<div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><div class="t m0 x0 h1 y0 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y1 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y2 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y3 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y4 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y5 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y6 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h1 y7 ff1 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x0 h2 y8 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x1 h2 y9 ff2 fs0 fc0 sc0 ls0 ws1"> Advanced Topics from "And Introduction to </div><div class="t m0 x1 h2 ya ff2 fs0 fc0 sc0 ls1 ws2"> Categorical Data Analysis" Chapters 7, 8, 9. </div><div class="t m0 x2 h2 yb ff2 fs0 fc0 sc0 ls2 ws3">By: Tyler Diestel </div><div class="t m0 x3 h2 yc ff2 fs0 fc0 sc0 ls5 ws4">A Statistics Se<span class="ls6 ws5">nior Project </span></div><div class="t m0 x3 h2 yd ff2 fs0 fc0 sc0 ls0 ws1">Cal Poly San Luis Obispo </div><div class="t m0 x4 h2 ye ff2 fs0 fc0 sc0 ls3 ws6">Winter 2011 </div><div class="t m0 x5 h2 yf ff2 fs0 fc0 sc0 ls7 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x6 y10 w1 h3" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg2.png"><div class="t m0 x5 h4 y11 ff3 fs1 fc0 sc0 lse">Introduction<span class="ff4 ls4">\ue003</span></div><div class="t m0 x5 h2 y12 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h2 y13 ff2 fs0 fc0 sc0 ls8 ws8"> <span class="_0 blank"> </span>After finishing Stat 418: An Introduction <span class="ls4 ws0">to Categorical Analysis in winter 2010, I </span></div><div class="t m0 x5 h2 y14 ff2 fs0 fc0 sc0 ls9 ws9">was curious about the many methods of categor<span class="ls5 ws4">ical analysis that <span class="ls8 wsa">we were unable to </span></span></div><div class="t m0 x5 h2 y15 ff2 fs0 fc0 sc0 ls0 wsa">cover. This was nothing more than me just<span class="ls8"> being curious, until <span class="lsa ws5">I began some of my </span></span></div><div class="t m0 x5 h2 y16 ff2 fs0 fc0 sc0 ls3 wsb">classes in the spring. Twice th<span class="ws6">roughout my<span class="_1 blank"></span> spring quarter th<span class="lsf wsc">ere were opportunities to use </span></span></div><div class="t m0 x5 h2 y17 ff2 fs0 fc0 sc0 lsb ws8">some of these advanced methods, yet I did not<span class="ls4 ws0"> know how to use them properly. It first </span></div><div class="t m0 x5 h2 y18 ff2 fs0 fc0 sc0 lsb wsd">occurred in my Stat 465: Statistical Consulti<span class="ls8 wse">ng class, when one of our clients had so<span class="_1 blank"></span>me </span></div><div class="t m0 x5 h2 y19 ff2 fs0 fc0 sc0 lsb wsf">before and after surveys from a camp. Thes<span class="ls1 ws2">e surveys could have been analyzed using </span></div><div class="t m0 x5 h2 y1a ff2 fs0 fc0 sc0 ls3 ws6">matched pair categorical analysis, yet I did <span class="lsf wsc">not know the m<span class="_1 blank"></span>ethods <span class="ls9 ws10">well enough at the time </span></span></div><div class="t m0 x5 h2 y1b ff2 fs0 fc0 sc0 ls9 ws11">to run them. An opportunity again arose in my <span class="_1 blank"></span><span class="ls8 ws12">Stat 427: Probability Th<span class="ls9 ws13">eory class to better </span></span></div><div class="t m0 x5 h2 y1c ff2 fs0 fc0 sc0 ls9 ws9">understand McNemar\u2019s test, but still I did not <span class="ls5 ws14">have any previous knowledge to bring to </span></div><div class="t m0 x5 h2 y1d ff2 fs0 fc0 sc0 ls9 wsb">the table. At last, I finally decided that I s<span class="ls1 ws2">hould learn these advanced categorical analysis </span></div><div class="t m0 x5 h2 y1e ff2 fs0 fc0 sc0 ls9 ws9">methods and write a tutorial for anyone else <span class="wsb">who might be in my shoes in the future. </span></div><div class="t m0 x5 h2 y1f ff2 fs0 fc0 sc0 lsb ws8"> <span class="_0 blank"> </span>In my Stat 418 class, by the end of<span class="ls4 ws15"> the year, we reach Section 7.1.3 in <span class="ff5 ls10 ws0">An </span></span></div><div class="t m0 x5 h2 y20 ff5 fs0 fc0 sc0 ls1 ws2">Introduction to Categorical Data Analysis<span class="ff2 ls3 ws8">, (Second Edition), by Alan Agresti. For m<span class="_1 blank"></span>y </span></div><div class="t m0 x5 h2 y21 ff2 fs0 fc0 sc0 ls9 ws13">senior project, I investigated<span class="ls1 ws16"> selected statistical methods <span class="ls3 wsb">from where Stat 418 ended to </span></span></div><div class="t m0 x5 h2 y22 ff2 fs0 fc0 sc0 ls9 ws17">Section 9.2 of Agresti\u2019s text. I have now co<span class="ws11">mpiled all of my notes, SAS code, and proofs </span></div><div class="t m0 x5 h2 y23 ff2 fs0 fc0 sc0 lsc ws18">into this tutoria<span class="_1 blank"></span>l. To be faithful to Agresti\u2019<span class="ls8 wsf">s text, I have kept all of the same chapter, </span></div><div class="t m0 x5 h2 y24 ff2 fs0 fc0 sc0 ls0 ws9">section, subsection, and table headers, so the <span class="ls5 ws14">reader can easily re<span class="ls3 ws19">ference the text. At </span></span></div><div class="t m0 x5 h2 y25 ff2 fs0 fc0 sc0 lsc ws10">times, I will reference dir<span class="_1 blank"></span>ect models that Ag<span class="lsb wsf">resti used. To indicate that I a<span class="_1 blank"></span>m directly </span></div><div class="t m0 x5 h2 y26 ff2 fs0 fc0 sc0 ls9 ws1a">referring to <span class="ff5 ls8 ws8">An Introduction to Categorical Data Analysis</span><span class="lsc ws1b">, I will use the abbreviation </span></div><div class="t m0 x5 h2 y27 ff2 fs0 fc0 sc0 ls8 ws12">\u2018ICDA.\u2019 </div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">2 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf3" class="pf w0 h0" data-page-no="3"><div class="pc pc3 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h5" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg3.png"><div class="t m0 x5 h2 y2a ff2 fs0 fc0 sc0 ls8 ws12"> <span class="_0 blank"> </span>This paper begins in Chapter 7 which deals with loglinear models for contingency </div><div class="t m0 x5 h2 y2b ff2 fs0 fc0 sc0 ls1 ws2">tables. I will start on Section 7.1.4, which explai<span class="lsf ws1c">ns how to create loglinear models for </span></div><div class="t m0 x5 h2 y2c ff2 fs0 fc0 sc0 ls9 wsd">three-way contingency tables. Section 7.1.5 will <span class="ls8 ws1d">then show how a model with two-factor </span></div><div class="t m0 x5 h2 y2d ff2 fs0 fc0 sc0 ls12 ws1e">parameters describes co<span class="_1 blank"></span>nditional associati<span class="_1 blank"></span><span class="ls0 wse">on. Then in Section 7.1.6, we will go over an </span></div><div class="t m0 x5 h2 y2e ff2 fs0 fc0 sc0 lsb ws1f">example to use all of the tools we have le<span class="ls8 ws20">arned thus far when analyzing data about </span></div><div class="t m0 x5 h2 y2f ff2 fs0 fc0 sc0 ls9 ws10">alcohol, cigarette and marijuana use. </div><div class="t m0 x5 h2 y30 ff2 fs0 fc0 sc0 ls9 ws1"> <span class="_0 blank"> </span>Section 7.2 describes how we can make in<span class="ls8 wsd">ferences from these loglinear models. </span></div><div class="t m0 x5 h2 y31 ff2 fs0 fc0 sc0 ls0 ws1">The first subsection deals with model fit. Th<span class="lsf wsc">en in Section 7.2.2 we <span class="ls13 ws21">start to analyze the </span></span></div><div class="t m0 x5 h2 y32 ff2 fs0 fc0 sc0 ls8 wsd">cell residuals of these models. In Section 7.2.3, we<span class="ls9 ws1a"> construct a test to determine if there is </span></div><div class="t m0 x5 h2 y33 ff2 fs0 fc0 sc0 lsb ws22">conditional association. We will then crea<span class="ws16">te condition<span class="_1 blank"></span>al odds ratios and confidence </span></div><div class="t m0 x5 h2 y34 ff2 fs0 fc0 sc0 ls0 ws9">intervals for them. Section 7.2.5, applies everythi<span class="ws23">ng that we have learned to m<span class="_1 blank"></span>odels that </span></div><div class="t m0 x5 h2 y35 ff2 fs0 fc0 sc0 lsc ws18">are larger than three-way table<span class="_1 blank"></span>s. We will then<span class="lsa ws24"> fo<span class="_1 blank"></span>llow this up with an example that deals </span></div><div class="t m0 x5 h2 y36 ff2 fs0 fc0 sc0 ls3 ws8">with automobile accidents in Section 7.2.6. In th<span class="lsb">e next section,<span class="_1 blank"></span> we will dive more into the </span></div><div class="t m0 x5 h2 y37 ff2 fs0 fc0 sc0 ls9 ws1a">understanding of the three-factor interacti<span