By Abramenko P., Van Maldeghem H.
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At the end of each chapter presenting a case study, we consider the similarities and diﬀerences in the results generated by the software procedures. We discuss reasons for any discrepancies, and make recommendations for use of the various procedures in diﬀerent settings. Appendix A presents several statistical software resources. 2), and HLM (Version 7) is provided in addition to links to other useful mixed modeling resources, including web sites for important materials from this book. Appendix B revisits the Rat Brain analysis from Chap- Introduction 5 ter 5 to illustrate the calculation of the marginal variance-covariance matrix implied by one of the LMMs considered in that chapter.
The following matrix is an example of an unstructured D matrix, in the case of an LMM having two random eﬀects associated with the i-th subject. D = V ar(ui ) = 2 σu1 σu1,u2 σu1,u2 2 σu2 In this case, the vector θ D contains three covariance parameters: ⎛ ⎞ 2 σu1 θD = ⎝ σu1,u2 ⎠ 2 σu2 We also deﬁne other more parsimonious structures for D by imposing certain constraints on the structure of D. A very commonly used structure is the variance components (or diagonal) structure, in which each random eﬀect in ui has its own variance, and all covariances in D are deﬁned to be zero.
In both of these conventions, the unit of analysis is indexed by i. 1 and in each of the chapters presenting example analyses. 1 General Speciﬁcation for an Individual Observation We begin with a simple and general formula that indicates how most of the components of an LMM can be written at the level of an individual observation in the context of a longitudinal two-level data set. 2. In the example chapters we proceed in a similar manner; that is, we specify the models at the level of an individual observation for ease of understanding, followed by elements of matrix notation.