Modelling Longitudinal Data

• Survival Analysis.

• Event History.

• Recurrent Events.

• A Final Point – and link to Multilevel Models (perhaps).

Yi 1 = ’Xi1 + i1

Vector of explanatory variables and estimates

Independent identifiably distributed error

Outcome 1 for individual i

Yi 2 = ’Xi2 + i2

Vector of explanatory variables and estimates

Independent identifiably distributed error

Outcome 2 for individual i

THE SAME AGAIN AT TIME 2

Yi 1 = ’Xi1 + i1 Yi 2 = ’Xi2 + i2

Considered together conventional regression analysis in NOT appropriate

Yi 2 - Yi 1 = ’(Xi2-Xi1) + (i2 - i1)

Change in Score

Here the ’ is simply a regression on the difference or change in scores.

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

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.

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.

Consider a binary outcome or two-state event

0 = Event has not occurred

1 = Event has occurred

Start of Study End of Study

0 1

0

0

1

1

t1 t2 t3

These durations are a continuous Y so why can’t we use standard

regression techniques?

Start of Study End of Study

0 1

0

0

1

1

1

0

CENSORED OBSERVATIONS

0

Start of Study End of Study

1

B

CENSORED OBSERVATIONS

A

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

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.

Start of Study End of Study

0 1

0

0

1

1

1

0

CENSORED OBSERVATIONS

0

UNEMPLOYMENT AND RETURNING TO WORK STUDY

0 = Unemployed; 1 = Returned to work

Y variable =

duration with censored observations

X1

X3

X2

A Statistical Model

Y variable =

duration with censored observations

Previous Occupation

Educational Qualifications

A Statistical Model

Length of Work experience

A continuous covariate

More complex event history analysis

Start of Study

End of Study

0

t1 t2 t3

0 = Unemployed; 1 = Returned to work

1 1

UNEMPLOYMENT AND RETURNING TO WORK STUDY

0

Start of Study

End of Study

0

t1

0 = Unemployed; 1 = Returned to work

UNEMPLOYMENT AND RETURNING TO WORK STUDY

Spell or Episode

Start of Study

End of Study

0

t1

0 = Unemployed; 1 = Returned to work

1

UNEMPLOYMENT AND RETURNING TO WORK STUDY

Transition = movement from one state to another

Recurrent Events Analysis

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.

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).

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.

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).

STATISTICAL ANALYSIS FOR BINARY RECURRENT

EVENTS (SABRE)

• Fits appropriate models for recurrent events.

• It is like GLIM.

• It can be downloaded free.

www.cas.lancs.ac.uk/software

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.

0 = Unemployed; 1 = Returned to work

UNEMPLOYMENT AND

RETURNING TO WORK STUDY –

A study for six months

Months1 2 3 4 5 6

obs 0 0 0 0 0 0

Constantly unemployed

Months1 2 3 4 5 6

obs 1 1 1 1 1 1

Constantly employed

Months1 2 3 4 5 6

obs 1 0 0 0 0 0

Employed in month 1 then unemployed

Months1 2 3 4 5 6

obs 0 0 0 0 0 1

Unemployed but gets a job in month six

Months1 2 3 4 5 6

obs 0 1 0 1 1 0obs 0 0 1 0 1 1obs 0 1 1 0 0 1obs 1 0 0 0 1 0

Mixed employment patterns

Here we have a binary outcome – so could we simply use logistic regression to model it?

Months1 2 3 4 5 6

obs 0 0 0 0 0 0

Yes and No!

SABRE fits two models that are appropriate to this analysis.

Model 1 = Pooled Cross-Sectional Logit Model

)'exp(1

)]'[exp()(

it

it

x

yxL

it

itB

POOLED CROSS-SECTIONAL

LOGIT MODEL

x it is a vector of explanatory variables and is a vector of

parameter estimates .

)'exp(1

)]'[exp()(

it

it

x

yxL

it

itB

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.

This approach can be regarded as a naïve solution to our data analysis problem.

We need to consider a number of issues….

MonthsY1 Y2

obs 0 0

Pickle’s 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.

SABRE fits two models that are appropriate to this analysis.

Model 2 = Random Effects Model

(or logistic mixture model)

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.

df(

)'exp(1

)]'[exp()(

1

it

iti

x

yxL

itT

t

itB

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.

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

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’.

Months1 2 3 4 5 6

obs 0 0 0 0 0 0

This person is a stayer!

Months1 2 3 4 5 6

obs 1 1 1 1 1 1

So is this person.

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.

Recurrent events may be analysed using other software but SABRE is specifically designed to handle stayers and this feature increases SABRE’s flexibility in representing residual heterogeneity (Barry, Francis, Davies, and Stott 1998).

Barry, J., Francis, B., Davies, R.B. and Stott,D. (1998) SABRE Users Guidehttp://www.cas.lancs.ac.uk/software/sabre3.1/sabreuse.html

Past BehaviourCurrent

Behaviour

STATE DEPENDENCE

UnemployedEmployed

Employed

Young People Aged 19

MAYAPRIL

Different Probabilities of Employment

This is called a MARKOV model

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

1tyγ'β -itx

ACCOUNTS FOR PREVIOUS

OUTCOME (yt-1)

UnemployedExplanatory

Variables Employed

EmployedExplanatory

Variables

The Model Provides TWO sets of estimates

MAY

APRIL

This is a ‘two-state’ MARKOV model

But we can make it more complicated.

MonthsY1 Y2

obs 0 0

First Order Markov Model

MonthsY1 Y2 Y3

obs 0 0 0

Second Order Markov Model

FINAL POINT – A THOUGHT!

Months1 2 3 4 5 6

obs 0 1 0 1 1 0obs 0 0 1 0 1 1obs 0 1 1 0 0 1obs 1 0 0 0 1 0

Mixed employment patterns

a b c d e f

1 2 3 4 1 2 1 2 3 1 2 3 1 2 1 2 3 1 2

g

Observations Months

Individuals

Hierarchical or Multilevel Data Structure

Is the recurrent events model simply a multilevel model fitted

at the single level?

A controversial point!

More later…..