Example 15 generalized likelihood ratio test let t lr

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Example 1.5 (Generalized Likelihood Ratio Test). Let T LR be the test statistic associated with a generalized likelihood ratio test of Erd¨ os– enyi versus a two-group stochastic block model , and T c LR correspond to an approximation obtained by spectral partitioning in place of the maximization over group membership. For the data of Section 1.4.1 , simulation yields a corresponding p-value of less than 10 3 with respect to T c LR , with Figure 1.5 confirming the power of this test. Our case study has so far yielded reassuring results. However, a closer look reveals that selecting appropriate network models and test statistics may require more careful consideration. Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671
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Inference for Graphs and Networks 15 Fig. 1.5. Receiver operating characteristic (ROC) curves corresponding to tests of the data of Section 1.4.1, with Erd¨os–R´ enyi null and two-group stochastic block model alter- nate. Test statistics T c LR and T d Var were calculated via simulation, with the ROC upper bound obtained using knowledge of the true group membership for each node. Example 1.6 (Degree Variance Test). Suppose we adopt instead the test statistic of Snijders (1981) : T d Var = 1 n 1 n i =1 n j =1 A ij 1 n n i =1 n j =1 A ij 2 , the sample variance of the observed degree sequence n j =1 A ij . A glance at the data of Section 1.4.1 indicates the poor fit of an Erd¨os–R´ enyi null , and indeed simulation yields a p-value of less than 10 3 . Figure 1.5 , however , reveals that T d Var possesses very little power. This dichotomy between a low p -value, and yet low test power, high- lights a limitation of the models exhibited thus far: in each case, both the expected degree sequence and the corresponding node connectivity proper- ties are determined by exactly the same set of model parameters. In this regard, test statistics depending on the data set only through its degree sequence can prove quite limiting, as the difference between the two mod- els under consideration lies entirely in their node connectivity properties, rather than the heterogeneity of their degree sequences. Indeed, significant degree variation is a hallmark of many observed network data sets, the data of Section 1.4.1 included; sometimes certain nodes are simply more connected than others. In order to conclude that Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law.
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