There they go Again

1
Initiative in Population ResearchCenter for
Human Resource Research The Conditional Frailty
Model an Analysis of Child Welfare DataJanet
M. Box-Steffensmeier, Suzanna DeBoef (PSU), Anand
Sokhey (OSU) And, Mel Moeschberger (OSU), Michael
Foster (UNC).
2
Outline

• What are repeated events processes?
• Why do they present modeling problems?
• The Options, old and new.
• Simulation Evidence.
• Some early conclusions, Foster Care.

3
One event Survival

• An observation (e.g., individual, country, etc.)
  is in some state until it dies. We want to know
  effect of some RX on risk of death for some
  interval of time, given that one is alive up
  until that time.
• Death of an alliance, tenure of office, demise of
  an interest group, passage of budget, length in
  foster care, etc.

4
Repeated Events

• An observation experiences the same event
  multiple times. We want to know the effect of
  some Rx on risk of experiencing an event for some
  interval of time.
• Repeat criminal offenders, repeat heart attack
  victims, repeated bouts of poverty, repeated
  spells of unemployment, repeated spells of foster
  care, etc.

5

• Repeated event processes ubiquitous in health,
  medical, public policy applications
• Different models give different results due to
  bias and inefficiency

6
Unique features of repeated events processes

• Event Dependence Once an observation experiences
  an event, it may become more or less likely to
  experience another (learning process or damage
  effects).
• Heterogeneity Some countries are more prone to
  experience events than others (unknown,
  unmeasured,unmeasurable).
• ? Repeated instances of the same event within a
  country are unlikely to be independent.

7

• That is, for recurrent events, correlation can
  come from 2 distinct sources
• Heterogeneity across individuals
• Event Dependence

8
How have we (typically) modeled repeated events
processes?

• Estimate a logit, ignoring duration dependence
  and losing censored cases.
• Estimate a parametric duration model, assuming a
  functional form for duration dependence, treating
  repeated observations as independent, but
  adjusting standard errors.
• Estimate a Cox model, treating repeated
  observations as independent, but adjusting
  standard errors.
• Estimate a Cox model, adding a frailty term.

9

• Models to be compared
• Variance Corrected
• Frailty
• Conditional Frailty

10
The Semiparametric Cox Model

• The hazard (or risk) that an event will occur for
  subject i is given by
• ?i(t) ?0(t)exp(Xi(t)ß)
• where ?0 is an unspecified nonnegative function
  of time called the baseline hazard and the X
  i(t)ß give the covariate effects.
• The model is a proportional hazard model
  covariates effects raise/lower the baseline
  hazard, they dont change its shape.
• If we have 2 individuals with covariate values X
  and X (in democracy or not), ß gives the
  relative risk (of being democratic or not).
• The Cox model imposes the assumption that
  events occur independently, i.e., that the timing
  and occurrence of repeated events is unrelated to
  the initial (and subsequent) occurrence(s) of an
  event.

11
The problem Most models assume independence of
events, which is unlikely to be true in repeated
events data! We are likely to have event
dependence, heterogeneity, or both.

• The Cost? Biased and Inefficient Estimates of
  the Effects we Care About!

12
Alternative 1 Robust or variance corrected models

• V-C models are fit as though the data consist of
  independent observations, and then robust
  standard errors are calculated post estimation.
• Robust standard errors are based on the idea that
  observations are independent across groups or
  clusters but not necessarily within groups.
• So in the repeated events context, the standard
  errors are adjusted to deal with the fact that
  observations within a country over time are not
  independent.
• No V-C models make allowance for biasing effects
  that can be produced by a lack of independence in
  event times due to heterogeneity.
• Some V-C models do account for event dependence.

