# Notes 8 - ’ & ST3241 Categorical Data Analysis I...

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Unformatted text preview: ’ & ST3241 Categorical Data Analysis I Loglinear Models 2 × 2 Models For Contingency Tables 1 ’ & Two-way Tables • Consider an I × J contingency table that crossclassifies a sample of n subjects on two categorical responses. • Let Y ij be the observed cell frequency and μ ij be the expected cell frequency of the ( i,j )-th cell. • Then we assume that the cell counts Y ij are independent having Poisson( μ ij ) distribution. • Note that, if π ij is the cell probability, then μ ij = nπ ij . 2 ’ & Independence Model • Under statistical independence of the row and column classifications, π ij = π i + π + j and hence μ ij = nπ i + π + j . • Denote the row variable by X and the column variable by Y . • The formula expressing independence is multiplicative. Thus, log μ ij is additive log μ ij = λ + λ X i + λ Y j for a row effect λ X i and a column effect λ Y j . • This is the loglinear model of independence. • The null hypothesis of independence between two categorical variables is simply the hypothesis that this model holds. 3 ’ & Example: Belief in Afterlife Observed Fitted Log Fitted Frequency Value Value 435 147 432.10 149.90 6.069 5.010 375 134 377.90 131.10 5.935 4.876 4 ’ & Example: Belief in Afterlife Parameter Set 1 Set 2 Set 3 λ 4.876 6.069 5.472 λ X 1 0.134 0.0067 λ X 2-0.134-0.067 λ Y 1 1.059 0.529 λ Y 2-1.059-0.529 5 ’ & Some SAS Codes data after; input female \$ belief \$ count; datalines; Female Yes 435 Female No 147 Male Yes 375 Male No 134 ; run; 6 ’ & SAS Codes proc genmod data=after order=data; class belief female; model count= female belief/ dist=poisson; output out=temp p=predict; run; proc print data=temp; var female belief count predict; run; 7 ’ & Partial Output Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 1 0.1620 0.1620 Scaled Deviance 1 0.1620 0.1620 Pearson Chi-Square 1 0.1621 0.1621 Scaled Pearson X2 1 0.1621 0.1621 Log Likelihood 5164.1959 Algorithm converged. 8 ’ & Partial Output Analysis Of Parameter Estimates StandardWald 95% Conf Chi- Parameter DFEstimate Error Limits SquarePr > ChiSq Intercept 1 4.8760 0.06794.7429 5.00905160.87 < .0001 female Female 1 0.1340 0.06070.0151 0.2530 4.88 0.0272 female Male 0 0.0000 0.00000.0000 0.0000-- belief Yes 1 1.0587 0.06920.9230 1.1944 233.83 < .0001 belief No 0 0.0000 0.00000.0000 0.0000-- Scale 0 1.0000 0.00001.0000 1.0000 9 ’ & Partial Output Obs female belief count predict 1 Female Yes 435 432.099 2 Female No 147 149.901 3 Male Yes 375 377.901 4 Male No 134 131.099 10 ’ & Interpretations of Parameters • For I × J tables, loglinear models treat the N = IJ cell counts as N independent observations of a Poisson random component....
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## This note was uploaded on 11/20/2011 for the course STATISTICS ST3241 taught by Professor Manwai's during the Spring '11 term at National University of Singapore.

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Notes 8 - ’ & ST3241 Categorical Data Analysis I...

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