class="ls8 wse">on. Lastly, in Section 7.2.8 we will discuss the </span></div><div class="t m0 x5 h2 y38 ff2 fs0 fc0 sc0 ls0 ws1">difference between statistical significance and practical s<span class="_1 blank"></span>ignificance. </div><div class="t m0 x5 h2 y39 ff2 fs0 fc0 sc0 ls3 ws16"> <span class="_0 blank"> </span>Section 7.3 ties Chapter 7 with Chapter 4 as<span class="ws25"> it relates loglinear <span class="ls9 ws1a">models to logistic </span></span></div><div class="t m0 x5 h2 y3a ff2 fs0 fc0 sc0 ls8 ws18">models. Section 7.3.1 uses logistic models to in<span class="ls3 ws26">terpret loglinear models<span class="ls4 ws0">. Then in the next </span></span></div><div class="t m0 x5 h2 y3b ff2 fs0 fc0 sc0 ls9 ws9">section we revisit the example in Section 7.2.6 <span class="ls10 ws23">and apply this new pe<span class="lsf wsc">rspective. Then in </span></span></div><div class="t m0 x5 h2 y3c ff2 fs0 fc0 sc0 ls1 ws2">Section 7.3.3 and Section 7.3.4 we discuss wh<span class="lsf wsb">en we would use loglinear models over </span></div><div class="t m0 x5 h2 y3d ff2 fs0 fc0 sc0 ls9 ws3">logistic models and vice versa. </div><div class="t m0 x5 h2 y3e ff2 fs0 fc0 sc0 ls8 wse"> <span class="_0 blank"> </span>We then jump to Chapter 8 where we cr<span class="ls0 ws9">eate models for matched pairs. Section 8.1 </span></div><div class="t m0 x5 h2 y3f ff2 fs0 fc0 sc0 ls3 ws26">compares dependent proportions. It begins <span class="ls9 ws8">by explaining the McNemar Test in Section </span></div><div class="t m0 x5 h2 y40 ff2 fs0 fc0 sc0 ls8 ws12">8.1.1 and then finishes by estimating the difference in marginal proportions. </div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">3 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf4" class="pf w0 h0" data-page-no="4"><div class="pc pc4 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h5" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg4.png"><div class="t m0 x5 h2 y2a ff2 fs0 fc0 sc0 ls4 ws0"> <span class="_0 blank"> </span>In Section 8.2 we construct logistic m<span class="ls3 wsf">odels to analyze matched pairs. We begin </span></div><div class="t m0 x5 h2 y2b ff2 fs0 fc0 sc0 lsa ws16">by creating marginal models for m<span class="_1 blank"></span>arginal <span class="ls4 ws20">proportions. Then we are introduced to </span></div><div class="t m0 x5 h2 y2c ff2 fs0 fc0 sc0 ls8 ws27">subjected-specific tables a<span class="ls1 ws2">nd population-average tables. In Section 8.2.3 we examine how </span></div><div class="t m0 x5 h2 y2d ff2 fs0 fc0 sc0 ls9 ws17">one can use conditional logistic<span class="ws1"> regression for matched paired<span class="ls1 ws2"> data. Section 8.2.4 is very </span></span></div><div class="t m0 x5 h2 y2e ff2 fs0 fc0 sc0 ls14 ws28">similar to the previous s<span class="_1 blank"></span>ecti<span class="lsf ws15">on but its data deals with case-<span class="ls4 ws0">controlled studies. Then in </span></span></div><div class="t m0 x5 h2 y2f ff2 fs0 fc0 sc0 ls3 ws6">Section 8.2.5 we make a connection betw<span class="ls2 ws29">een the McNem<span class="_1 blank"></span>ar Test and the Cochran-</span></div><div class="t m0 x5 h2 y30 ff2 fs0 fc0 sc0 ls0 ws1">Mantel-Haenszel Test. </div><div class="t m0 x5 h2 y31 ff2 fs0 fc0 sc0 ls8 wsd"> <span class="_0 blank"> </span>Section 8.3 discusses how to<span class="ws12"> interpret margins of square contingency tables. The </span></div><div class="t m0 x5 h2 y32 ff2 fs0 fc0 sc0 lsa ws24">first section explains marginal homogeneity<span class="ls4 ws29"> and nom<span class="_1 blank"></span>inal classifi<span class="ls0 ws24">cation. This is then </span></span></div><div class="t m0 x5 h2 y33 ff2 fs0 fc0 sc0 ls0 ws1a">followed up by an example which deals with di<span class="ls15 ws20">fferent brands of co<span class="ls8 ws12">ffee. In Section 8.3.3 </span></span></div><div class="t m0 x5 h2 y34 ff2 fs0 fc0 sc0 ls8 ws12">we discuss how to analyze ordinal paired da<span class="ls4 ws0">ta. Then we use this new method to analyze </span></div><div class="t m0 x5 h2 y35 ff2 fs0 fc0 sc0 ls0 ws9">data in an example in Section 8.3.4. </div><div class="t m0 x5 h2 y36 ff2 fs0 fc0 sc0 ls0 wsf"> <span class="_0 blank"> </span>Section 8.4 deals with symmetry and <span class="ws2a">quasi-symm<span class="_1 blank"></span>etry models. In the first </span></div><div class="t m0 x5 h2 y37 ff2 fs0 fc0 sc0 ls8 wsa">subsection we are introduced <span class="ls4 ws0">to the symmetry model. Then in Section 8.4.2 we create the </span></div><div class="t m0 x5 h2 y38 ff2 fs0 fc0 sc0 lsb ws1f">quasi-symmetry model, which is a more re<span class="_1 blank"></span><span class="ls3 ws19">alistic version of th<span class="ws2b">e symmetry model. </span></span></div><div class="t m0 x5 h2 y39 ff2 fs0 fc0 sc0 ls1 wse"> <span class="_0 blank"> </span>We then skip to Section 8.6 where we <span class="ws2">learn the Bradley-Terry model that ranks </span></div><div class="t m0 x5 h2 y3a ff2 fs0 fc0 sc0 ls4 ws0">subjects in paired situations. In Section 8.6.2, <span class="ls1 ws9">we are able to run th<span class="ls3 wsb">is model on a data set </span></span></div><div class="t m0 x5 h2 y3b ff2 fs0 fc0 sc0 ls9 ws1a">of men\u2019s tennis players. Next in Secti<span class="wsb">on 8.6.2.a I have added my own example which </span></div><div class="t m0 x5 h2 y3c ff2 fs0 fc0 sc0 ls3 ws20">analyzes data from the 2010 Major League Baseball season. </div><div class="t m0 x5 h2 y3d ff2 fs0 fc0 sc0 ls8 ws10"> <span class="_0 blank"> </span>The last chapter we will look at will <span class="ls9 ws1a">be Chapter 9, which creates models for </span></div><div class="t m0 x5 h2 y3e ff2 fs0 fc0 sc0 ls3 ws16">correlated and clustered responses. Secti<span class="ls2">on 9.1 discusses m<span class="_1 blank"></span>arginal and conditional </span></div><div class="t m0 x5 h2 y3f ff2 fs0 fc0 sc0 ls8 ws20">models. This has three subsections. The first <span class="ws12">explains how marginal models can be used </span></div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">4 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf5" class="pf w0 h0" data-page-no="5"><div class="pc pc5 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h5" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg5.png"><div class="t m0 x5 h2 y2a ff2 fs0 fc0 sc0 ls0 ws9">for data with a clustered binary response. <span class="lsa ws15">We will then run an exam<span class="_1 blank"></span>ple in the next </span></div><div class="t m0 x5 h2 y2b ff2 fs0 fc0 sc0 ls0 ws24">section, and lastly, introd<span class="lsb ws1f">uce conditional models fo</span><span class="ws1">r a repeated response. </span></div><div class="t m0 x5 h2 y2c ff2 fs0 fc0 sc0 ls3 ws24"> <span class="_0 blank"> </span>Section 9.2 will be our last section. Here <span class="ls8 ws8">we will first explain the quasi-likelihood </span></div><div class="t m0 x5 h2 y2d ff2 fs0 fc0 sc0 ls8 ws12">approach, and then relate th<span class="ls3 ws26">at to the General Estimati</span><span class="wsd">ng Equation (GEE) methodology in </span></div><div class="t m0 x5 h2 y2e ff2 fs0 fc0 sc0 ls3 ws9">Section 9.2.2. The next two sections will be ex<span class="ls8 ws12">amples to help us understand GEE. Lastly, </span></div><div class="t m0 x5 h2 y2f ff2 fs0 fc0 sc0 ls2 ws29">we will find out the limitations that the G<span class="ls3 ws6">EE has com<span class="_1 blank"></span>pared to the maximum likelihood </span></div><div class="t m0 x5 h2 y30 ff2 fs0 fc0 sc0 ls9 ws2c">method. <span class="_2 blank"> </span> </div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">5 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf6" class="pf w0 h0" data-page-no="6"><div class="pc pc6 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h5" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg6.png"><div class="t m0 x5 h4 y41 ff3 fs1 fc0 sc0 ls16 ws2d">Table<span class="ff4 ls17">\ue003</span><span class="ls19">of<span class="ff4 ls4 ws2e">\ue003</span><span class="ls18 ws2f">Contents<span class="ff4 ls4">\ue003</span></span></span></div><div class="t m0 x5 h2 y42 ff2 fs0 fc0 sc0 lsb ws30">Introduction<span class="ls4 ws31"> ......................................................................................................................... <span class="_3 blank"> </span>2<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y43 ff2 fs0 fc0 sc0 ls8 ws12">7.1.4 Loglinear Models for Three-Way Tables<span class="ls4 ws32"> .......................................................... <span class="_1 blank"></span>8<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y44 ff2 fs0 fc0 sc0 ls8 ws12">7.1.5 Two-Factor Parameters Describe Conditional Association<span class="ls4 ws33"> ............................. 