Notes 4 - $ ST3241 Categorical Data Analysis I Three-way...

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& $ % ST3241 Categorical Data Analysis I Three-way Contingency Tables An Introduction: Conditional Associations 1
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& $ % Example: Death Penalty Data Death Penalty Victims’ Defendant’s Race Race Yes No White White 53 414 Black 11 37 Black White 0 16 Black 4 139 2
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& $ % 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
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& $ % 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
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& $ % Example: Death Penalty Data Death Penalty Victims’ Defendant’s Race Race Yes No White White 53 414 Black 11 37 Black White 0 16 Black 4 139 5
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& $ % Example: Death Penalty Data Death Penalty Defendant’s Race Yes No White 53 430 Black 15 176 6
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& $ % 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
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& $ % 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 0 16 0.0 Black 4 139 2.8 Total White 53 430 11.0 Black 15 176 7.9 8
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& $ % 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 0 s paradox . This result applies to quantitative as well as categorical variables. 9
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& $ % 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.
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