Example 14 contingency table test consider the data

Info icon This preview shows pages 13–15. Sign up to view the full content.

Example 1.4 (Contingency Table Test). Consider the data of Sec- tion 1.4.1. When categorical covariates are known , a contingency table test for independence between rows and columns may be performed according to the data shown in Table 1.1. The Pearson test statistic T χ 2 in this case evaluates to over 47 , and with only 2 degrees of freedom , the corresponding p-value for these data is less than 10 3 . In this case, the null hypothesis – that the Erd¨ os–R´ enyi model’s sole Bernoulli parameter can be used to describe both inter- and intra-subgroup connection probabilities – can clearly be rejected. As in the case of Zheng et al. (2006) and others, this χ 2 approach has been generally used to reject an Erd¨os–R´ enyi null when given network data include a categorical covariate for each node. (A cautionary reminder is in order: employing this method when covariates are inferred from data corresponds to a misuse of maximally selected statistics (Altman et al. , 1994).) Of course, in cases where it is computationally feasible, we may 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
Image of page 13

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

14 B. P. Olding and P. J. Wolfe instead use simulation to determine the exact distribution of any chosen test statistic T under whichever null model is assumed. 1.4.3. The case of latent categorial covariates The Erd¨ os–R´ enyi model of Definition 1.1 clearly implies a lack of network structure through its nodal properties, thus supporting its use as a null model in cases such as Example 1.4 and those described above. In contrast, the partial exchangeability exhibited by the stochastic block model of Def- inition 1.2 suggests its use as an alternate model that explicitly exhibits network structure. To this end, the usual Neyman–Pearson logic implies the adoption of a generalized likelihood ratio test statistic: T LR = sup p i>j P ( A ij ; p ) max c sup p 00 ,p 01 ,p 11 i>j P ( A ij ; p 00 , p 01 , p 11 , c ( i ) , c ( j )) = i>j p A ij (1 p ) 1 A ij max c sup p 00 ,p 01 ,p 11 i>j ( p c ( i ) c ( j ) ) A ij (1 p c ( i ) c ( j ) ) 1 A ij . As we have seen in Section 1.3.2, however, maximizing the likelihood of the covariate vector c ∈ { 0 , 1 } n in general requires an exhaustive search. Faced with the necessity of approximate inference, we recall that the spec- tral partitioning algorithms outlined earlier in Section 1.3.2 provide an alternative to exact likelihood maximization in c . The resultant test statistic T c LR is computationally feasible, though with reduced power, and to this end we may test the data of Section 1.4.1 as follows.
Image of page 14
Image of page 15
This is the end of the preview. Sign up to access the rest of the document.
  • Spring '12
  • Kushal Kanwar
  • Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern