The dispersion parameter was fixed at one in Models 1-3. However, its estimated value is also given in brackets in Table 3.3. In Model 1 it is very close to one and fixing it at one has made very little difference to the fixed effects standard errors. In Models 2 and 3 the dispersion parameter is well below one. This is largely due to the influence of the uniform categories and thus the dispersion parameter estimate can be considered as another indicator of their presence.
The dispersion parameters in Model 4 and 5 are both notably below one due to the presence of uniform centre effect categories. These parameters help to overcome the discrepancy in the mean/variance relationship caused by the random effects estimates being shrunken compared with their raw means. If dispersion parameters were omitted, it is likely that a downward bias in the centre and centre-treatment variance components would have occurred. However, it is difficult to tell how adequately the dispersion parameter has overcome this potential problem. Our own view is that the results are likely to be satisfactory. We can draw some comfort from the fact that the results from Model 4 are similar to those from Model 1, which does not suffer from the problems associated with uniform effects categories.
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