Multilevel/Mixed Models/A mixed model paradox

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With a clustered sample one could think of estimating the mean of a population by using the overall mean of the sample or by using the mean of cluster means. The former method gives the same weight to each unit but less weight to cluster means of small clusters. The latter method gives more weight to units from small clusters. A third possibility is to use a mixed model estimate that gives each unit a weight that is between what it gets with the two former methods. One might therefore expect that the mixed model estimate will be also lie between the estimates from these methods.

That this is not necessarily the case results from the fact that the weights for the mixed model estimate are not a convex combination of the weights from the two original methods. The mixed model estimate will often lie between the original estimates but may not, particularly if there is a strong relationship between the response level and cluster size. Unless the mixed model weights are an exact convex combination of the two original set of weights -- i.e. each unit's mixed model weight is the same convex linear combination of that units weights for an overall mean and for a mean of cluster means -- there must be a set of cluster means that will produce the paradoxical result that the mixed model estimate will not lie in the interval extending from the overall mean to the mean of means.

I will try to construct a example in which the cluster means are strongly related to cluster size.

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