We have already met a number of models which have been described as mixed models, but it may not be clear what unites them. The key distinguishing feature of mixed models compared with fixed effects models is that they are able to model data in which the observations are not independent. To express this more positively, we say that mixed models are able to model the covariance structure of the data.
A simple type of mixed model is the random effects model which was introduced in Sections 1.2 and 1.3. Here, certain effects in the model are assumed to have arisen from a distribution and thus give rise to another source of random variation in addition to the residual variation. These effects are referred to as random effects. For example, when patient effects were fitted in the trial introduced in Section 1.2, random variation occurred both between patients and as residual variation. Any number of random effects can be specified in a model; for example, in a multicentre trial (as in Section 1.3) both centre and centre-treatment effects can be fitted as random, giving rise to two additional sources of variation.
In random coefficients models a covariate effect is allowed to vary randomly. For example, in the repeated measures hypertension data considered in Section 1.4, interest might centre on the rate of change of DBP measured over the four treatment visits in the three arms of the trial. The random coefficients model allows this rate of change (or slope) to vary randomly between patients. This is achieved technically by fitting patients and the patient-slope interaction as random and these effects are referred to as random coefficients.
The covariance pattern model, introduced in Section 1.4, is a third type of mixed model which directly models a pattern of correlations between observations. For example, in repeated measures trials interest is focused on several observations of the response variable made over a period of time and we can allow for the correlations (or, equivalently, covariances) between these observations. Suitable mixed models lead to more appropriate estimates of fixed effects and can investigate the nature of these covariances.
Random effects models, random coefficients models and covariance pattern models form three categories of mixed models. Mixed models can also be defined with combinations of random effects, random coefficient effects and covariance patterns. The choice will depend on the application and the objectives of the analysis.
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