By Shelby J. Haberman (auth.), Neil J. Dorans, Sandip Sinharay (eds.)
In 2006, Paul W. Holland retired from academic checking out provider (ETS) after a profession spanning 5 many years. In 2008, ETS backed a convention, Looking Back, honoring his contributions to utilized and theoretical psychometrics and information. Looking Back attracted a wide viewers that got here to pay homage to Paul Holland and to listen to shows by way of colleagues who labored with him in specific methods over these forty+ years. This publication comprises papers according to those shows, in addition to vignettes supplied through Paul Holland sooner than each one section.
The papers during this booklet attest to how Paul Holland's pioneering rules stimulated and proceed to steer a number of fields resembling social networks, causal inference, merchandise reaction idea, equating, and DIF. He utilized statistical pondering to a vast variety of ETS actions in try out improvement, statistical research, try out safeguard, and operations. the unique papers contained during this publication offer ancient context for Paul Holland’s paintings along remark on a few of his significant contributions by means of noteworthy statisticians operating today.
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Additional resources for Looking Back: Proceedings of a Conference in Honor of Paul W. Holland
Sample text
Fienberg et al. distribution with size 1 and À Á class probabilities pij ð0; 0Þ; pij ð1; 0Þ; pij ð0; 1Þ; pij ð1; 1Þ. Thus, X is composed of n2 independent multinomial draws, one for each pair of nodes. Note that the expected value of X is p. By standard theory of exponential families (see Barndorff-Nielsen, 1978; Brown, 1986), the vector T ¼ AX contains the sufficient statistics for the model parameters. ) Unlike Holland and Leinhardt (1981), who encode the observed network using the nðn À 1Þ off-diagonal elements of the incidence matrix, À Á we choose to represent the observed network using a vector of dimension 4 n2 ¼ 2nðn À 1Þ.
Borrowing the terminology from Eriksson, Fienberg, Rinaldo, and Sullivant (2006), we call the polytope S the marginal polytope of the model. From the theory of exponential families (see Barndorff-Nielsen, 1978; Brown, 1986), the marginal polytope is the convex support of the family of distributions representing the p1 model whose design matrix is A. Furthermore, the MLE p^ of the vector of probabilities p exists if and only if the vector t of observed sufficient statistics is in the interior of S.
Every other vector in the kernel is a multiple of this one, so this binomial is enough À to Á encode the whole toric ideal. We represent the observed network as a 4 n2 -dimensional vector X with 0=1 entries, so that the cardinality of the sample space is 2nðnÀ1Þ . E. Fienberg et al. distribution with size 1 and À Á class probabilities pij ð0; 0Þ; pij ð1; 0Þ; pij ð0; 1Þ; pij ð1; 1Þ. Thus, X is composed of n2 independent multinomial draws, one for each pair of nodes. Note that the expected value of X is p.