Overview
Survival Analysis in the 21st Century:
Rewriting Event History with Latent Variables
Katherine E. Masyn, Ph.D.
Graduate School of Education
Harvard University
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Survival Basics
Discrete-time Survival Analysis (DTSA)
Vintage DTSA
Modern DTSA
•• DTSA as SEM
•• DTSA as LTA
•• DTSA as LCA
Texas Tech University
December 6, 2013
Time-to-event data
Survival Basics
Event history: A record of whether
and when an event occurred for
each individual in a sample during
a finite observation period., e.g.,
time of death, grade of school
dropout, age of first alcohol use in
school-aged children, etc.
Example
ƒ One of the oft-cited factors that influence the quality
and equality of public education in the U.S. is the
lack of well-trained, high-quality teachers.
ƒ Originally, this problem was classified at a supplyside issue, i.e., more people need to be recruited
and effectively inducted into the teaching profession.
ƒ Then this problem was classified as a retention
issue.
ƒ To further understand teacher retention/attrition and
its correlates, we need a way to model both whether
and when individuals exit the teaching profession.
Target event
ƒ An event for which each person in the
population is theoretically at-risk, i.e., has a
non-zero probability of experiencing the
event at some point in time.
–– At what age
ƒ Did you first drink alcohol without parent permission?
ƒ Were you suspended from school for first time?
ƒ Were you first arrested by the police?
ƒ The risk for the event as well as the actual
timing of the event varies across
individuals.
Key Information
ƒ What is the target event, i.e., for what is
the individual at-risk?
ƒ What defines risk onset, i.e., t = 0:00:00?
ƒ What is the metric for measuring time
from risk onset to event occurrence?
ƒ Under what circumstances is the event
time of an individual unknown or not
observed, i.e., missing?
Risk onset
ƒ Start the clock when everybody is initially
at-risk to experience the event
–– At what age
ƒ Did you first drink alcohol without parent permission? Æ Time 0 = Birth
ƒ Were you suspended from school for first time? Æ Time 0 = age of first
school enrollment.
ƒ Were you first arrested by the police? Æ Time 0 = ?
ƒ Start the clock at the occurrence of a
precipitating event
Time metric
ƒ Continuous
The ““exact”” time of an event for each
subject is known, e.g., time of death
ƒ Discrete
1) The timing of an event is continuous
but is only recorded for an interval of time,
e.g., grade of school dropout.
2) The timing of an event is itself discrete,
e.g., grade retention.
Missing data
ƒ Missing data is endemic in longitudinal studies;
survival studies are no exception.
ƒ Various mechanisms for missing data in the
survival context are referred to under the unifying
term, censoring, indicating that the event times
for some subjects are unknown to the researcher.
ƒ Censoring is usually assumed to be
noninformative which means that the distribution
of censoring times is independent of event times,
conditional on the set of observed covariates.
(Think: Missing-at-random)
Example (cont’d)
ƒ The most typical survival data is rightcensored and this type of missingness is
the easiest to deal with in the data analysis.
ƒ Right censoring occurs when a subject in
the sample has not experienced the event
of interest at the end of the observation
period. It is assumed that the event
eventually occurs sometime after the end of
the study.
ƒ The target event is first departure from fulltime classroom teaching.
ƒ Risk for all subjects begins on the morning of
the first day of the first year of teaching.
ƒ Events are assumed to be discretely occurring
between school years.
ƒ Once an individual has left teaching for the
first time, he/she is no longer at-risk.
ƒ Event time is unknown if individual is lost to
follow-up or the time is greater than 5 years.
Survival probability
Let T be the time interval of the event
where T{1,2,……,J}
Discrete-time
Survival Analysis
(DTSA)
PS(j), called the survival probability, is
defined as the probability of ““surviving””
beyond time interval j, i.e., the
probability that the event occurs after
interval j:
PS(j) = P(T > j).
Hazard probability
Ph(j), called the hazard probability, is
defined as the probability of the event
occurring in the time interval j, provided it
has not occurred prior to j:
Ph(j) = P(T = j | T t j).
Essentially, Ph(j) is the probability of the
event occurring in time interval j among
those at-risk in j.
Use of Hazard Probabilities
ƒ Identifies especially risky time periods, i.e.,
times when the event is particularly likely
to occur.
ƒ Characterizes the shape of the hazard
function –– determining whether risk
increases, decreases, or remains constant
over time.
