Notes 4 - ’ & ST3241 Categorical Data Analysis I...

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Unformatted text preview: ’ & ST3241 Categorical Data Analysis I Three-way Contingency Tables An Introduction: Conditional Associations 1 ’ & Example: Death Penalty Data Death Penalty Victims’ Defendant’s Race Race Yes No White White 53 414 Black 11 37 Black White 16 Black 4 139 2 ’ & Objectives • To find association between X and Y by controlling other covariates that can influence the association. • We study the effect of X on Y by fixing such covariates constant. • In other words, study the association between X and Y given the levels of Z . 3 ’ & Partial Tables • Two-way tables between X and Y at separate levels of Z . • The two-way contingency table obtained by combining the partial tables is called the X- Y marginal table. • Each cell count in the marginal table is a sum of counts from the same cell location in the partial tables. • The marginal table, rather than controlling Z , ignores it and does not contain any information about Z . 4 ’ & Example: Death Penalty Data Death Penalty Victims’ Defendant’s Race Race Yes No White White 53 414 Black 11 37 Black White 16 Black 4 139 5 ’ & Example: Death Penalty Data Death Penalty Defendant’s Race Yes No White 53 430 Black 15 176 6 ’ & Notes • The associations in partial tables are called conditional associations . • Conditional associations in partial tables can be quite different from associations in marginal tables. 7 ’ & Example: Death Penalty Data Death Penalty Victims’ Defendant’s Percentage Race Race Yes No Yes White White 53 414 11.3 Black 11 37 22.9 Black White 16 0.0 Black 4 139 2.8 Total White 53 430 11.0 Black 15 176 7.9 8 ’ & Simpson’s Paradox • This death penalty data is an example of Simpson’s paradox. • The result that a marginal association can have different direction from the conditional associations is called Simpson s paradox . • This result applies to quantitative as well as categorical variables. 9 ’ & Odds Ratios • Consider 2 × 2 × K tables, where K denotes the number of levels of a control variable Z . • Let { n ijk } denote the observed frequencies and let { μ ijk } denote their expected frequencies. • Within a fixed level k of Z , θ XY ( k ) = μ 11 k μ 22 k μ 12 k μ 21 k describes conditional X- Y association. • We refer to them as the X- Y conditional odds ratios ....
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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 4 - ’ & ST3241 Categorical Data Analysis I...

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