Modelling Longitudinal Data

1
Modelling Longitudinal Data

• General Points
• Single Event histories (survival analysis)
• Multiple Event histories

2
Motivation

• Attempt to go beyond more simple material in the
  first workshop.
• Begin to develop an appreciation of the notation
  associated with these techniques.
• Gain a little hands-on experience.

3
Statistical Modelling FrameworkGeneralized
Linear Models

• An interest in generalized linear models is
  richly rewarded. Not only does it bring together
  a wealth of interesting theoretical problems but
  it also encourages an ease of data analysis sadly
  lacking from traditional statistics.an added
  bonus of the glm approach is the insight provided
  by embedding a problem in a wider context. This
  in itself encourages a more critical approach to
  data analysis.
• Gilchrist, R. (1985) Introduction GLIM and
  Generalized Linear Models, Springer Verlag
  Lecture Notes in Statistics, 32, pp.1-5.

4
Statistical Modelling

• Know your data.
• Start and be guided by substantive theory.
• Start with simple techniques (these might
  suffice).
• Remember John Tukey!
• Practice.

5
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).
6
As social scientists we are often substantively
interested in whether a specific event has
occurred.
7
Survival Data Time to an event

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

8
Survival Data Time to an event

• Social Sciences
• Duration of unemployment.
• Duration of housing tenure.
• Duration of marriage.
• Time to conception.
• Time to orgasm.

9
Consider a binary outcome or two-state event

• 0 Event has not occurred
• 1 Event has occurred

10
A
0
1
B
1
0
C
1
0
t1 t2 t3
End of Study
Start of Study
11
These durations are a continuous Y so why cant
we use standard regression techniques?
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These durations are a continuous Y so why cant
we use standard regression techniques?

• We can. It might be better to model the log of Y
  however. These models are sometimes known as
  accelerated life models.

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1946 Birth Cohort Study
Research Project 2060 (1st August 2032 VG
retires!)
1946
A
0
1
B
1
0
C
1
0
t1 t2 t3 t4
Start of Study
1Death
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Breast Feeding Study Data Collection
Strategy1. Retrospective questioning of
mothers2. Data collected by Midwives 3.
Health Visitor and G.P. Record
15
Breast Feeding Study
2001
Age 6
Start of Study
Birth 1995
16
Breast Feeding Study
2001
0
1
Age 6
1
0
1
0
t1 t2 t3
Start of Study
Birth 1995
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Accelerated Life Model

• Loge ti b0 b1x1iei

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Accelerated Life Model

• Loge ti b0 b1x1iei

error term
Beware this is log t
explanatory variable
constant
19
At this point something should dawn on you like
fish scales falling from your eyes like pennies
from Heaven.
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b0 b1x1iei is the r.h.s.

• Think about the l.h.s.
• Yi - Standard liner model
• Loge (odds) Yi - Standard logistic model
• Loge ti - Accelerated life model
• We can think of these as a single class of
  models and (with a little care) can interpret
  them in a similar fashion (as Ian Diamond of the
  ESRC would say this is phenomenally groovy).

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0
1
0
0
1
CENSORED OBSERVATIONS
1
0
1
0
Start of Study
End of Study
22
A
1
B
CENSORED OBSERVATIONS
Start of Study
End of Study
23
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(proportional hazard model)

• 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.

25
Cox, D.R. (1972) Regression models and life
tables JRSS,B, 34 pp.187-220.
26
Childcare Study Studying a cohort of women who
returned to work after having their first child.

• 24 month study
• The focus of the study was childcare spell 2
• 341 Mothers (and babies)

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Variables

• ID
• Start of childcare spell 2 (month)
• End of childcare spell 2 (month)
• Gender of baby (male female)
• Type of care spell 2 (a relative childminder
  nursery)
• Family income (crude measure)

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Describes the decline in the size of the risk
set over time.

• Survival Function
• (or survival curve)

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S(t) 1 F(t) Prob (Tgtt) alsoS(t1)
S(t2) for all t2 gt t1

• Survival Function

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S(t) 1 F(t) Prob (Tgtt)

• Survival Function

survival probability
Cumulative probability
event
time
complement
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S(t1) S(t2) for all t2 gt t1

• Survival Function

All this means is once youve left the risk set
you cant return!!!
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Median Survival Times
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Too hard to interpret except for the Rain Man
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HAZARD

• In advanced analyses researchers sometimes
  examine the shape of something called the hazard.
  In essence the shape of this is not constrained
  like the survival function. Therefore it can
  potentially tell us something about the social
  process that is taking place.

37
For the very keen

• Hazard
• the rate at which events occur
• Or
• the risk of an event occurring at a particular
  time, given that it has not happened before t

38
For the even more keen

• Hazard
• The conditional probability of an event occurring
  at time t given that it has not happened before.
  If we call the hazard function h(t) and the pdf
  for the duration f(t)
• Then,
• h(t) f(t)/S(t)

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A Statistical Model
Y variable duration with censored observations
X1
X2
X3
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A Statistical Model
Family income
Y variable duration with censored observations
Gender of baby
Type of childcare
Mothers age
A continuous covariate
42
For the keen..Cox Proportional Hazard Model

• h(t)h0(t)exp(bx)

43
Cox Proportional Hazard Model
exponential

• h(t)h0(t)exp(bx)

X var
estimate
hazard
baseline hazard(unknown)
44
For the very keen..Cox Proportional Hazard
Model can be transformed into an additive model

• log h(t)a(t) bx
• Therefore

45
For the very keen..Cox Proportional Hazard
Model

• log h(t)b0(t) b1 x1
• This should look distressingly familiar!

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Define the code for the event (i.e. 1 if
occurred 0 if censored)
48
Enter explanatory variables (dummies and
continuous)
49
Chi-square related
X var
Standard error
Estimate
Un-logged estimate
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What does this mean?

• Our Y the duration of childcare spell 2.
• Note we are modelling the hazard!

51
Significant Variables

• Family income plt.001
• Gender baby p.696
• Mothers age p.262
• Childminder plt.001
• Nursery plt.001

52
Effects on the hazard

• Family income plt.001
• 30K
• Up to 30K
• Childminder plt.001
• Nursery plt.001