Modelling Longitudinal Data

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Modelling Longitudinal Data

• Survival Analysis.
• Event History.
• Recurrent Events.
• A Final Point and link to Multilevel Models
  (perhaps).

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Vector of explanatory variables and estimates
Yi 1 bXi1 ei1
Outcome 1 for individual i
Independent identifiably distributed error
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THE SAME AGAIN AT TIME 2
Vector of explanatory variables and estimates
Yi 2 bXi2 ei2
Outcome 2 for individual i
Independent identifiably distributed error
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Considered together conventional regression
analysis in NOT appropriate
Yi 1 bXi1 ei1
Yi 2 bXi2 ei2
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Change in Score
Yi 2 - Yi 1 b(Xi2-Xi1) (ei2 - ei1)
Here the b is simply a regression on the
difference or change in scores.
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As social scientists we are often substantively
interested in whether a specific event has
occurred.
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Survival Data Time to an event

• In the medical area
• Duration from treatment to death.
• Time to return of pain after taking a pain
  killer.

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Survival Data Time to an event

• Social Sciences
• Duration of unemployment.
• Duration of time on a training scheme.
• Duration of housing tenure.
• Duration of marriage.
• Time to conception.

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Consider a binary outcome or two-state event

• 0 Event has not occurred
• 1 Event has occurred

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0
1
1
0
1
0
t1 t2 t3
End of Study
Start of Study
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These durations are a continuous Y so why cant
we use standard regression techniques?
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0
1
0
0
1
CENSORED OBSERVATIONS
1
0
1
0
Start of Study
End of Study
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A
1
B
CENSORED OBSERVATIONS
Start of Study
End of Study
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These durations are a continuous Y so why cant
we use standard regression techniques?

• What should be the value of Y for person A and
  person B at the end of our study (when we fit the
  model)?

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Cox Regression

• is a method for modelling time-to-event data in
  the presence of censored cases.
• Explanatory variables in your model (continuous
  and categorical).
• Estimated coefficients for each of the
  covariates.
• Handles the censored cases correctly.

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UNEMPLOYMENT AND RETURNING TO WORK STUDY
0 Unemployed 1 Returned to work
0
1
0
0
1
CENSORED OBSERVATIONS
1
0
1
0
Start of Study
End of Study
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A Statistical Model
Y variable duration with censored observations
X1
X2
X3
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A Statistical Model
Previous Occupation
Y variable duration with censored observations
Educational Qualifications
Length of Work experience
A continuous covariate
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More complex event history analysis
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UNEMPLOYMENT AND RETURNING TO WORK STUDY
0 Unemployed 1 Returned to work
1
1
0
0
t1 t2 t3
End of Study
Start of Study
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UNEMPLOYMENT AND RETURNING TO WORK STUDY
0 Unemployed 1 Returned to work
Spell or Episode
0
t1
End of Study
Start of Study
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UNEMPLOYMENT AND RETURNING TO WORK STUDY
0 Unemployed 1 Returned to work
Transition movement from one state to another
1
0
t1
End of Study
Start of Study
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Recurrent Events Analysis
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The structure of many large-scale studies results
in survey data being collected at a number of
discrete occasions. In this situation, rather
than being continuous, time lends itself to be
conceptualized as a sequence of discrete events.
Furthermore, social scientists are often
substantively interested in whether a specific
event has occurred. Taken together, these two
issues appeal to the adoption of a discrete-time
or event history approach.
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Recurrent events are merely outcomes that can
take place on a number of occasions. A simple
example is unemployment measured month by month.
In any given month an individual can either be
employed or unemployed. If we had data for a
calendar year we would have twelve discrete
outcome measures (i.e. one for each month).
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Social scientists now routinely employ
statistical models for the analysis of discrete
data, most notably logistic and log-linear
models, in a wide variety of substantive areas. I
believe that the adoption of a recurrent events
approach is appealing because it is a logical
extension of these models.
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Willet and Singer (1995) conclude that
discrete-time methods are generally considered to
be simpler and more comprehensible, however,
mastery of discrete-time methods facilitates a
transition to continuous-time approaches should
that be required.Willet, J. and Singer, J.
(1995) Investigating Onset, Cessation, Relapse,
and Recovery Using Discrete-Time Survival
Analysis to Examine the Occurrence and Timing of
Critical Events. In J. Gottman (ed) The Analysis
of Change (Hove Lawrence Erlbaum Associates).
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STATISTICAL ANALYSIS FOR BINARY RECURRENT EVENTS
(SABRE)

• Fits appropriate models for recurrent events.
• It is like GLIM.
• It can be downloaded free.

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www.cas.lancs.ac.uk/software
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Consider a binary outcome or two-state event

• 0 Event has not occurred
• 1 Event has occurred
• In the cross-sectional situation we are used to
  modelling this with logistic regression.