10<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y45 ff2 fs0 fc0 sc0 ls8 wsa">7.1.6 Example: Alcohol, Ciga<span class="ls0 ws24">rette and Marijuana Use<span class="ls4 ws34"> ............................................ <span class="_4 blank"> </span>11<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x9 h2 y46 ff2 fs0 fc0 sc0 ls8 ws12">7.2 Inference for Loglinear Models<span class="ls4 ws34"> .............................................................................. <span class="_4 blank"> </span>15<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y47 ff2 fs0 fc0 sc0 ls9 ws13">7.2.1 Chi-Squared Goodness-of-Fit Tests<span class="ls4 ws35"> ................................................................. <span class="_5 blank"> </span>15<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y48 ff2 fs0 fc0 sc0 ls1 ws2">7.2.2 Loglinear Cell Residuals<span class="ls4 ws36"> .................................................................................. <span class="_3 blank"> </span>16<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y49 ff2 fs0 fc0 sc0 ls0 ws1">7.2.3 Test about Conditional Associations<span class="ls4 ws37"> ............................................................... 17<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y4a ff2 fs0 fc0 sc0 ls1 ws2">7.2.4 Confidence Intervals fo<span class="ls0 ws24">r Conditional Odds Ratios<span class="ls4 ws38"> .......................................... <span class="_6 blank"> </span>18<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y4b ff2 fs0 fc0 sc0 ls8 ws12">7.2.5 Loglinear Models for Higher Dimensions<span class="ls4 ws39"> ....................................................... <span class="_7 blank"> </span>18<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y4c ff2 fs0 fc0 sc0 lsb ws1f">7.2.6 Example: Automobile Accidents and Seat B<span class="_1 blank"></span>elts<span class="ls4 ws34"> ............................................. <span class="_4 blank"> </span>19<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y4d ff2 fs0 fc0 sc0 ls8 ws12">7.2.7 Three-Factor Interaction<span class="ls4 ws3a"> .................................................................................. 23<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y4e ff2 fs0 fc0 sc0 ls4 ws8">7.2.8 Large Samples and Statistical <span class="ls8 ws12">Versus Practical Significance</span><span class="ws3b"> .......................... <span class="_7 blank"> </span>24<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x9 h2 y4f ff2 fs0 fc0 sc0 ls4 ws0">7.3 The Loglinear-Logistic Connection<span class="ws33"> ........................................................................ <span class="_6 blank"> </span>27<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y50 ff2 fs0 fc0 sc0 ls4 ws15">7.3.1 Using Logistic Models to Interpret Loglinear Models<span class="ws32"> .................................... <span class="_1 blank"></span>27<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y51 ff2 fs0 fc0 sc0 ls0 ws1">7.3.2 Example: Auto Accident Data Revisited<span class="ls4 ws3c"> ......................................................... <span class="_7 blank"> </span>28<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y52 ff2 fs0 fc0 sc0 lsf wsc">7.3.3 Correspondence between Loglinear and Logistic Models<span class="ls4 ws3d"> ............................... <span class="_3 blank"> </span>28<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y53 ff2 fs0 fc0 sc0 ls4 ws0">7.3.4 Strategies in Model Selection<span class="ws3a"> .......................................................................... 29<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x5 h2 y54 ff2 fs0 fc0 sc0 ls0 ws1">Chapter 8 Models for Matched Pairs<span class="ls4 ws3e"> ................................................................................ <span class="_4 blank"> </span>31<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x9 h2 y55 ff2 fs0 fc0 sc0 ls0 ws1">8.1 Comparing Dependent Proportions<span class="ls4 ws3f">......................................................................... 31<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y56 ff2 fs0 fc0 sc0 lsb ws1f">8.1.1 McNemar Test Compari<span class="ls1 ws2">ng Marginal Proportions<span class="ls4 ws39"> ........................................... <span class="_7 blank"> </span>33<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y57 ff2 fs0 fc0 sc0 ls0 ws1">8.1.2 Estimating Differences of Proportions<span class="ls4 ws40"> ............................................................. <span class="_5 blank"> </span>35<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x9 h2 y58 ff2 fs0 fc0 sc0 ls1 ws22">8.2 Logistic Regressi<span class="lsb ws41">on for Matched Pairs<span class="ls4 ws42"> ................................................................... <span class="_7 blank"> </span>37<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y59 ff2 fs0 fc0 sc0 ls1 ws9">8.2.1 Marginal Models for Marginal Proportions<span class="ls4 ws38"> ..................................................... <span class="_6 blank"> </span>37<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y5a ff2 fs0 fc0 sc0 lsf wsc">8.2.2 Subject-Specific and P<span class="lsb ws1f">opulation-Averaged Tables<span class="ls4 ws32"> ......................................... <span class="_1 blank"></span>38<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y5b ff2 fs0 fc0 sc0 ls1 ws2">8.2.3 Conditional Logistic Regr<span class="ls8 ws12">ession for Matched-Pairs<span class="ls4 ws36"> ........................................ <span class="_6 blank"> </span>39<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y5c ff2 fs0 fc0 sc0 ls0 ws1">8.2.4 Logistic Regression for Matched Case-Control Studies<span class="ls4 ws43"> .................................. <span class="_6 blank"> </span>41<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y5d ff2 fs0 fc0 sc0 ls1 ws44">8.2.5 Connection between McNemar a<span class="ls4 ws0">nd Cochran-Mantel-Haenszel Test<span class="ws45"> .............. <span class="_3 blank"> </span>42<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x9 h2 y5e ff2 fs0 fc0 sc0 ls3 ws16">8.3 Comparing Margins of S<span class="lsb ws46">quare Contingency Tables<span class="ls4 ws47"> ............................................... <span class="_3 blank"> </span>43<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y5f ff2 fs0 fc0 sc0 ls0 ws1">8.3.1 Marginal Homogeneity and Nominal Classifications<span class="ls4 ws36"> ...................................... <span class="_6 blank"> </span>43<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y60 ff2 fs0 fc0 sc0 ls8 ws12">8.3.2 Example: Coffee Brand Market Share<span class="ls4 ws48"> ............................................................. 43<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y61 ff2 fs0 fc0 sc0 ls8 ws12">8.3.3 Marginal Homogeneity and Order Categories<span class="ls4 ws38"> ................................................. <span class="_6 blank"> </span>46<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y62 ff2 fs0 fc0 sc0 ls9 ws20">8.3.4 Example: Recycle or Drive Less to Help Environment<span class="ls4 ws3d"> ................................... <span class="_6 blank"> </span>48<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">6 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div><a class="l" 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data-dest-detail='[43,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:102.000000px;bottom:151.896000px;width:422.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[43,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:133.056000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[43,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:114.276000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[46,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:95.496000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[48,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:76.716000px;width:410.001000px;height:15.768100px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf7" class="pf w0 h0" data-page-no="7"><div class="pc pc7 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h5" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg7.png"><div class="t m0 x9 h2 y63 ff2 fs0 fc0 sc0 ls1 ws2">8.4 Symmetry and Quasi-Symmetr<span class="ls9 ws10">y Models for Square Tables<span class="ls4 ws49"> .................................. 