Modelling negative correlation

Usually, a negative variance component is an underestimate of a small or zero variance component. However, occasionally it can indicate negative correlation between observations within the same random effects category. This is an unlikely scenario in most clinical trials; for example, it would be hard to imagine a situation where observations taken on the same patient could be more variable than observations taken on different patients. However, in the following veterinary example, negative correlation is more feasible. Imagine an animal feeding experiment where animals are grouped in cages. Here, it is possible that the greediest animals in a cage eat more than other animals, from a finite food supply, causing animal weight to become more variable within cages than between cages. In a model fitting cage effects as random this would lead to a negative variance component for cage effects, indicating negative correlation between animal weights in the same cage.

To model this negative correlation the model can be redefined as a covariance pattern model. Here the random effects (e.g. cages) are omitted, but correlation within the random effects is modelled by including covariance parameters in the residual variance matrix, R. Thus negative as well as positive correlation is allowed within the random effects (cages). We illustrate this model redefinition usingthe multi-centredatausedtodescribe the mixed model in Section 2.1. Recall that a random effects model was specified with centre effects fitted as random and that the variance matrix, V, was given by V = ZGZ' + R

 + a 2 a2 a2 ac2 0 0 0 0 0 a2 a2 + a2 a2 ac2 0 0 0 0 0 a2 a2 a2 + a 2 ac2 0 0 0 0 0 a2 a2 ac2 a2 + a 2 0 0 0 0 0 0 0 0 0 a2 + a 2 ac2 0 0 0 0 0 0 0 ac2 a2 + a2 0 0 0 0 0 0 0 0 0 a2 + a 2 ac2 a2 0 0 0 0 0 0 ac2 a2 + a 2 a2 0 0 0 0 0 0 ac2 ac2 a2 + a

When the model is redefined as a covariance pattern model, centre effects are excluded from the model but covariance is allowed in the R matrix between observations at the same centre. A constant covariance can be obtained by using a compound symmetry covariance pattern which gives the V matrix as

 a2 pa2 pa2 pa2 0 0 0 0 0 ^ pa2 a2 pa2 pa2 0 0 0 0 0 pa2 pa2 a2 pa2 0 0 0 0 0 pa2 pa2 pa2 a2 0 0 0 0 0 0 0 0 0 a2 pa2 0 0 0 0 0 0 0 pa2 a2 0 0 0 0 0 0 0 0 0 a2 pa2 pa2 0 0 0 0 0 0 pa2 a2 pa2 0 0 0 0 0 0 pa2 pa2 a2

where p = the correlation between patients at the same centre.

Thus, V has an identical form to the random effects model except that it is parameterised differently. A negative covariance of observations at the same centre, pa2, is now permissible.