ƒ Most survival models are specified in
terms of the hazard probabilities.
Hazard and Survival Probabilities
The relationship between PS(j) and Ph(j) is given by
ƒ When the hazard is high, the survival
function drops more rapidly.
ƒ When the hazard is low, the survival
function drops slowly.
ƒ When hazard is zero, the survival function
is constant.
ƒ Both Ph and PS give us important
information: PS tells who is at risk at a
given time and Ph tells what their risk is.
Proportional Hazard Odds Model
Vintage DTSA
Estimated using MLE within a traditional
logistic regression or fixed-effects
multilevel logistic regression framework.
f= continuous latent variable; c=categorical latent variable
y= continuous observed variable; u=categorical observed variable
T= continuous survival time; x= observed covariate
y
f
Modern DTSA
T
x
c
u
WITHIN
BETWEEN
Advantages of Unifying Framework
ƒ ML estimation
ƒ Allows for different measurement models for
predictors
ƒ Allows for multilevel data structure
ƒ Facilitates the modeling of joint processes
–– Accounts for shared variance between processes——
observed and unobserved
–– Allows for moderation or mediation of observed
covariates by process-level (latent) variables
–– More accurately reflects the complexity and
interconnectedness of longitudinal processes
ƒ Easily accommodates extensions:
–– Time-varying covariates
–– Non-proportional hazard, i.e., time-varying
covariates effects
–– Random effects, i.e., individually-varying
covariate effects
–– Recurring events
–– Competing risks
e1
E1
DTSA as SEM
e2
E2
e3
…
eJ
E3
EJ
x
Np1
zp1
Np2
zp2
NpJ
Np3
zp3
…
zpJ
DTSA as SEM Extensions
ƒ Multivariate DTSA
–– Recurring Events
–– Multiple Spell
–– Competing risk
ƒ
ƒ
ƒ
ƒ
Event history outcomes in mediation model
Multilevel DTSA
Semi-parametric frailty models
Latent class predictors of survival
Multivariate DTSA
ƒ Commonly there is only one event state but
more states are possible.
ƒ Event type
–– Single, non-recurring
–– Recurrent events (multiple spells)
Same outcome that may occur more than once
–– Competing risks
More than one possible outcome
–– Parallel process
Multiple event processes occurring at the
same time with independent event-specific
risk status but shared variance
Multivariate
Recurring Event: Professional role changes
for educators, e.g., classroom teacher,
administrator, researcher, etc.
Research question: What is the relationship
between time spent as a classroom teacher
and time spent in subsequent roles in or
outside of Education?
e11
e12
…
e20
e1J
z1
e21
…
e2J-
z2
c
z3
e11 e21
e12 e22
e30
e31
e13 e23
…
e3J-
…
e1J
e2J
Competing Risks: Professional roles
following departure from full-time classroom
teaching, e.g., administrator, researcher,
etc.
Research question: How does administrative
support influence the hazard of leaving
teaching for another role in Education
compared to the hazard of leaving teaching
for a professional role outside Education?
Mediation w/ Event History Outcome
Research question: How does first grade
aggression mediate the effect of gender on
time to first school removal?
K
x
c
zp1
zp2
zp3
…
zpJ
Multilevel DTSA
ƒ Event: First school removal (suspension or
expulsion)
ƒ Research question: How does individuallevel aggressive-disruptive behavior and
classroom-level aggressive-disruptive
behavior influence the hazard of school
removal?
Unobserved heterogeneity
Often referred to as frailty in the
continuous-time survival literature, this
involves the idea that there may be
variability in individuals’’ underlying
(baseline) risk for an event that is not
directly measurable, i.e., some individuals
are more ““prone”” to an event than others.
Modeling approaches
ƒ Parametric: Assume some underlying
parametric distribution for K and
maximize the likelihood.
ƒ Nonparametric: Use a finite mixture,
i.e., latent classes (C = 1,……,K), to
nonparametrically approximate the
distribution of K. (Long-term survivor
model is a special case.)
ƒ Events: 1st arrest, 2nd arrest, 3rd arrest
ƒ Research question: How do individuals
differ with respect to underlying
susceptibility to arrest and what predicts
that individual-level variability?