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UNEMPLOYMENT AND RETURNING TO WORK STUDY A
study for six months
0 Unemployed 1 Returned to work
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Months 1 2 3 4 5 6obs 0 0 0 0 0 0Constantly
unemployed
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Months 1 2 3 4 5 6obs 1 1 1 1 1 1Constantly
employed
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Months 1 2 3 4 5 6obs 1 0 0 0 0 0Employed
in month 1 then unemployed
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Months 1 2 3 4 5 6obs 0 0 0 0 0 1Unemployed
but gets a job in month six
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Months 1 2 3 4 5 6obs 0 1 0 1 1 0obs 0 0 1 0
1 1obs 0 1 1 0 0 1obs 1 0 0 0 1 0Mixed
employment patterns
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Here we have a binary outcome so could we
simply use logistic regression to model
it?Months 1 2 3 4 5 6obs 0 0 0 0 0 0
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Yes and No!
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SABRE fits two models that are appropriate to
this analysis.

• Model 1 Pooled Cross-Sectional Logit Model

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POOLED CROSS-SECTIONAL LOGIT MODEL

x it is a vector of explanatory variables and b
is a vector of parameter estimates .
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POOLED CROSS-SECTIONAL LOGIT MODEL

In conventional logistic regression models, where
each observation is assumed to be independent, a
logistic link function is used, the contribution
to the likelihood by the ith case and the t th
event is given by the equation above.
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This approach can be regarded as a naïve solution
to our data analysis problem.
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We need to consider a number of issues.
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Months Y1 Y2 obs 0 0
Pickles tip - In repeated measured analysis we
would require something like a paired t test
rather than an independent t test because we
can assume that Y1 and Y2 are related.
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SABRE fits two models that are appropriate to
this analysis.

• Model 2 Random Effects Model
• (or logistic mixture model)

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• Repeated measures data violate an important
  assumption of conventional regression models.
• The responses of an individual at different
  points in time will not be independent of each
  other.
• This problem has been overcome by the inclusion
  of an additional, individual-specific error term.

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• The random effects model extends the pooled
  cross-sectional model to include a case-specific
  random error term to account for residual
  heterogeneity.
• For a sequence of outcomes for the ith case, the
  basic random effects model has the integrated (or
  marginal likelihood) given by the equation.

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Davies and Pickles (1985) have demonstrated that
the failure to explicitly model the effects of
residual heterogeneity may cause severe bias in
parameter estimates. Using longitudinal data the
effects of omitted explanatory variables can be
overtly accounted for within the statistical
model. This greatly improves the accuracy of the
estimated effects of the explanatory variables
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Movers and Stayers

• When considering data on recurrent events there
  will be individuals for whom there will be zero
  (or very low) probabilities of change in outcome
  from one event to the next. These individuals are
  termed as stayers.

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Months 1 2 3 4 5 6obs 0 0 0 0 0 0This
person is a stayer!
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Months 1 2 3 4 5 6obs 1 1 1 1 1 1So is this
person.
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• An awareness of the issue of stayers is
  important for technical reasons. A limitation of
  a parametric modelling approach is that the tail
  behaviour of the normal distribution is
  inconsistent with stayers and they will tend to
  be underestimated (see Spilerman 1972).
• Spilerman, S. (1972) Extensions of the
  Mover-Stayer Model, American Journal of
  Sociology, 78, pp.599-626.

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• Recurrent events may be analysed using other
  software but SABRE is specifically designed to
  handle stayers and this feature increases SABREs
  flexibility in representing residual
  heterogeneity (Barry, Francis, Davies, and Stott
  1998).
• Barry, J., Francis, B., Davies, R.B. and Stott,D.
  (1998) SABRE Users Guide
• http//www.cas.lancs.ac.uk/software/sabre3.1/sabre
  use.html

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STATE DEPENDENCE
Past Behaviour
Current Behaviour
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Different Probabilities of Employment
Young People Aged 19
APRIL
MAY
Employed
Unemployed
Employed
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This is called a MARKOV model

• A Markov model helps to control for a previous
  outcome (or behaviour).

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ACCOUNTS FOR PREVIOUS OUTCOME (yt-1)
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The Model Provides TWO sets of estimates
APRIL
MAY
Unemployed Explanatory Variables
Employed
Employed Explanatory Variables
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This is a two-state MARKOV model

• But we can make it more complicated.

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First Order Markov Model
Months Y1 Y2 obs 0 0
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Second Order Markov Model
Months Y1 Y2 Y3 obs 0 0 0
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FINAL POINT A THOUGHT!
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Months 1 2 3 4 5 6obs 0 1 0 1 1 0obs 0 0 1 0
1 1obs 0 1 1 0 0 1obs 1 0 0 0 1 0Mixed
employment patterns
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Hierarchical or Multilevel Data Structure
Individuals
f
g
a
b
c
d
e
1 2 3 4 1 2 1 2 3 1 2 3 1
2 1 2 3 1 2
Observations Months
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Is the recurrent events model simply a multilevel
model fitted at the single level?

• A controversial point!
• More later..