Up until this point, when several series were available, I have assumed that they arose from the same stochastic process (unless some covariates were available to distinguish among them). The one exception was Section 7.3 in which I introduced frailty models for recurrent events. It is now time to look at some standard methods for handling differences among series when adequate covariate information to describe those differences is not available.

Random effects

Suppose that we observe a number of series having the same basic characteristics. If we examine several responses from one of these series, we can expect them to be related more closely, independently of their distance apart in time, than if we take one response from each of the series. Indeed, generally we assume the responses from separate series to be independent, whereas those on the same series usually are not. If inadequate information is available from covariates in order to be able to model the differences among the series, then other methods must be used. The usual approach is to assume that one or more parameters of the distribution describing the stochastic process differ randomly (because we do not have appropriate covariates to describe systematic change) among the series. This type of construction yields a mixture distribution. However, in contrast to the dynamic models of Chapters 7 and 11, here the random parameters will be static, only varying among series and not over time.