Latent variable predictors
Indirectly measured risk factors for an
event over time. Such a risk factors
could be modeled as a categorical or
continuous latent variables. These
risk factors could be time-varying or
time-invariant.
y1
Event: First departure from full-time classroom
teaching.
Predictor: School climate* measured by a set of
indicators related to administration and student
characteristics along with school resources.
Time-varying since teachers may switch
schools.
Research question: How does school climate
affect if and when teachers leave the
classroom?
*Could be teacher-specific or could use multilevel
formulation if teachers are nested within schools.
y2
T
T
K
c
yp
y1
y2
yp
Growth process: Teacher-rated aggression
in grades 1-6.
Event: School removal in grades 7-12.
Research question: What is the relationship
between aggression across grades 1-6
and the risk of school removal across
grades 7-12?
Aggression
y1 – y10
e8 – e18
Ky
Ke
School removal
Low aggression
Med aggression
High aggression (10%)
ce
cy
Med aggression (48%)
High aggression
Low aggression (43%)
x
DTSA as LTA
ƒ Event occurrence represents an
individual’’s transition from one ““state”” (preevent) to another ““state”” (post-event).
ƒ States can be
–– Physical (e.g., living in a homeless shelter)
–– Psychological (e.g., depressed or healthy)
–– Social (e.g., married or divorced)
Transition model:
Restrictions for T=j to T=j+1 (jt2):
Time 0
Time 1
Event
Pre-event
Time 2
Time 3
Post-event
Event
Pre-event
Pre-event
DTSA as LTA Extensions
ƒ Multiple indicators of event status
ƒ Dual process DTSA
Post-event
Wj,j+1
ej+1 = 0
ej+1 = 1
ej+1 = 2
ej = 0
1––Ph(j+1)
Ph(j+1)
0
ej = 1
0
0
1
ej = 2
0
0
1
Event
Pre-event
Multiple indicators of event occurrence
ƒ In some applications, event occurrence is
inferred through indirect observation of the
presence/absence of one or more symptoms
that are used collectively (e.g., behavior
checklist) to arrive at a ““definitive”” clinical
diagnosis.
ƒ In some settings, multiple measures of onset
are used but obtain conflicting information,
e.g., ““have you ever had a drink?””, ““at what
age did you first drink?””, ““number of drinks
days in past month?””
The trouble with error
Quantifying error
u1j
u2j
ƒ Ignoring measurement error on event
occurrence (and, thus, risk status) can
results in either upward- or downwardbiased hazard probability estimates.
ƒ The impact of measurement error on the
baseline hazard estimates depends on
u3j
ej
Kj=3
ƒ Symptom sensitivity:
ƒ Symptom specificity:
–– Number of symptoms
–– Sensitivity and specificity of each symptom
–– ““True”” baseline rate
P(upj = 1 | ej = 1)
P(upj = 0 | ej z 1)
Dual Process DTSA
u11
u12
u13
u21
u22
c0
c1
c2
K0=1
K1=2
K2=3
u23
u31
u32
c3
K3=3
u33
Unlike time-varying predictors of events or
competing risks, dual process DTSA
models do not prioritize one event over
another.
Events: High school dropout and onset of
illicit drug use.
Research question: What is the relationship
between the risk of high school dropout and
the risk of illicit drug use over time?
Time 0
A
Time 1
Time 2
Time 3
Event
Post-event
Post-event
Pre-event
Pre-event
Event
B
Pre-event
Pre-event
Event
Event
Pre-event
Pre-event
Post-event
Post-event
Event
Event
Pre-event
Pre-event
Event history model
DTSA as LCA
DTSA as LCA Extensions
ƒ Event history mediation
–– Event timing mediates the effect of risk factor
on outcome
ƒ Onset-to-growth models (““Launch”” models)
Each ““latent”” class is a subgroup of adolescents that share
the same age of onset, e.g., Class 3 consists of individuals
that had their first real drink of alcohol at age 11. Class
membership is known for all individuals with
observed/reported onset age and partially known for rightcensored individuals.
Onset-to-growth
At each time point there are two types
of zeros:
1) Pre-onset zeros (individuals who
have not initiated use).
2) Transient abstainers following onset.
Intra-individual change
For onset-to-growth, there are two process
models
ƒ Event history –– describes the time-to-event
process
–– Models the probability that an individual will
transition from ““at risk”” for onset to initiating
alcohol use.
