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Modelling Longitudinal Data. General Points Single Event histories (survival analysis) Multiple Event histories. Motivation. Attempt to go beyond more simple material in the first workshop. Begin to develop an appreciation of the notation associated with these techniques.

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modelling longitudinal data
Modelling Longitudinal Data
  • General Points
  • Single Event histories (survival analysis)
  • Multiple Event histories
motivation
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.
statistical modelling framework generalized linear models

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.

statistical modelling
Statistical Modelling
  • Know your data.
  • Start and be guided by ‘substantive theory’.
  • Start with simple techniques (these might suffice).
  • Remember John Tukey!
  • Practice.
slide5

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).

as social scientists we are often substantively interested in whether a specific event has occurred

As social scientists we are often substantively interested in whether a specific event has occurred.

survival data time to an event
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.
survival data time to an event8
Survival Data – Time to an event

Social Sciences…

  • Duration of unemployment.
  • Duration of housing tenure.
  • Duration of marriage.
  • Time to conception.
  • Time to orgasm.
consider a binary outcome or two state event

Consider a binary outcome or two-state event

0 = Event has not occurred

1 = Event has occurred

slide10

A

0

1

B

1

0

C

1

0

t1 t2 t3

End of Study

Start of Study

these durations are a continuous y so why can t we use standard regression techniques12

These durations are a continuous Y so why can’t 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’.

slide13

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

1=Death

slide14

Breast Feeding Study –Data Collection Strategy1. Retrospective questioning of mothers2. Data collected by Midwives 3. Health Visitor and G.P. Record

slide15

Breast Feeding Study –

2001

Age 6

Start of Study

Birth

1995

slide16

Breast Feeding Study –

2001

0

1

Age 6

1

0

1

0

t1 t2 t3

Start of Study

Birth

1995

accelerated life model

Accelerated Life Model

Loge ti = b0 + b1x1i+ei

accelerated life model18

Accelerated Life Model

error term

Beware this is log t

explanatory variable

Loge ti = b0+ b1x1i+ei

constant

slide19

At this point something should dawn on you – like fish scales falling from your eyes – like pennies from Heaven.

b 0 b 1 x 1 i e i is the r h s
b0 + b1x1i+ei 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”).

slide21

0

1

0

0

1

CENSORED OBSERVATIONS

1

0

1

0

Start of Study

End of Study

slide22

A

1

B

CENSORED OBSERVATIONS

Start of Study

End of Study

these durations are a continuous y so why can t we use standard regression techniques23

These durations are a continuous Y so why can’t 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)?

cox regression proportional hazard model

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.

childcare study studying a cohort of women who returned to work after having their first child
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)
variables
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)
describes the decline in the size of the risk set over time

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

Survival Function

(or survival curve)

s t 1 f t prob t t

S(t) = 1 – F(t) = Prob (T>t)

survival probability

Cumulative probability

event

Survival Function

time

complement

s t 1 s t 2 for all t 2 t 1

S(t1) S(t2) for all t2 > t1

All this means is… once you’ve left the risk set you can’t return!!!

Survival Function

hazard

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.

for the very keen

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

for the even more keen

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)

slide40

A Statistical Model

Y variable = duration with censored observations

X1

X2

X3

slide41

A Statistical Model

Family income

Y variable = duration with censored observations

Gender of baby

Type of childcare

Mother’s age

A continuous covariate

cox proportional hazard model

Cox Proportional Hazard Model

exponential

X var

estimate

h(t)=h0(t)exp(bx)

hazard

baseline hazard(unknown)

for the very keen cox proportional hazard model can be transformed into an additive model

For the very keen..Cox Proportional Hazard Model can be transformed into an additive model

log h(t)=a(t) + bx

Therefore…

for the very keen cox proportional hazard model

For the very keen..Cox Proportional Hazard Model

log h(t)=b0(t) + b1 x1

This should look distressingly familiar!

slide47

Define the code for the event

(i.e. 1 if occurred – 0 if censored)

slide48

Enter explanatory variables

(dummies and continuous)

slide49

Chi-square related

X var

Standard error

Estimate

Un-logged estimate

what does this mean

What does this mean?

Our Y the duration of childcare spell #2.

Note we are modelling the hazard!

significant variables
Family income p<.001

Gender baby p=.696

Mother’s age p=.262

Childminder p<.001

Nursery p<.001

Significant Variables
effects on the hazard
Effects on the hazard
  • Family income p<.001

£30K +

Up to £30K

Childminder p<.001

Nursery p<.001