More complex event history analysis

Start of Study End of Study 0 t1 0 = Unemployed; 1 = Working UNEMPLOYMENT AND RETURNING TO WORK STUDY Spell or Episode

Start of Study End of Study 0 t1 t2 t3 0 = Unemployed; 1 = Working 11 UNEMPLOYMENT AND RETURNING TO WORK STUDY 0

Start of Study End of Study 0 t1 0 = Unemployed; 1 = Working 1 UNEMPLOYMENT AND RETURNING TO WORK STUDY Transition = movement from one state to another

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.

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 = Working UNEMPLOYMENT AND RETURNING TO WORK STUDY – A study for six months

Months obs Constantly unemployed

Months obs Constantly employed

Months obs Employed in month 1 then unemployed

Months obs Unemployed but gets a job in month six

Here we have a binary outcome – so could we simply use logistic regression to model it? Yes and No – We need to think about this issue.

Appropriate Software STATISTICAL ANALYSIS FOR BINARY RECURRENT EVENTS (SABRE) Fits appropriate models for recurrent events. It is like GLIM. It can be downloaded free.

SABRE fits two models that are appropriate to this analysis. Model 1 = Pooled Cross-Sectional Logit Model Think of this as being the same as a logistic regression in any software package.

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

We could fit a pooled cross- sectional model to our recurrent events data. This approach can be regarded as a naïve solution to our data analysis problem.

We need to consider a number of issues….

Months Y 1 Y 2 obs00 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 Y 1 and Y 2 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.

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 i th 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

An example – see Davies, Elias & Penn (1992). A study of wive’s employment status. Y (femp) 0 = wife unemployed 1 = wife employed X 1 (fmune)0 = husband employed 1 = husband unemployed X 2 (fund1) 0 = no child under 1 year 1 = child under 1 year

Results of various models ModelX VarsDevianced.f. Pooled Pooledfmune Pooledfmune + fund Random effects fmune + fund

Deviance = on 1576 residual degrees of freedom dis e Parameter Estimate S. Error ___________________________________________________ int fmune ( 1) E+00 ALIASED [I] fmune ( 2) fund1 ( 1) E+00 ALIASED [I] fund1 ( 2) scale Random effect

Past Behaviour Current Behaviour STATE DEPENDENCE

Unemployed Employed MAY APRIL STATE DEPENDENCE

Months Y 1 Y 2 obs00 Lag Model

ACCOUNTS FOR PREVIOUS OUTCOME ( y t - 1 )

This is called a Lagged model A Lagged model helps to control for a previous outcome (or behaviour).

ModelX VarsDevianced.f. Random effects fmune + fund Drop yfmune + fund Lagfmune + fund Results of models – with state dependence

Deviance = on 1420 residual degrees of freedom Deviance decrease = on 1 residual degree of freedom dis e Parameter Estimate S. Error ___________________________________________________ int fmune ( 1) E+00 ALIASED [I] fmune ( 2) fund1 ( 1) E+00 ALIASED [I] fund1 ( 2) lag scale

State dependence can be explored further by the estimation of a a ‘two-state’ MARKOV model.

Unemployed Explanatory Variables Employed Explanatory Variables The Model Provides TWO sets of estimates MAY APRIL

Results of models – with state dependence ModelX VarsDevianced.f. Drop yfmune + fund Lagfmune + fund Markovfmune + fund

Parameter Estimate S. Error ___________________________________________________ Unemployed Women at t-1 _______ int fmune ( 1) E+00 ALIASED [I] fmune ( 2) fund1 ( 1) E+00 ALIASED [I] fund1 ( 2) scale Employed Women at t-1 _______ int fmune ( 1) E+00 ALIASED [I] fmune ( 2) fund1 ( 1) E+00 ALIASED [I] fund1 ( 2) scale E

SABRE – Good Points Fits appropriate models for recurrent events. It is like GLIM. It can be downloaded free. There is a users list. Uses the deviance to compare models (correct likelihood). Fits the Markov model. Fits a range of other models (e.g. loglinear + ordinal). Can do more advance analysis (e.g. Mover/Stayers).

SABRE – Bad Points It is like GLIM – you need to understand a prog. Syntax. Data management and handling are poor. There are few users.

Alternatives to SABRE STATA – Does not fit the full range of models. Multilevel model software – Okay up to a point but check that the likelihood is correct (complicated). No software other than SABRE fits a continuation ratio model (ordinal), Markov model or the mover/stayer.