–– Marks the beginning of the use trajectories, Ti=0
ƒ Post-onset use trajectories
–– Models the expected level of use across time as
a function of time, conditional on initiation.
Inter-individual differences
ƒ Event history –– individual differences in the
hazard of onset
ƒ Growth trajectories–– individual differences
in growth trajectory parameters (e.g.,
intercept and slope growth factors)
Hazard of Alcohol Use Onset
Survival of Alcohol Use Onset
0.3000
1.000
0.900
0.2500
0.800
0.700
0.2000
0.600
0.1500
0.500
0.400
0.1000
0.300
0.200
0.0500
0.100
0.0000
0
2
4
6
8
10
12
14
16
18
20
0.000
0.00
2.00
4.00
Age
6.00
8.00
10.00
12.00
14.00
16.00
18.00
20.00
Age
Multi-faceted Longitudinal Process
Model-estimated Levels of Use by Onset Age
# of drinking days (in past 30 days)
4
3
Timing of first lapse
Use vs. non-use
following lapse
onset ” 9
onset 10
onset 11
onset 12
2
onset 13
onset 14
onset 15
onset 16
onset 17
1
Overall
0
15
16
17
Age
18
Risk and protective
factors
Frequency of use
following first use,
i.e., % days drinking
Intensity of use
following first use,
e.g., drinks per
drinking day
Time(1st Lapse)
Risk and
protective factors
Frequent-heavy: 6%
ƒ Coping skills at intake were related to timing of
first lapse and variability within post-lapse
drinking classes.
Infrequent-moderate: 82%
Prolapsing: 12%
ƒ Alcohol Dependence Scores were related to
timing of first lapse, variability within post-lapse
drinking classes, and post-lapse drinking class
membership.
ƒ The timing of first lapse was related to post-lapse
drinking class membership.
Exciting New Extensions
ƒ Intersection with methods for intensive
longitudinal data (e.g., micro-coded
behavior observations of mother-child
interactions)
ƒ Causal effects of event timing on life
course trajectories
Select References
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Fairchild, A., Gottschall, A., Masyn, K., Prinz, R. (Under review). Estimating mediation effect on discrete-time
survival outcomes using a structural equation modeling approach.
Masyn, K. Advances in event history and survival models for prevention science. (In press). In Z. Sloboda & H.
Petras (Eds.), Advances in Prevention Science.
Malone, P. S., Northrup, T. F., Masyn, K. E., Lamis, D. A., & Lamont, A. E. (2012). Initiation and persistence of
alcohol use in United States Black, Hispanic, and White male and female youth. Addictive Behaviors, 37, 299-305.
Petras, H., Masyn, K., Buckley, J., Ialongo, N., Clark, M., & Kellam, S. (2011). Who is most at-risk for school
removal? An application of multilevel discrete-time survival analysis to understand individual and contextual-level
influences. Journal of Educational Psychology, 103(1), 223-237.
Malone, P., Lamis, D., Masyn, K., & Northrup, T. (2010). Modeling of the gateway drug hypothesis: A dual-process
discrete-time survival analysis approach. Multivariate Behavioral Research, 45(5), 790-805.
Masyn, K. (2009). Discrete-time survival factor mixture analysis for low-frequency recurrent event histories.
Research in Human Development, 6(2), 165-194.
Masyn, K. (2008). Modeling measurement error in event occurrence for single, non-recurring events in discretetime survival analysis. In G.R. Hancock, G.R. & K.M. Samuelsen (Eds.). Advances in latent variable mixture models
(pp. 105-145). Charlotte, NC: Information Age Publishing, Inc.
Quartz, K.H., Thomas, A., Anderson, L., Masyn, K., Lyons, K.B., & Olsen, B. (2008). Careers in motion: A
longitudinal retention study of role changing among early career urban educators. Teachers College Record, 10(1),
218-250.
Witkiewitz, K. & Masyn, K.E. (2008). Drinking trajectories following an initial lapse. Psychology of Addictive
Behaviors, 22(2), 157––167.
Asparouhov, T., Masyn, K., & Muthén, B. (2006). Continuous time survival in latent variable models. In ASA
Biometrics Section: Proceedings of the Joint Statistical Meeting, August 6-10 (pp. 180-187). Seattle, WA.
Muthén, B., & Masyn, K. (2005). Discrete-time survival mixture analysis. Journal of Educational and Behavioral
Statistics, 30(1), 27-58.
Thank you -