50<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y64 ff2 fs0 fc0 sc0 ls8 wsa">8.4.1 Symmetry as a <span class="ls10 ws23">Logistic Model<span class="ls4 ws34"> ........................................................................ <span class="_4 blank"> </span>50<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y65 ff2 fs0 fc0 sc0 ls8 ws12">8.4.2 Quasi-Symmetry<span class="ls4 ws3e"> .............................................................................................. <span class="_4 blank"> </span>51<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x9 h2 y66 ff2 fs0 fc0 sc0 ls1 ws2">8.6 Bradley-Terry Model for Paired Preferences<span class="ls4 ws45"> .......................................................... <span class="_3 blank"> </span>52<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y67 ff2 fs0 fc0 sc0 ls8 wsa">8.6.1 The Bradley-Terry Model<span class="ls4 ws4a"> ................................................................................ <span class="_8 blank"> </span>52<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y68 ff2 fs0 fc0 sc0 lsb ws22">8.6.2 Example: Ranking Men Tennis Players<span class="ls4 ws47"> ........................................................... <span class="_3 blank"> </span>53<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y69 ff2 fs0 fc0 sc0 ls8 ws12">8.6.2.a Example: MLB-National League West<span class="ls4 ws4b"> ......................................................... <span class="_3 blank"> </span>56<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x5 h2 y6a ff2 fs0 fc0 sc0 ls1 ws2">Chapter 9 Modeling Correlated, Clustered Responses<span class="ls4 ws34"> ..................................................... <span class="_4 blank"> </span>60<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x9 h2 y6b ff2 fs0 fc0 sc0 ls3 ws6">9.1 Marginal Models versus Conditional Models<span class="ls4 ws31"> ......................................................... <span class="_6 blank"> </span>60<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y6c ff2 fs0 fc0 sc0 ls1 ws2">9.1.1 Marginal Models for a Clustered Binary Response<span class="ls4 ws33"> ......................................... <span class="_8 blank"> </span>61<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y6d ff2 fs0 fc0 sc0 ls0 ws1">9.1.2 Example: Longitudinal Study of Treatments for Depression<span class="_1 blank"></span><span class="ls4 ws34"> .......................... <span class="_4 blank"> </span>61<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y6e ff2 fs0 fc0 sc0 ls1 ws2">9.1.3 Conditional Models for a Repeated Response<span class="ls4 ws4a"> ................................................. <span class="_8 blank"> </span>64<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x9 h2 y6f ff2 fs0 fc0 sc0 ls0 wsb">9.2 Marginal Modeling: The Generalized<span class="ls8 ws12"> Estimating Equations (GEE) Approach<span class="ls4 ws3c"> ..... <span class="_7 blank"> </span>65<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y70 ff2 fs0 fc0 sc0 ls0 ws1e">9.2.1 Quasi-Likelihood Methods<span class="ls4 ws49"> .............................................................................. 65<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y71 ff2 fs0 fc0 sc0 ls3 ws6">9.2.2 Generalized Estimating Equa<span class="ls1 ws2">tion Methodology: Basic Ideas<span class="_1 blank"></span><span class="ls4 ws32"> ......................... <span class="_1 blank"></span>66<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y72 ff2 fs0 fc0 sc0 ls8 ws12">9.2.3 GEE for Binary Data: Depression Study<span class="ls4 ws34"> ......................................................... <span class="_4 blank"> </span>67<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x8 h2 y73 ff2 fs0 fc0 sc0 lsa ws15">9.2.4 Example: Terato<span class="ls1 ws2">logy Overdispersion<span class="_1 blank"></span><span class="ls4 ws34"> .............................................................. <span class="_4 blank"> </span>71<span class="ff6 fs2">\ue003</span></span></span></div><div class="t m0 x8 h2 y74 ff2 fs0 fc0 sc0 ls9 ws1a">9.2.5 Limitations of GEE Compared with ML<span class="ls4 ws3e"> ......................................................... <span class="_4 blank"> </span>74<span class="ff6 fs2">\ue003</span></span></div><div class="t m0 x5 h2 y75 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h6 y76 ff1 fs3 fc0 sc0 ls1a ws0"> </div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">7 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div><a class="l" data-dest-detail='[50,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:102.000000px;bottom:705.216000px;width:422.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[50,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:686.436000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[51,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:667.596000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[52,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:102.000000px;bottom:648.816000px;width:422.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[52,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:630.036000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[53,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:611.196000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[56,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:592.416000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[60,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:90.000000px;bottom:573.636000px;width:434.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[60,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:102.000000px;bottom:554.796000px;width:422.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[61,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:536.016000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[61,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:517.236000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[64,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:498.456000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[65,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:102.000000px;bottom:479.616000px;width:422.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[65,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:460.836000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[66,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:442.056000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[67,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:423.216000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[71,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:404.436000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a><a class="l" data-dest-detail='[74,"XYZ",0,792,null]'><div class="d m1" style="border-style:none;position:absolute;left:114.000000px;bottom:385.656000px;width:410.001000px;height:15.768000px;background-color:rgba(255,255,255,0.000001);"></div></a></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf8" class="pf w0 h0" data-page-no="8"><div class="pc pc8 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h7" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg8.png"><div class="t m0 x5 h8 y77 ff7 fs4 fc0 sc0 ls1b ws4c">7.1.4\ue003Loglinear\ue003Models\ue003for\ue003Three\u2010Way\ue003Tables<span class="ls4">\ue003</span></div><div class="t m0 x5 h2 y78 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 xa h2 y79 ff2 fs0 fc0 sc0 lsa ws4">A three-way table can be interpre<span class="_1 blank"></span>ted much<span class="ls9 ws10"> like a two-way table in which we would </span></div><div class="t m0 x5 h2 y7a ff2 fs0 fc0 sc0 ls9 ws8">like to test independence. To examine inde<span class="ls1 ws2">pendence, we are going to look at several </span></div><div class="t m0 x5 h2 y7b ff2 fs0 fc0 sc0 lsf wsc">models used to find expected cell counts. </div><div