Chapter 9Multilevel event history models9.1 Event history modelsThis class of models, also known as survival time models or event duration models, have as the response variable the length of time between events. Such events may be, for example, birth and death, or the beginning and end of a period of employment with corresponding times being length of life or duration of employment. There is a considerable theoretical and applied literature, especially in the field of biostatistics and a useful summary is given by Clayton (1988). We consider two basic approaches to the modelling of duration data. The first is based upon proportional hazard models. The second is based upon direct modelling of the log duration, often known as accelerated life models. In both cases we may wish to include explanatory variables. The multilevel structure of such models arises in two general ways. The first is where we have repeated durations within individuals, analogous to our repeated measures models of chapter 5. Thus, individuals may have repeated spells of various kinds of employment of which unemployment is one. In this case we have a 2-level model with individuals at level 2, often referred to as a renewal process. We can include explanatory dummy variables to distinguish these different kinds of employment or states. The second kind of model is where we have a single duration for each individual, but the individuals are grouped into level 2 units. In the case of employment duration the level 2 units would be firms or employers. If we had repeated measures on individuals within firms then this would give rise to a 3-level structure.9.2 CensoringA characteristic of duration data is that for some observations we may not know the exact duration but only that it occurred within a certain interval, known as interval censored data, was less than a known value, left censored data, or greater than a known value, right censored data. For example, if we know at the time of a study, that someone entered her present employment before a certain date then the information available is only that the duration is longer than a known value. Such data are known as right censored. In another case we may know that someone entered and then left employment between two measurement occasions, in which case we know only that the duration lies in a known interval. The models described in this chapter have procedures for dealing with censoring In the case of the parametric models, where there are relatively large proportions of censored data the assumed form of the distribution of duration lengths is important, whereas in the partially parametric models the distributional form is ignored. It is assumed that the censoring mechanism is non informative, that is independent of the duration lengths.In some cases, we may have data which are censored but where we have no duration information at all. For example, if we are studying the duration of first marriage and we end the study when individuals reach the age of 30, all those marrying for the first time after this age will be excluded. To avoid bias we must therefore ensure that age of marriage is an explanatory variable in the model and report results conditional on age of marriage.There is a variety of models for duration times. In this chapter we show how some of the more frequently used models can be extended to handle multilevel data structures. We consider first hazard based models.9.3 Hazard based models in continuous timeThe underlying notions are those of survivor and hazard functions. Consider the (single level) case where we have measures of length of employment on workers in a firm. We define the proportion of the workforce employed for periods greater than t as the survivor function and denote it by where is the density function of length of employment. The hazard function is defined as and represents the instantaneous risk, in effect the (conditional) probability of someone who is employed at time t, ending employment in the next (small) unit interval of time. The simplest model is one which specifies an exponential distribution for the duration time, which gives , so that the hazard rate is constant and . In general, however, the hazard rate will change over time and a number of alternative forms have been studied (see for example, Cox and Oakes, 1984). A common one is based on the assumption of a Weibull distribution, namely or the associated extreme value distribution formed by replacing by . Another approach to incorporating time-varying hazards is to divide the time scale into a number of discrete intervals within which the hazard rate is assumed constant, that is we assume a piecewise exponential distribution. This may be useful where there are natural units of time, for example based on menstrual cycles in the analysis of fertility, and this can be extended by classifying units by other factors where time varies over ca