Crossclassified And Multiple Membership Structures

Across a wide range of disciplines it is commonly the case that data have a structure that is not purely hierarchical. Individuals may be clustered not only into hierarchically ordered units (for example occasions nested within patients nested within clinics), but may also belong to more than one type of unit at a given level of a hierarchy. Consider the example of a livestock animal such as a cow where there are a large number of mothers, each producing several female offspring that are eventually reared for milk on different farms. Thus, an offspring might be classified as belonging to a particular combination of mother and farm, in which case they will be identified by a cross-classification of these.

Raudenbush [26] and Rasbash and Goldstein [27] present the general structure of a model for handling complex hierarchical structuring with random cross-classifications. For example, assuming that we wish to formulate a linear model for the milk yield of offspring taking into account both the mother and the farm, then we have a cross-classified structure, which can be modelled as follows:

in which the yield of offspring i, belonging to the combination of mother j1 and farm j2, is predicted by a set of fixed coefficients (X^)i(j1j2). The random part of the model is given by two level-2 residual terms, one for the mother (uj1) and one for the farm (uj2), together with the usual level-1 residual term for each offspring. Decomposing the variation in such a fashion allows us to see how much of it is due to the different classifications. This particular example is somewhat oversimplified, since we have ignored paternity and we would also wish to include factors such as age of mother, parity of offspring etc. An application of this kind of modelling to a more complex structure involving Salmonella infection in chickens is given by Rasbash and Browne [28].

Considering now just the farms, and ignoring the mothers, suppose that the offspring often change farms, some not at all and some several times. Suppose also that we know, for each offspring, the weight wij2, associated with the j2th farm for offspring i with 2/2= Wy2 = 1. These weights, for example, may be proportional to the length of time an offspring stays in a particular farm during the course of our study. Note that we allow the possibility that for some (perhaps most) animals only one farm is involved so that one of these probabilities is one and the remainder are zero. Note that when all level-1 units have a single non-zero weight of 1 we obtain the usual purely hierarchical model. We can write for the special case of membership of up to two farms {1,2}:

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