class="t m0 xa h2 y7c ff2 fs0 fc0 sc0 ls3 wsb">The first model we are going to look at is the<span class="ff5 ls4 ws0"> mutual independence model<span class="ff2 ls2c">: </span></span></div><div class="t m0 x3 h9 y7d ff2 fs0 fc0 sc0 lsb ws30">log(<span class="ff8 ls4 ws4d">\u03bc</span><span class="fs5 ls1c ws4e v1">ijk</span><span class="ls1d wsb">) = <span class="ff8 ls4">\u03bb</span><span class="ls2d ws0"> +<span class="ff7 ls1e">\u03bb<span class="fs6 ls4 v2">\uebdc</span></span></span></span></div><div class="t m0 xb ha y7e ff7 fs6 fc0 sc0 ls1f">\uebd1<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 xc hb y7f ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 xd ha y7e ff7 fs6 fc0 sc0 ls20">\uebd2<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls4 v4">\uebde</span></div><div class="t m0 xe hb y80 ff7 fs6 fc0 sc0 ls21">\uebd3<span class="ff2 fs0 ls4 ws0 v3">. </span></div><div class="t m0 x5 h2 y81 ff2 fs0 fc0 sc0 ls2 wsd">This model is like the two-way m<span class="_1 blank"></span>odel of i<span class="ls4 ws0">ndependence equation 7.1 in ICDA. It treats </span></div><div class="t m0 x5 h2 y82 ff2 fs0 fc0 sc0 ls3 ws16">each variable as conditionally independent <span class="ls9 ws10">and marginally independent. F<span class="_1 blank"></span>or review, </span></div><div class="t m0 x5 h2 y83 ff5 fs0 fc0 sc0 ls1 ws2">conditional independence<span class="ff2 ls8 ws12"> is when the conditional odds<span class="ws27"> ratio is equal to 1, and </span></span><span class="ls9 ws0">marginal </span></div><div class="t m0 x5 h2 y84 ff5 fs0 fc0 sc0 ls8">independence<span class="ff2 wse"> is when the marginal odds ratio is equa<span class="ls9 ws1a">l to 1. This model is best used when </span></span></div><div class="t m0 x5 h2 y85 ff2 fs0 fc0 sc0 ls9 wsd">each of the X, Y, and Z variables is unaffect<span class="lsc ws4">ed by the other variab<span class="_1 blank"></span>les, thus causing them<span class="_1 blank"></span> </span></div><div class="t m0 x5 h2 y86 ff2 fs0 fc0 sc0 ls4 ws0">to be independent of each other. Below is a <span class="ws6a">proof of why this is <span class="lsb ws1f">the model used for the </span></span></div><div class="t m0 x5 h2 y87 ff2 fs0 fc0 sc0 lsb ws1f">independence model: </div><div class="t m0 xf h9 y88 ff7 fs0 fc0 sc0 ls0 ws4f">\ue737\ue74a\ue740\ue741\ue74e\ue003\ue750 \ue741<span class="_4 blank"> </span>\ue003\ue73d<span class="_9 blank"> </span>\ue74c\ue750<span class="_a blank"> </span>\ue003\ue74b\ue742<span class="_b blank"> </span>\ue741\ue74c\ue741 \ue740\ue741\ue74a\ue73f\ue741<span class="_4 blank"> </span>,<span class="ff2 ls4 ws0"> <span class="_c blank"></span><span class="ff7 ls0 ws50">\ue744<span class="_d blank"> </span>\ue74f\ue74f\ue751\ue749 \ue745\ue74b\ue74a<span class="_e blank"> </span>\ue003\ue745\ue74a\ue740<span class="_f blank"> </span>\ue74a</span></span></div><div class="t m0 x10 h9 y89 ff7 fs0 fc0 sc0 ls2e ws51">\uf23a<span class="_10 blank"></span>\ue7e8 <span class="fs6 ls2f ws52 v5">\ueb3e \ueb3e\ueb3e</span></div><div class="t m0 x11 h9 y89 ff7 fs0 fc0 sc0 ls1e">\ue7e8<span class="fs6 ls22 ws53 v5">\uebdc\uebdd\uebde </span><span class="ls23">\ued4c<span class="fs6 ls30 ws54 v5">\uebdc\ueb3e<span class="_1 blank"></span>\ueb3e<span class="fs0 ls4 ws4d v6">\uf23b\uf23a\ue7e8<span class="_4 blank"></span></span><span class="ls2f ws55">\ueb3e\uebdd <span class="fs0 ls4 ws56 v6">\uf23b\uf23a\ue7e8 </span><span class="ls24">\uebde<span class="fs0 ls4 v6">\uf23b<span class="ff2 ws0"> </span></span></span></span></span></span></div><div class="t m0 x12 hb y8a ff7 fs6 fc0 sc0 ls30 ws57">\uebdc\ueb3e<span class="_1 blank"></span>\ueb3e <span class="ls2f ws52">\ueb3e\uebdd<span class="_4 blank"></span>\ueb3e \ueb3e\ueb3e\uebde</span></div><div class="t m0 x13 hc y8b ff7 fs0 fc0 sc0 ls4 ws4d">\u03bc<span class="fs6 ls22 ws58 v5">\uebdc\uebdd\uebde </span><span class="ls2e ws59 v0">\ued4c\ue74a<span class="_1 blank"></span>\ue5db<span class="_11 blank"></span>\ue7e8</span></div><div class="t m0 x14 hd y8a ff7 fs6 fc0 sc0 ls31 ws5a">\uebdc\uebdd\uebde <span class="fs0 ls2e ws5b v6">\ued4c\ue74a<span class="_10 blank"></span>\uf23a<span class="_10 blank"></span>\ue7e8 <span class="ls4 ws5c">\uf23b\uf23a\ue7e8 \uf23b\uf23a\ue7e8<span class="_12 blank"> </span>\uf23b<span class="ff2 ws0"> </span></span></span></div><div class="t m0 x15 h9 y8c ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls1e v7">\ue7e8</span><span class="fs6 ls25 v2">\uebdc</span>\uf23a<span class="ls1e v7">\ue7e8</span><span class="fs6 ls2f v2">\ueb3e\ueb3e\uebde</span></div><div class="t m0 x16 he y8d ff7 fs0 fc0 sc0 ls5 ws5d">log\ued6b\u03bc<span class="fs6 ls22 ws5e v5">\uebdc\uebdd\uebde </span><span class="ls32 ws5f">\ued6f\ued4cl<span class="_10 blank"></span>o<span class="_10 blank"></span>g<span class="_13 blank"></span>\uf240<span class="_10 blank"></span>\ue003<span class="_10 blank"></span>\ue74a <span class="fs6 ls30 ws54 v5">\ueb3e\ueb3e</span><span class="ls4 ws4d v8">\uf23b</span><span class="ls2 ws60 v0">\ued6b\ue7e8<span class="fs6 ls2f ws61 v5">\ueb3e\uebdd \ueb3e </span></span><span class="ls26">\ued6f<span class="ls4 ws4d v8">\uf23b</span><span class="ls4">\uf241<span class="ff2 ws0"> </span></span></span></span></div><div class="t m0 x17 h9 y8e ff7 fs0 fc0 sc0 ls5 ws5d">log\ued6b\u03bc<span class="fs6 ls22 ws5e v5">\uebdc\uebdd\uebde </span><span class="ls32 ws62">\ued6f\ued4cl<span class="_10 blank"></span>o \ue748<span class="_14 blank"> </span><span class="ls33 ws63">\ued6b<span class="_15 blank"></span>\ue7e8 <span class="ls27 ws64">\ued6f\ued45\ue748<span class="_15 blank"></span>\ue74b<span class="_15 blank"></span>\ue743</span></span></span></div><div class="t m0 x18 h9 y8f ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls28 v7">\ue7e8</span><span class="fs6 ls2f ws65 v2">\ueb3e\ueb3e\uebde </span><span class="v0">\uf23b<span class="ff2 ws0 v7"> <span class="_16 blank"></span><span class="ff7 ls32 ws66">g<span class="_10 blank"></span>\uf23a<span class="_10 blank"></span>\ue74a<span class="_10 blank"></span>\uf23b<span class="_1 blank"></span>\ued45 \ue74b<span class="_10 blank"></span>\ue743</span></span></span></div><div class="t m0 x19 h9 y90 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls1e v7">\ue7e8</span><span class="fs6 ls30 ws54 v2">\uebdc\ueb3e<span class="_1 blank"></span>\ueb3e<span class="fs0 ls29 v9">\uf23b<span class="ls33 ws67 v7">\ued45\ue748<span class="_15 blank"></span>\ue74b<span class="_15 blank"></span>\ue743 <span class="fs6 ls2f ws68 v5">\ueb3e\uebdd<span class="_4 blank"> </span>\ueb3e</span></span></span></span></div><div class="t m0 x1a h9 y91 ff7 fs0 fc0 sc0 ls5 ws5d">log\ued6b\u03bc<span class="fs6 ls22 ws5e v5">\uebdc\uebdd\uebde </span><span class="ls32 ws69">\ued6f\ued4c\u03bb<span class="_10 blank"></span>\ue003<span class="_11 blank"></span>\ued45<span class="_1 blank"></span>\u03bb</span></div><div class="t m0 x1b hb y92 ff7 fs6 fc0 sc0 ls4">\uebdc</div><div class="t m0 x1b hf y93 ff7 fs6 fc0 sc0 ls2a">\uebd1<span class="fs0 ls34 v3">\ued45\ue003\u03bb</span></div><div class="t m0 x1c hb y92 ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 x1d hf y93 ff7 fs6 fc0 sc0 ls2b">\uebd2<span class="fs0 ls34 v3">\ued45\ue003\u03bb</span></div><div class="t m0 x1e hb y94 ff7 fs6 fc0 sc0 ls4">\uebde</div><div class="t m0 x1e hb y95 ff7 fs6 fc0 sc0 ls21">\uebd3<span class="ff2 fs0 ls4 ws0 v3"> </span></div><div class="t m0 x5 h2 y96 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h2 y97 ff2 fs0 fc0 sc0 ls9 ws1a">We use (X,Y,Z) to denote this model. </div><div class="t m0 x1f h2 y98 ff2 fs0 fc0 sc0 ls9 ws1a">The next model allows for independence <span class="lsf wsc">between only two of <span class="ls13 ws6b">the factors instead </span></span></div><div class="t m0 x5 h2 y99 ff2 fs0 fc0 sc0 ls2 wse">of all three: </div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">8 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pf9" class="pf w0 h0" data-page-no="9"><div class="pc pc9 w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h5" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bg9.png"><div class="t m0 x20 h9 y9a ff2 fs0 fc0 sc0 lsb ws30">log(<span class="ff8 ls4 ws4d">\u03bc</span><span class="fs5 ls1c ws4e v1">ijk</span><span class="ls1d wsb">) = <span class="ff8 ls4">\u03bb</span><span class="ls2d ws0"> +<span class="ff7 ls1e">\u03bb<span class="fs6 