We develop a dependent Dirichlet process (DDP) model for repeated measuresmultiple membership (MM) data. This data structure arises in studies underwhich an intervention is delivered to each client through a sequence ofelements which overlap with those of other clients on different occasions. Ourinterest concentrates on study designs for which the overlaps of sequencesoccur for clients who receive an intervention in a shared or grouped fashionwhose memberships may change over multiple treatment events. Our motivatingapplication focuses on evaluation of the effectiveness of a group therapyintervention with treatment delivered through a sequence of cognitivebehavioral therapy session blocks, called modules. An open-enrollment protocolpermits entry of clients at the beginning of any new module in a manner thatmay produce unique MM sequences across clients. We begin with a model thatcomposes an addition of client and multiple membership module random effectterms, which are assumed independent. Our MM DDP model relaxes the assumptionof conditionally independent client and module random effects by specifying acollection of random distributions for the client effect parameters that areindexed by the unique set of module attendances. We demonstrate how thisconstruction facilitates examining heterogeneity in the relative effectivenessof group therapy modules over repeated measurement occasions.
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