13
V-C Model Alternatives vary based on assumptions
about the risk set and event dependence

• Cox with robust standard errors (aka
  Andersen-Gill) one baseline hazard, but only at
  risk for the k1th event after the kth event.
• Conditional Models The baseline hazard varies by
  k. Only at risk for the k1th event after the kth
  event. Estimation is thus conditioned on the
  number of events experienced.
• ?i(t) ?0k(t)exp(Xi(t)ß)
• Marginal Models No distinction is made for the
  number of events experienced when identifying the
  risk set. Always at risk for all k events.
• Models may be estimated in gap time or elapsed
  time. The conditional model in gap time is ?i(t)
  ?0k(t-tk-1)exp(Xi(t)ß)

14
Alternative 2 Frailty or Random Effects Models

• Frailty models make assumptions about the nature
  of the heterogeneity, specifically its
  distribution, and incorporate it into model
  estimates (use robust standard errors).
• The assumption made is that some
  observations/countries are intrinsically more or
  less prone to experiencing the event than are
  others, and that the distribution of these
  individual-specific effects can be known
  (approximated). Individuals are assumed to be a
  member of an identifiable family with which
  proneness is shared, but whose source is unknown
  or unmeasured.
• Frailty models do not account for event
  dependence.

15
Random Effects Model

• ?i(t) ?0(t)exp(Xi(t)ß Z i?)
• Here a unit scores a 1 on Z if it is a member of
  group j, with which it shares some frailty (some
  shared proneness). That frailty is added to
  the hazard. If there is enough variation unique
  to the family, then the variance of the random
  effect will be significant.
• A parametric assumption must be made for the
  distribution of the frailties, for ?.

16
A 3rd Alternative The Conditional Frailty Model

• ?i(t) ?0k(t)exp(Xi(t)ß Z i?)
• The conditional frailty model incorporates key
  features of repeated events processes into the
  model
• Event dependence is allowed by varying the
  baseline hazard with k
• Heterogeneity is allowed by including the random
  effect.
• We also define the risk set such that cases are
  not at risk for the k1th event until they have
  had the kth.
• May be estimated in either gap or elapsed time.

17
Simulations The Data Generating Process

• We draw the time to an individual is kth
  event --tik-- from an exponential distribution
  with rate (risk) ?ik(t)
• ?ik(t) ?0k(t)exp(Xi(t)ß µi)
• Where
• ?0k(t) ?0 No Event Dependence
• ?0k(t) k?0 Event Dependence
• and
• µi 0 No Heterogeneity
• µi N(0,1) Heterogeneity

18
Models

• We estimate 7 Models varying the existence of
  event dependence, heterogeneity, and definitions
  of the risk set.
• Andersen-Gill, Conditional Elapsed/Gap, Frailty
  (Gauss/Gamma), Conditional Frailty (Gauss/Gamma).
• ß -1.0, baseline .10, N100, M1000, klt15,
  follow up time is 50 periods.

19
(No Transcript)
20
(No Transcript)
21
(No Transcript)
22
(No Transcript)
23
(No Transcript)
24
Simulation Findings

• If event dependence exists
• MUST estimate baseline hazards that vary with k.
• Estimating a random effect INSTEAD causes big
  problems.
• Conditional Frailty Model is at least as good as
  alternatives
• If heterogeneity exists
• MUST estimate random effect.
• Conditional Frailty Model is at least as good as
  alternatives
• Estimating varying baseline hazards INSTEAD
  causes big problems.
• If both heterogeneity and event dependence exist
• The conditional frailty model captures both
  effects better than any alternative.
• Interestingly, as long as there is some
  heterogeneity, the AG model works pretty well
  (even with event dependence).
• Sensitivity of results? To baseline hazard, rare
  events, other?

25
Foster Care Data from the Chapin Hall Center for
Children, University of Chicago

• The Conditional Frailty model is useful because
  it can disentangle
• 1) the role of event dependence, i.e., the effect
  of repeated spells of time in the child welfare
  system
• 2) heterogeneity in terms of unmeasured child
  level characteristics

26
Motivation Instability in Foster Care

• Ideal children placed in state custody would be
  returned to parents or placed for adoption in a
  relatively short period of time. While in state
  custody, stable placements with foster parents or
  in community-based institutional settings (such
  as group homes).
• Reality 2 decades of research show substantial
  departure from the ideal. Central theme is
  negative effects of the instability weakened
  attachment to care givers, emotional and
  behavioral problems, school failure, criminal
  activity, and early parenthood. Heightened
  concern over instability gaining momentum over
  time.
• Lawsuits and legislation landmark Adoption and
  Safe Families Act of 1997. Eventually, look for
  intervention effect of the legislation and state
  differences in implementation.

27
Data

• Placement data for children who enter the foster
  care system for the first time in 2000 and 2001.
  Observed through December 2003.
• Multiple placements within their first spell.
• Limited number of covariates, at this time, yet
  does illustrate the usefulness of the conditional
  frailty model.

28
(No Transcript)
29
(No Transcript)
30
Implications

• ? There is both event dependence and
  heterogeneity in the foster care data so a
  conditional frailty model is required.
• ? Placements are event dependent, with more
  frequent placements leading to further
  disruptions and more movement via placements.
• ? Our work has the potential to guide policy
  makers with respect to the investment of
  resources by determining what child and/or system
  level characteristics produce different risks.

31
Extensions

• Substantive
• -- more covariates of interest, e.g.,
  characteristics of the child, care givers, and
  agency responsible for the childs care.
• -- NIH grant, ex. Does use of mental health
  serves reduce the likelihood of churning within
  the child welfare system?
• Methodological
• -- Multilevel, example administrative levels
• -- Competing Risks, multiple placements and
  different types of placements.

32
Parting Advice?

• Think about the process you care about. Is it
  likely to be plagued by
• Event dependence?
• Heterogeneity?
• Both?
• Then pick a model that allows for these features.
• Do we care about risk since the last event or
  since the beginning? Pick the relevant time
  scale.
• What about the distribution of events? How many
  cases will contribute information to higher
  strata? Is it enough for estimation?

33
Thank you!
34
An Application Conflict!

• We model the hazard of a militarized
  international conflict in a (politically
  relevant) dyad as a function of 6 covariates
  (1950-85)
• Democracy (polity 3) (-)
• Level of economic growth (-)
• Presence of an alliance in the dyad (-)
• Contiguity status ()
• Military Capability ratio (-)
• Extent of bilateral trade (-)
• Estimate each of the models identified above.

35
Data Organization Counting Process Notation
36

• ANDERSEN GILL coxph(formula Surv(starta,
  stopa, dispute, type "counting") democ
  growth allies contig capratio trade, data
  bzorn, na.action na.exclude, method
  "efron", robust T)
• coef exp(coef) se(coef) robust se z
  p
• democ -0.439 .644 0.0998 0.0952
  -4.62 3.9e-06
• growth -3.227 .0397 1.2279 1.3011
  -2.48 1.3e-02
• allies -0.414 .0661 0.1107 0.1133
  -3.65 2.6e-04
• contig 1.214 3.37 1.1209 0.1266
  9.58 0.0e00
• capratio -0.214 .0807 0.0514 0.0632
  3.39 7.1e-04
• trade -13.162 1.92e-06 10.3265 11.4066
  -1.15 2.5e-01
• Rsquare 0.013 (max possible 0.227 ) N20448
• Likelihood ratio test 272 on 6 df, p0
• Wald test 203 on 6 df, p0
• Score (logrank) test 262 on 6 df, p0,
  Robust 221 p0

37
(No Transcript)
38

• CONDITIONAL ELAPSED. coxph(formula Surv(starta,
  stopa, dispute, type "counting") democ
  growth allies contig capratio trade
  strata(sumdisp) cluster(dyadid), data bzorn,
  na.action na.exclude, method "efron", robust
  T)
• coef exp(coef) e(coef) robust se
  z p
• democ 0.1615 1.1753 0.1123 0.1024
  1.577 0.11000
• growth -3.7687 0.0231 1.2444 1.0633
  -3.544 0.00039
• allies 0.1439 1.1547 0.1167 0.1079
  1.333 0.18000
• contig 0.2866 1.3318 0.1243 0.1108
  2.586 0.00970
• capratio 0.0595 1.0613 0.0441 0.0289
  2.057 0.04000
• trade 6.1495 468.461 8.1422 6.5343
  0.941 0.35000
• Rsquare 0.001 (max possible 0.117 ) N20448
• Likelihood ratio test 25.5 on 6 df,
  p0.000276
• Wald test 34.7 on 6 df,
  p5.01e-06
• Score (logrank) test 26.1 on 6 df,
  p0.000211, Robust 29.5 p4.86e-05