ls4 v2">\uebdc</span></span></span></span></div><div class="t m0 x10 hf y9b ff7 fs6 fc0 sc0 ls1f">\uebd1<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x21 hb y9c ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 x22 hf y9b ff7 fs6 fc0 sc0 ls20">\uebd2<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls4 v4">\uebde</span></div><div class="t m0 xb h10 y9d ff7 fs6 fc0 sc0 ls21">\uebd3<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebde</span></div><div class="t m0 xd hf y9b ff7 fs6 fc0 sc0 ls3e ws6c">\uebd1\uebd3 <span class="ff2 fs0 lsa ws15 v3"> + <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x23 hb y9c ff7 fs6 fc0 sc0 ls3f">\uebdd\uebde</div><div class="t m0 x24 hb y9b ff7 fs6 fc0 sc0 ls40 ws6d">\uebd2\uebd3 <span class="ff2 fs0 ls9 ws1a v3"> (7.4). </span></div><div class="t m0 x5 h2 y9e ff2 fs0 fc0 sc0 lsc ws10">This model allows there to be an asso<span class="_1 blank"></span>ciat<span class="lsf wsc">ion between X and Z controlling for Y, which </span></div><div class="t m0 x5 h9 y9f ff2 fs0 fc0 sc0 ls2 ws13">can be seen in the <span class="ff7 ls1e">\u03bb<span class="fs6 ls31 v2">\uebdc\uebde</span></span></div><div class="t m0 x25 hb ya0 ff7 fs6 fc0 sc0 ls41 ws6e">\uebd1\uebd3 <span class="ff2 fs0 ls9 ws1a v3"> term. Likewise, this model allows there to be an association </span></div><div class="t m0 x5 h9 ya1 ff2 fs0 fc0 sc0 lsf ws1a">between Y and Z controlling for X, which can be seen in the <span class="ff7 ls4">\u03bb</span></div><div class="t m0 x26 hb ya2 ff7 fs6 fc0 sc0 ls42">\uebdd\uebde</div><div class="t m0 x27 hb ya3 ff7 fs6 fc0 sc0 ls3e ws6f">\uebd2\uebd3 <span class="ff2 fs0 ls2 wse v3"> term. This model also </span></div><div class="t m0 x5 h2 ya4 ff2 fs0 fc0 sc0 lsf wsc">shows conditional independence between X and Y, <span class="ls0 ws1">controlling for Z. This is why there is </span></div><div class="t m0 x5 h9 ya5 ff2 fs0 fc0 sc0 ls3 ws6">no term for the X and Y interaction because <span class="ff7 ls1e">\u03bb<span class="fs6 ls31 v2">\uebdc\uebdd</span></span></div><div class="t m0 x28 hb ya6 ff7 fs6 fc0 sc0 ls41 ws70">\uebd1\uebd2 <span class="ff2 fs0 ls4 ws6 v3">= 0 in this case. We use (XZ, YZ) to </span></div><div class="t m0 x5 h2 ya7 ff2 fs0 fc0 sc0 ls1 ws22">denote this model. </div><div class="t m0 x1f h2 ya8 ff2 fs0 fc0 sc0 lsc ws16">To describe an interactio<span class="_1 blank"></span>n between all th<span class="ls3 ws19">ree pairs of variable<span class="ls9 ws10">s we would use the </span></span></div><div class="t m0 x5 h2 ya9 ff2 fs0 fc0 sc0 ls8 ws7f">homogeneous association model: </div><div class="t m0 x29 h9 yaa ff2 fs0 fc0 sc0 lsb ws30">log(<span class="ff8 ls4 ws4d">\u03bc</span><span class="fs5 ls1c ws4e v1">ijk</span><span class="ls1d wsb">) = <span class="ff8 ls4">\u03bb</span><span class="ls2d ws0"> +<span class="ff7 ls1e">\u03bb<span class="fs6 ls4 v2">\uebdc</span></span></span></span></div><div class="t m0 x2a ha yab ff7 fs6 fc0 sc0 ls1f">\uebd1<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x2b hb yac ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 x2c ha yab ff7 fs6 fc0 sc0 ls20">\uebd2<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls4 v4">\uebde</span></div><div class="t m0 x0 h10 yad ff7 fs6 fc0 sc0 ls21">\uebd3<span class="ff2 fs0 ls13 v3">+<span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebdd</span></div><div class="t m0 x2d ha yab ff7 fs6 fc0 sc0 ls3e ws71">\uebd1\uebd2 <span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebde</span></div><div class="t m0 x2e ha yab ff7 fs6 fc0 sc0 ls3e ws6c">\uebd1\uebd3 <span class="ff2 fs0 lsa ws15 v3"> + <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x2f hb yac ff7 fs6 fc0 sc0 ls42">\uebdd\uebde</div><div class="t m0 x30 hb yab ff7 fs6 fc0 sc0 ls40 ws6d">\uebd2\uebd3 <span class="ff2 fs0 ls43 ws2a v3"> (7.5). </span></div><div class="t m0 x5 h2 yae ff2 fs0 fc0 sc0 lsb ws10">This allows all three pair of<span class="ws15"> variables to have con<span class="_1 blank"></span>ditional association, which m<span class="_1 blank"></span>eans that </span></div><div class="t m0 x5 h2 yaf ff2 fs0 fc0 sc0 ls1 ws15">all odds ratios for any two variables are equal at<span class="ls3 ws28"> all levels of the th<span class="ls4 ws0">ird variable. This is </span></span></div><div class="t m0 x5 h2 yb0 ff2 fs0 fc0 sc0 ls4 ws0">known as <span class="ff5 ls8 ws12">homogeneous association</span><span class="ls3 wsb">. To denote this model we will use (XY, XZ, YZ). </span></div><div class="t m0 x1f h2 yb1 ff2 fs0 fc0 sc0 ls2 wsb">The last model contains a term for each <span class="_1 blank"></span><span class="lsb wsd">variable, each pair of variables, and a </span></div><div class="t m0 x5 h2 yb2 ff2 fs0 fc0 sc0 ls13 ws41">term that explains an intera<span class="ls0 wsb">ction between all three variable<span class="_1 blank"></span><span class="ls4 ws0">s. This is known as the </span></span></div><div class="t m0 x5 h2 yb3 ff2 fs0 fc0 sc0 ls2 wse">saturated model: </div><div class="t m0 x29 hc yb4 ff2 fs0 fc0 sc0 lsb ws30">log(<span class="ff8 ls4 ws4d">\u03bc</span><span class="fs5 ls1c ws4e v1">ijk</span><span class="ls1d wsb v0">) = <span class="ff8 ls4">\u03bb</span><span class="ls2d ws0"> +<span class="ff7 ls1e">\u03bb<span class="fs6 ls4 v2">\uebdc</span></span></span></span></div><div class="t m0 x2a ha yb5 ff7 fs6 fc0 sc0 ls1f">\uebd1<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x2b hb yb6 ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 x2c ha yb5 ff7 fs6 fc0 sc0 ls20">\uebd2<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls4 v4">\uebde</span></div><div class="t m0 x0 h11 yb7 ff7 fs6 fc0 sc0 ls21">\uebd3<span class="ff2 fs0 ls13 v3">+<span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebdd</span></div><div class="t m0 x2d ha yb5 ff7 fs6 fc0 sc0 ls3e ws71">\uebd1\uebd2 <span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebde</span></div><div class="t m0 x2e ha yb5 ff7 fs6 fc0 sc0 ls3e ws6c">\uebd1\uebd3 <span class="ff2 fs0 ls13 v3">+<span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x1e hb yb6 ff7 fs6 fc0 sc0 ls42">\uebdd\uebde</div><div class="t m0 x1e ha yb5 ff7 fs6 fc0 sc0 ls3e ws6f">\uebd2\uebd3 <span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls31 ws72 v4">\uebdc\uebdd\uebde</span></div><div class="t m0 x31 hb yb5 ff7 fs6 fc0 sc0 ls41 ws73">\uebd1\uebd2\uebd3 <span class="ff2 fs0 ls4 ws0 v3"> </span></div><div class="t m0 x5 h2 yb8 ff2 fs0 fc0 sc0 ls2 wse">The model is the most general m<span class="_1 blank"></span>odel and has a <span class="ls8 ws12">perfect fit because it <span class="ls4 ws0">describes all of the </span></span></div><div class="t m0 x5 h2 yb9 ff2 fs0 fc0 sc0 ls4 ws20">possible interactions. Below is proof of <span class="ls9 ws8">why this must be the saturated model: </span></div><div class="t m0 x32 he yba ff7 fs0 fc0 sc0 ls44 ws74">\ue72b\ue74a\ue003\ue750\ue744\ue745\ue74f \ue003\ue74f\ue73d\ue750\ue751\ue74e\ue73d\ue750\ue741\ue740 \ue003\ue749\ue74b\ue740\ue741\ue748 ,<span class="_3 blank"> </span>\ue750\ue744\ue741\ue74e\ue741 \ue003\ue745\ue74f \ue003\ue73d\ue003\ue74f\ue745\ue74a\ue743\ue748\ue741 \ue003\ue73f\ue74b\ue74a\ue74f\ue750\ue73d\ue74a\ue750 \ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue003<span class="ls4 ws4d v8">\uf23a</span><span class="ls35">\ue7e3<span class="ls4 ws4d v8">\uf23b</span><span class="lsf">,\ue003<span class="ff2 ls4 ws0"> </span></span></span></div><div class="t m0 x33 h9 ybb ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls36 ws75 v7">\ue72b\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x34 h9 ybb ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ls1 ws76 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d <span class="ls37">1<span class="ls3 ws77">\ue74a\ue750\ue003\u03bb</span></span></span></div><div class="t m0 x35 hb ybc ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 x36 ha ybd ff7 fs6 fc0 sc0 ls38">\uebd2<span class="fs0 ls44 ws78 v3">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f,\ue003<span class="ff2 ls4 ws0"> <span class="_17 blank"></span><span class="ff7 ls1 ws79">\ue74a\ue750\ue003\u03bb<span class="fs6 ls4 v2">\uebdc</span></span></span></span></div><div class="t m0 x1a h12 ybd ff7 fs6 fc0 sc0 ls39">\uebd1<span class="fs0 ls44 ws7a v3">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f, </span><span class="fs0 ls4 ws4d va">\uf23a</span><span class="fs0 ls3d ws7b v3">\ue72c\ued46 </span><span class="fs0 ls4 ws4d va">\uf23b<span class="ls3 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d</span></span></div><div class="t m0 x5 h2 ybe ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h13 ybf ff2 fs0 fc0 sc0 ls3a ws0"> <span class="ff7 ls4 ws4d vb">\uf23a<span class="ls3b ws7c v7">\ue72d\ued46<span class="_1 blank"></span>1</span></span></div><div class="t m0 x37 h9 yc0 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ls1 ws79 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d\ue74a\ue750\ue003\u03bb</span><span class="fs6 vc">\uebde</span></div><div class="t m0 x38 h14 yc1 ff7 fs6 fc0 sc0 ls3c">\uebd3<span class="fs0 ls44 ws78 v3">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f,\ue003<span class="_4 blank"></span><span class="ff2 ls4 ws0"> </span></span></div><div class="t m0 x25 h9 yc2 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls36 ws75 v7">\ue72b\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x39 h9 yc2 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3d ws7d v7">\ue72c\ued461</span></div><div class="t m0 x3a h9 yc2 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ls1 ws79 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d\ue74a\ue750\ue003\u03bb</span><span class="fs6 ls31 vc">\uebdc\uebdd</span></div><div class="t m0 x3b ha yc3 ff7 fs6 fc0 sc0 ls3e ws7e">\uebd1\uebd2 <span class="fs0 ls44 ws78 v3">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f<span class="_4 blank"></span>,\ue003<span class="ff2 ls4 ws0"> </span></span></div><div class="t m0 x5 h2 yc4 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x7 h2 y28 ff2 fs0 fc0 sc0 ls11 ws3">9 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div> <div id="pfa" class="pf w0 h0" data-page-no="a"><div class="pc pca w0 h0"><img fetchpriority="low" loading="lazy" class="bi x6 y10 w1 h15" alt="" src="https://files.passeidireto.com/f05d7538-f13d-4e4f-b017-dd840c9b7583/bga.png"><div class="t m0 x3c h9 yc5 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls45 ws80 v7">\ue72b\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x3d h9 yc5 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3b ws7c v7">\ue72d\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x3e h9 yc5 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ls1 ws81 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d\ue74a\ue750 </span><span class="fs6 ls31 vc">\uebdc\uebde</span></div><div class="t m0 x3f ha yc6 ff7 fs6 fc0 sc0 ls3e ws82">\uebd1\uebd3 <span class="fs0 ls44 ws78 v3">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f,\ue003<span class="ff2 ls4 ws0"> </span></span></div><div class="t m0 x40 hb yc7 ff7 fs6 fc0 sc0 ls3e">\uebd2\uebd3</div><div class="t m0 xc h9 yc8 ff7 fs0 fc0 sc0 ls1">\ue003\u03bb</div><div class="t m0 x41 h9 yc9 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls3b ws7c v7">\ue72d\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x42 h9 yc9 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3d ws7d v7">\ue72c\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x3 h9 yc9 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ls3 ws77 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d\ue74a\ue750\ue003\u03bb</span></div><div class="t m0 x43 h16 yca ff7 fs6 fc0 sc0 ls42 ws83">\uebdd\uebde <span class="fs0 ls44 ws84 v9">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f,<span class="_5 blank"> </span>\ue73d\ue74a\ue740 <span class="ff2 ls4 ws0"> </span></span></div><div class="t m0 x44 h9 ycb ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls45 ws85 v7">\ue72b\ued46<span class="_1 blank"></span>1 <span class="ls44 ws78">\ue750\ue741\ue74e\ue74f.<span class="ff2 ls4 ws0"> </span></span></span></div><div class="t m0 x5 h2 ycc ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x45 h9 ycb ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3d ws7d v7">\ue72c\ued461</span></div><div class="t m0 x46 h9 ycb ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3b ws7c v7">\ue72d\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x47 h9 ycb ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ls3 ws77 v7">\ue74a\ue74b\ue74a\ue74e\ue741\ue740\ue751\ue74a\ue740\ue73d\ue74a\ue750\ue003\u03bb</span><span class="fs6 ls22 vc">\uebdc\uebdd\uebde</span></div><div class="t m0 x48 ha ycd ff7 fs6 fc0 sc0 ls41 ws86">\uebd1\uebd2\uebd3 <span class="fs0 ls44 v3">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741</span></div><div class="t m0 x5 h2 yce ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x49 h9 ycf ff7 fs0 fc0 sc0 ls1 ws79">\ue735\ue74b<span class="_4 blank"></span>\ue003\ue750\ue744\ue741\ue003\ue750\ue74b\ue750\ue73d\ue748<span class="_4 blank"> </span>\ue003\ue74a\ue751\ue749\ue73e\ue741\ue74e<span class="_4 blank"> </span>\ue003\ue74b\ue742<span class="_4 blank"></span>\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f<span class="_4 blank"></span>\ue003\ue741\ue74d\ue751\ue73d\ue748\ue74f<span class="_4 blank"></span>:<span class="ff2 ls4 ws0"> </span></div><div class="t m0 x4a he yd0 ff7 fs0 fc0 sc0 ls51 ws87">1\ued45<span class="ls4 ws4d v8">\uf23a</span><span class="ls36 ws75">\ue72b\ued46<span class="_1 blank"></span>1</span></div><div class="t m0 x4b h9 yd1 ff7 fs0 fc0 sc0 ls46">\uf23b<span class="ls47 v7">\ued45</span><span class="ls4 ws4d">\uf23a</span><span class="ls3d ws7d v7">\ue72c\ued461</span></div><div class="t m0 x45 h9 yd1 ff7 fs0 fc0 sc0 ls29">\uf23b<span class="ls48 v7">\ued45</span><span class="ls4 ws4d">\uf23a</span><span class="ls3b ws88 v7">\ue72d\ued46<span class="_1 blank"></span>1 <span class="ls36 ws75">\ued46<span class="_18 blank"></span>1</span></span></div><div class="t m0 x4c h9 yd1 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3b ws7c v7">\ue72d\ued46<span class="_18 blank"></span>1</span></div><div class="t m0 x4d h9 yd1 ff7 fs0 fc0 sc0 ls29">\uf23b<span class="ls48 v7">\ued45</span><span class="ls4 ws4d">\uf23a</span><span class="ls3d ws7d v7">\ue72c\ued461</span></div><div class="t m0 x4e h9 yd1 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls49 ws89 v7">\ue73c\ued461</span></div><div class="t m0 x4f h9 yd1 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ff2 ws0 v7"> </span></div><div class="t m0 x5 h2 yd2 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x50 h9 yd1 ff7 fs0 fc0 sc0 ls29">\uf23b<span class="ls48 v7">\ued45</span><span class="ls4 ws4d">\uf23a</span><span class="ls36 ws75 v7">\ue72b\ued46<span class="_18 blank"></span>1</span></div><div class="t m0 x51 h9 yd1 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls4a ws8a v7">\ue72c\ued461</span></div><div class="t m0 x43 h9 yd1 ff7 fs0 fc0 sc0 ls29">\uf23b<span class="ls48 v7">\ued45</span><span class="ls4 ws4d">\uf23a<span class="ls36 v7">\ue72b</span></span></div><div class="t m0 x5 h2 yd3 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x52 he yd4 ff7 fs0 fc0 sc0 ls4 ws4d">\ued45<span class="v8">\uf23a</span><span class="ls36 ws75">\ue72b\ued46<span class="_18 blank"></span>1</span></div><div class="t m0 x19 h9 yd5 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls4a ws8a v7">\ue72c\ued461</span></div><div class="t m0 x53 h9 yd5 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b\uf23a<span class="ls3b ws7c v7">\ue72d\ued46<span class="_18 blank"></span>1</span></div><div class="t m0 x48 h9 yd5 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ff2 ws0 v7"> </span></div><div class="t m0 x54 h9 yd6 ff7 fs0 fc0 sc0 ls4b">\ue55c<span class="ls4 ws4d">\uf23b<span class="ff2 ws0"> </span></span></div><div class="t m0 x5 h2 yd7 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x55 he yd6 ff7 fs0 fc0 sc0 ls4c ws8b">\ue72b<span class="_15 blank"></span>\ue72c\ued45\ue72d \ued461\ued45\ue72b<span class="_15 blank"></span>\ue72d \ued46\ue72b \ued46\ue72d \ued451\ued45\ue72c<span class="_15 blank"></span>\ue72d \ued46\ue72c\ued46\ue72d \ued451\ued45<span class="ls4 ws4d v8">\uf23a</span><span class="ls33 ws8c">\ue72b<span class="_15 blank"></span>\ue72c<span class="_4 blank"></span>\ued46\ue72b<span class="_4 blank"></span>\ued46\ue72c\ued451</span></div><div class="t m0 x56 h9 yd8 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23b<span class="ws8d v7">\uf23a\ue747<span class="_19 blank"> </span>\ued46 1</span></div><div class="t m0 x55 h9 yd9 ff7 fs0 fc0 sc0 ls33 ws8e">\ue55c 1\ued46\ue72d<span class="_4 blank"></span>\ued46\ue72b<span class="_4 