of siblings)which well use
    in a few minutes

(ALDA, Section 12.3, p 428)
9
Including a time-varying predictor in the
person-period data set
ID 40 Reported first depression onset at 23
first parental divorce at age 9
ID PERIOD PD FEMALE NSIBS EVENT 40 4
0 1 4 0 40 5 0 1
4 0 40 6 0 1 4 0
40 7 0 1 4 0 40 8
0 1 4 0 40 9 1 1
4 0 40 10 1 1 4 0
40 11 1 1 4 0 40 12
1 1 4 0 40 13 1 1
4 0 40 14 1 1 4 0
40 15 1 1 4 0 40 16
1 1 4 0 40 17 1 1
4 0 40 18 1 1 4 0
40 19 1 1 4 0 40 20
1 1 4 0 40 21 1 1
4 0 40 22 1 1 4 0
40 23 1 1 4 1
(ALDA, Section 12.3, p 428)
10
Including a time-varying predictor in the
discrete-time hazard model
(ALDA, Section 12.3.1, p 428-434)
11
Interpreting a fitted DT hazard model that
includes a TV predictor
(ALDA, Section 12.3.2, pp 434-440)
12
Using time-varying predictors to test competing
hypotheses about a predictors effectThe long
term vs short term effects of parental death on
first depression onset
ID PERIOD PDEATH1 PDEATH2 40 4
0 0 40 5 0 0 40 6 0
0 40 7 0 0 40 8 0
0 40 9 1 1 40 10
1 0 40 11 1 0 40 12
1 0 40 13 1 0 40 14
1 0 40 15 1 0 40 16
1 0 40 17 1 0 40
18 1 0 40 19 1 0 40
20 1 0 40 21 1
0 40 22 1 0 40 23 1
0
13
The linear additivity assumption Uncovering
violations and simple solutions
Linear additivity assumption Unit differences in
a predictortime-invariant or time-varyingcorresp
ond to fixed differences in logit-hazard.
  • Data source Nina Martin Margaret Keiley
    (2002)
  • Sample 1,553 adolescents (n887, 57.1 had been
    abused as children)
  • Research design
  • Incarceration history from age 8 to 18
  • n342 (22.0.8) had been arrested.
  • RQs
  • Whats the effect of abuse on the risk of arrest?
  • Whats the effect of race?
  • Does the effect of abuse differ by race (or
    conversely, does the effect of race differ by
    abuse status)?

(ALDA, Section 12.4, pp 443)
14
Evidence of an interaction between ABUSE and RACE
What is the shape of the logit hazard
functions? For all groups, Risk of 1st arrest is
low during childhood, accelerates during the teen
years, and peaks between 14-17
How does the level differ across groups? While
abused children appear to be consistently at
greater risk of 1st arrest, but the differential
is especially pronounced among Blacks
(ALDA, Section 12.4.1, pp 444-447)
15
Interpreting the interaction between ABUSE and
RACE
  • In comparison to a White child who had not been
    abused, the odds of 1st arrest are
  • 28 higher for Blacks who had not been abused
    (note this is not stat sig.)
  • 43 higher for Whites who had been abused (this
    is stat sig.)
  • Nearly 3 times higher for Blacks who had been
    abused.

This is not the only way to violate the linear
additivity assumption
(ALDA, Section 12.4.1, pp 444-447)
16
Checking the linear additivity assumption Is
the effect of NSIBS on depression onset linear?
Use all your usual strategies for checking
non-linearity transform the predictors, use
polynomials, re-bin the predictor, .
All models include a cubic effect of TIME, and
the main effects of FEMALE and PD
(ALDA, Section 12.4.2, pp 447-451)
17
The proportionality assumptionIs a predictors
effect constant over time or might it vary?
(ALDA, Section 12.5.1, pp 451-456)
18
Discrete-time hazard models that do not invoke
the proportionality assumption
(ALDA, Section 12.5.1, pp 454-456)
19
The proportionality assumption Uncovering
violations and simple solutions
  • Data source Suzanne Graham (1997) dissertation
  • Sample 3,790 high school students who
    participated in the Longitudinal Survey of
    American Youth (LSAY)
  • Research design
  • Tracked from 10th grade through 3rd semester of
    collegea total of 5 periods
  • Only n132 (3.5) took a math class for all of
    the 5 periods!
  • RQs
  • When are students most at risk of dropping out of
    math?
  • Whats the effect of gender?
  • Does the gender differential vary over time?

(ALDA, Section 12.4, pp 443)
20
Checking the proportionality assumption Is the
effect of FEMALE constant over time?
All models include a completely general
specification for TIME using 5 time dummies
HS11, HS12, COLL1, COLL2, and COLL3
(ALDA, Section 12.5.2, pp 456-460)
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