The random effects are also assumed to be fixed effects in the observed response, which is expressed as a fixed effect of the latent variable. In effect, the fixed effects arise as the mean of the latent variable.
In RE models, the full model is specified with individual-level random effects, and standard deviations can be estimated. In FE models, the full model is specified with individual-level fixed effects, and as standard approaches, variances are estimated. So, in the absence of RE, FE and RE models are identical.
As in social sciences, many applied researchers prefer to use FE models, because of their simplicity and computational speed, and so researchers have tried to extend FE models to allow for random effects. This has led to the development of mixed-effects models with random effects. We next consider the theoretical underpinnings of random-effects models.
A key assumption is that the individual-level random effects follow a Normal distribution with constant variance. However, such models are not compatible with a latent variable framework, and cannot be expressed as a GLS model. We consider this in more detail next.
The traditional approach of random effects is based on Gaussianity of latent (unobserved) individual-level effects, and a simple model is used for the observed responses. This is illustrated in Box 2.1, which shows the theoretical underpinnings of the two approaches.
A latent variable framework is illustrated in Box 2.2. Individual-level random effects are represented by random variables, the values of which are unobserved, and it is assumed that these values are distributed as a Normal or specific Normal distribution, with constant variance. Additional assumptions are that the random effects are independent across individuals and measurements, and that they have zero correlation.
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