blank"></span>\ued46\ue72c\ued45\ue72b<span class="_15 blank"></span>\ue72c \ued45<span class="_1 blank"></span>\ue72b<span class="_15 blank"></span>\ue72d<span class="_4 blank"></span>\ued45\ue72c<span class="_15 blank"></span>\ue72d<span class="_1a blank"> </span>\ued46\ue72b<span class="_15 blank"></span>\ue72d<span class="_4 blank"></span>\ued46\ue72c<span class="_15 blank"></span>\ue72d<span class="_4 blank"></span>\ued45\ue72d<span class="_4 blank"></span>\ued46\ue72b<span class="_15 blank"></span>\ue72c\ued45\ue72b \ued45\ue72c\ued461<span class="_15 blank"></span><span class="ff2 ls4 ws0"> </span></div><div class="t m0 x5 h2 yda ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x57 h9 yd9 ff7 fs0 fc0 sc0 ls33 ws8c">\ued45\ue72b<span class="_15 blank"></span>\ue72c<span class="_15 blank"></span>\ue72d</div><div class="t m0 x5 h2 ydb ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x58 h9 ydc ff7 fs0 fc0 sc0 ls52 ws8f">\ue55c\ue72b<span class="_10 blank"></span>\ue72c<span class="_10 blank"></span>\ue72d<span class="_1b blank"></span><span class="ff2 ls4 ws0"> </span></div><div class="t m0 x5 h2 ydd ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h17 yde ff2 fs0 fc0 sc0 ls4d ws0"> <span class="ff7 lsf ws90 vb">\ue735\ue74b \ue003\ue750\ue744\ue745\ue74f\ue003\ue749\ue74b\ue740\ue741\ue748 \ue003\ue744\ue73d\ue74f \ue003\ue73d\ue74f\ue003\ue749\ue73d\ue74a\ue755 \ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f \ue003\ue73d\ue74f\ue003\ue74b\ue73e\ue74f\ue741\ue74e\ue752\ue741\ue740 \ue003\ue73f\ue741\ue748\ue748 \ue003\ue73f\ue74b\ue751\ue74a\ue750\ue74f .<span class="_3 blank"> </span>\ue72b\ue750 <span class="ls53 ws91">\ue003\ue745\ue74f\ue003\ue750\ue744\ue741<span class="_4 blank"></span>\ue003</span></span><span class="ls4 vb"> </span></div><div class="t m0 x5 h2 ydf ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h9 ye0 ff7 fs0 fc0 sc0 ls4 ws92">\ue74f\ue73d\ue750\ue751\ue74e\ue73d\ue750\ue741\ue740 \ue003\ue748\ue74b\ue743\ue748\ue745\ue74a\ue741\ue73d\ue74e \ue003\ue749\ue74b\ue740<span class="_1c blank"> </span>\ue741 \ue003\ue74a\ue751\ue749\ue73e\ue741\ue74e\ue003\ue74b\ue742 <span class="ls44 ws78">\ue003\ue74c\ue73d\ue74e\ue73d\ue749\ue741\ue750\ue741\ue74e\ue74f\ue003</span><span class="ff2 ws0"> <span class="_1d blank"></span><span class="ff7 ws93">\ue741\ue748<span class="_4 blank"></span>, \ue744\ue73d\ue752\ue745\ue74a\ue743\ue003\ue750\ue744\ue741<span class="_4 blank"> </span>\ue003\ue749\ue73d\ue754\ue745\ue749\ue751\ue749<span class="_4 blank"></span>\ue003\ue74c\ue74b\ue74f\ue74f\ue745\ue73e\ue748</span></span></div><div class="t m0 x59 h9 ye1 ff7 fs0 fc0 sc0 lsf ws94">\ue751\ue74a\ue745\ue74d\ue751\ue741\ue748\ue755\ue003\ue741\ue74f\ue750\ue745\ue749\ue73d\ue750\ue741\ue740 \ue003\ue73e\ue755 \ue003\ue750\ue744\ue741\ue003\ue740\ue73d\ue750\ue73d .<span class="ff2 ls4 ws0"> </span></div><div class="t m0 x5 h2 ye2 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h2 ye3 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h2 ye4 ff2 fs0 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h8 ye5 ff7 fs4 fc0 sc0 ls1b ws4c">7.1.5\ue003Two\u2010Factor\ue003Parameters\ue003De<span class="ls4e ws95">scribe\ue003Conditional\ue003Association<span class="ls4">\ue003</span></span></div><div class="t m0 x5 h6 ye6 ff1 fs3 fc0 sc0 ls4 ws0"> </div><div class="t m0 x5 h2 ye7 ff2 fs0 fc0 sc0 ls9 ws8"> <span class="_1e blank"> </span>This next section dives further into<span class="wsd"> the homogeneous association model (Equation </span></div><div class="t m0 x5 h2 ye8 ff2 fs0 fc0 sc0 ls44 ws11">7.5 ICDA) </div><div class="t m0 x5a hc ye9 ff2 fs0 fc0 sc0 lsb ws30">log(<span class="ff8 ls4 ws4d">\u03bc</span><span class="fs5 ls1c ws4e v1">ijk</span><span class="ls1d wsb v0">) = <span class="ff8 ls4">\u03bb</span><span class="ls2d ws0"> +<span class="ff7 ls1e">\u03bb<span class="fs6 ls4 v2">\uebdc</span></span></span></span></div><div class="t m0 x5b ha yea ff7 fs6 fc0 sc0 ls1f">\uebd1<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x5c hb yeb ff7 fs6 fc0 sc0 ls4">\uebdd</div><div class="t m0 x5d ha yea ff7 fs6 fc0 sc0 ls20">\uebd2<span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls4 v4">\uebde</span></div><div class="t m0 x1b h11 yec ff7 fs6 fc0 sc0 ls21">\uebd3<span class="ff2 fs0 ls13 v3">+<span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebdd</span></div><div class="t m0 xc ha yea ff7 fs6 fc0 sc0 ls3e ws71">\uebd1\uebd2 <span class="ff2 fs0 ls13 ws0 v3">+ <span class="ff7 ls1e">\u03bb</span></span><span class="ls31 v4">\uebdc\uebde</span></div><div class="t m0 x5e ha yea ff7 fs6 fc0 sc0 ls3e ws6c">\uebd1\uebd3 <span class="ff2 fs0 lsa ws15 v3"> + <span class="ff7 ls4">\u03bb</span></span></div><div class="t m0 x5f hb yeb ff7 fs6 fc0 sc0 ls42">\uebdd\uebde</div><div class="t m0 x60 hb yea ff7 fs6 fc0 sc0 ls40 ws6d">\uebd2\uebd3 <span class="ff2 fs0 ls4 v3">.<span class="ff5 ws0"> </span></span></div><div class="t m0 x5 h2 yed ff2 fs0 fc0 sc0 ls1 wsf">A proof below has been provided to show that<span class="ls13 ws3"> this model has the characteristic that the </span></div><div class="t m0 x5 h2 yee ff2 fs0 fc0 sc0 ls8 ws20">conditional odds ratios between any two variable<span class="ls4 ws0">s are the same at each level of the third </span></div><div class="t m0 x5 h2 yef ff2 fs0 fc0 sc0 ls3 ws25">variable. </div><div class="t m0 x61 h9 yf0 ff7 fs0 fc0 sc0 ls54 ws96">\ue72e \ue74f<span class="_1f blank"> </span>,<span class="ff2 ls4 ws0"> </span></div><div class="t m0 x33 h9 yf1 ff7 fs0 fc0 sc0 ls15">\ue748\ue74b\ue743\ue7e0</div><div class="t m0 x55 hd yf2 ff7 fs6 fc0 sc0 ls41 ws97">\uebd1\uebd2\uf23a\uebde<span class="_4 blank"> </span>\uf23b <span class="fs0 ls2e ws59 v6">\ued4cl<span class="_10 blank"></span>o<span class="_10 blank"></span>g</span></div><div class="t m0 x62 h18 yf3 ff7 fs0 fc0 sc0 ls13">\ued6c<span class="ls4 ws4d vd">\ue7e4<span class="_1 blank"></span><span class="fs6 ls55 ws98 v5">\ueb35\ueb35\uebde <span class="fs0 ls56 v6">\ue5db\ue7e4</span></span></span></div><div class="t m0 x61 hb yf4 ff7 fs6 fc0 sc0 ls57">\ueb36\ueb36\uebde</div><div class="t m0 x63 h9 yf5 ff7 fs0 fc0 sc0 ls4 ws4d">\ue7e4<span class="_1 blank"></span><span class="fs6 ls55 ws98 v5">\ueb35\ueb36\uebde <span class="fs0 ls56 v6">\ue5db\ue7e4</span></span></div><div class="t m0 x61 hb yf6 ff7 fs6 fc0 sc0 ls57">\ueb36\ueb35\uebde</div><div class="t m0 x64 h9 yf0 ff7 fs0 fc0 sc0 ls54 ws94">\ue741\ue750 \ue003\ue756<span class="_20 blank"> </span>\ued4c<span class="_21 blank"> </span>\ue747 \ue003<span class="_22 blank"> </span>\ue74b \ue003\ue750\ue744\ue73d\ue750\ue003\ue756 \ue003\ue745\ue74f \ue003\ue742\ue745\ue754\ue741\ue740</div><div class="t m0 x65 h19 yf3 ff7 fs0 fc0 sc0 ls4f">\ued70<span class="ls2e ws59 v0">\ued4cl<span class="_10 blank"></span>o<span class="_10 blank"></span>g</span></div><div class="t m0 x19 hc yf7 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="v7">\ue7e4<span class="_1 blank"></span><span class="fs6 ls57 ws99 v5">\ueb35\ueb35\uebde <span class="fs0 ls29 v9">\uf23b<span class="ls33 ws8c v7">\ued45l<span class="_1b blank"></span>o<span class="_15 blank"></span>g</span></span></span></span></div><div class="t m0 x1c h9 yf8 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="ls1e v7">\ue7e4</span><span class="fs6 ls55 ws9a v2">\ueb36\ueb36\uebde </span><span class="ls46">\uf23b<span class="ls58 ws9b v7">\ued46l<span class="_15 blank"></span>o<span class="_15 blank"></span>g</span></span></div><div class="t m0 x66 h9 yf8 ff7 fs0 fc0 sc0 ls4 ws4d">\uf23a<span class="v7">\ue7e4<span class="_1 blank"></span><span class="fs6 ls55 ws9a v5">\ueb35\ueb36\uebde <span class="fs0 ls29 v9">\uf23b<span class="ls33 ws8c v7">\ued46l<span class="_1b blank"></span>o<span class="_15 blank"></span>g<span class="_23 blank"></span>\ue003<span class="_15 blank"></span>\uf23a<span class="_1b blank"></span>\ue7e4</span></span></span></span></div><div class="t m0 x67 hd yf2 ff7 fs6 fc0 sc0 ls55 ws9c">\ueb36\ueb35\uebde <span class="fs0 ls4 v6">\uf23b<span class="ff2 ws0"> </span></span></div><div class="t m0 x68 h2 y28 ff2 fs0 fc0 sc0 ls50 ws24">10 | <span class="fc1 lsd ws7">Page</span><span class="ls4 ws0"> </span></div><div class="t m0 x5 h2 y29 ff2 fs0 fc0 sc0 ls4 ws0"> </div></div><div class="pi" data-data='{"ctm":[1.000000,0.000000,0.000000,1.000000,0.000000,0.000000]}'></div></div>
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