39

• CONDITIONAL GAP coxph(formula Surv(start,
  stop, dispute, type "counting") democ growth
  allies contig capratio trade
  strata(sumdisp) cluster(dyadid), data bzorn,
  na.action na.exclude, method "efron",
  robust T)
• coef exp(coef) se(coef) robust se
  z p
• democ 0.0987 1.1038 0.1089 0.0746
  1.323 1.9e-01
• growth -3.4328 0.0323 1.2384 1.2402
  -2.768 5.6e-03
• allies -0.2022 0.8169 0.1151 0.0939
  2.154 3.1e-02
• contig 0.6177 1.8546 0.1225 0.1036
  5.961 2.5e-09
• capratio 0.0557 1.0573 0.0463 0.0253
  2.202 2.8e-02
• trade 0.8258 2.2837 11.6276 9.5944
  0.086 9.3e-01
• Rsquare 0.002 (max possible 0.142 ) N20448
• Likelihood ratio test 36.4 on 6 df,
  p2.29e-06
• Wald test 51.2 on 6 df,
  p2.69e-09
• Score (logrank) test 36.9 on 6 df,
  p1.81e-06, Robust 43.4 p9.78e-08

40

• FRAILTY GAMMA. coxph(formula Surv(starta,
  stopa, dispute, type "counting") democ
  growth allies contig capratio trade
  frailty.gamma(dyadid), data bzorn, na.action
  na.exclude, method "efron")
• coef exp(coef) se(coef) se2 Chisq DF
  p
• democ -0.365 .69420 0.1309
  0.1108 7.78 1 5.3e-03
• growth -3.685 .02511 1.3457
  1.2991 7.50 1 6.2e-03
• allies -0.370 .69073 0.1685
  0.1252 4.82 1 2.8e-02
• contig 1.199 3.3168 0.1673
  0.1310 51.41 1 7.5e-13
• capratio -0.199 .81955 0.0547
  0.0495 13.29 1 2.7e-04
• trade -3.039 .047883 12.0152
  10.3084 0.06 1 8.0e-01
• frailty.gamma(dyadid)
  708.95 394 0.0e00
• Iterations 7 outer, 27 Newton-Raphson
• Variance of random effect 2.42
  I-likelihood -2399.4
• Degrees of freedom for terms 0.7 0.9 0.6
  0.6 0.8 0.7 394.2
• Rsquare 0.052 (max possible 0.227 )
  N20448
• Likelihood ratio test 1089 on 399 df, p0
• Wald test 121 on 399 df, p1

41
CONDITIONAL FRAILTY GAMMA. coxph(formulaSurv(star
t, stop, dispute, type"counting")democgrowthal
liescontigcapratiotradestrata(sumdisp)
frailty.gamma(dyadid), data bzorn, na.action
na.exclude, method "efron") coef
exp(coef) se(coef) se2 Chisq DF p
democ 0.0988 1.1038 0.1089
0.1089 0.82 1 3.6e-01 growth
-3.4225 .03263 1.2389 1.2389 7.63 1
5.7e-03 allies -0.2022
.8169 0.1151 0.1151 3.09 1 7.9e-02 contig
0.6178 1.8548 0.1225
0.1225 25.43 1 4.6e-07 capratio
0.0557 1.0573 0.0463 0.0463 1.45 1
2.3e-01 trade 0.8119 2.2522
11.6292 11.6292 0.00 1 9.4e-01 frailty.gamm
a(dyadid) 0.00 0
9.1e-01 Iterations 6 outer, 26 Newton-Raphson
Variance of random effect 5e-07
I-likelihood -1549.1 Degrees of freedom for
terms 1 1 1 1 1 1 0 Rsquare 0.002 (max
possible 0.142 ) N20448 Likelihood ratio
test 36.4 on 6 df, p2.29e-06 Wald test
36.2 on 6 df, p2.54e-06 Warning
messages Loglik converged before variable 6
beta may be infinite. in fitter(X, Y, strats,
offset, init, control, weights weights,
42
(No Transcript)
43
What did we learn?

• Signs and levels of significance change for some
  variables.
• Contiguity has a positive effect with varying
  magnitude
• Capability ratio has a small effect that flips
  signs with the models.
• Effects of growth and being in an alliance are
  robust to model choice.

44
(No Transcript)
45
(No Transcript)
46
(No Transcript)