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Notes 2 - $ ST3241 Categorical Data Analysis I Two-way...

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& $ % ST3241 Categorical Data Analysis I Two-way Contingency Tables 2 × 2 Tables, Relative Risks and Odds Ratios 1
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& $ % What Is A Contingency Table (p.16) Suppose X and Y are two categorical variables X has I categories Y has J categories Display the IJ possible combinations of outcomes in a rectangular table having I rows for the categories of X and J columns for the categories of Y . A table of this form in which the cells contain frequency counts of outcomes is called a contingency table . 2
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& $ % Example: Belief In Afterlife Data (p.18) Belief in Afterlife Gender Yes No or Undecided Female 435 147 Male 375 134 A contingency table that cross classifies two variables is called a two - way table . A table which cross classifies three variables is called a three - way table . A two-way table having I rows and J columns is called an I × J table. 3
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& $ % Some Notations, Definitions · · · π ij = P [ X = i, Y = j ] = probability that ( X, Y ) falls in the cell in row i and column j . The probabilities { π ij } form the joint distribution of X and Y . Note that, I X i =1 J X j =1 π ij = 1 4
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& $ % Marginal Distributions (p.17) The marginal distribution of X is π i + , which is obtained by the row sums, that is, π i + = J X j =1 π ij The marginal distribution of Y is π + j , which is obtained by the column sums, that is π + j = I X i =1 π ij For example, for a 2 × 2 table π 1+ = π 11 + π 12 , π +1 = π 11 + π 21 5
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& $ % Notations For The Data Cell counts are denoted by { n ij } , with n = I X i =1 J X j =1 n ij Cell proportions are p ij = n ij n The marginal frequencies are row totals { n i + } and column totals { n + j } 6
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& $ % Example Belief in Afterlife Gender Yes No or Undecided Total Female n 11 =435 n 12 =147 n 1+ =582 Male n 21 =375 n 22 =134 n 2+ =509 Total n +1 =810 n +2 =281 n =1091 7
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& $ % Example: Sample Proportions Belief in Afterlife Gender Yes No or Undecided Total Female p 11 =0.398 p 12 =0.135 p 1+ =0.533 Male p 21 =0.344 p 22 =0.123 p 2+ =0.467 Total p +1 =0.742 p +2 =0.258 p =1.00 8
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& $ % Conditional Probabilities Let Y be a response variable and X be an explanatory variable. It is informative to construct separate probability distributions for Y at each level of X . Such a distribution consists of conditional probabilities for Y given the level of X and is called a conditional distribution . 9
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& $ % Example: Sample Conditional Distributions For females, Proportion of yes responses = 0.747 Proportion of no responses = 0.253 For males, Proportion of yes responses = 0.737 Proportion of no responses = 0.263 10
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& $ % Independence Is the belief in afterlife is independent of gender?
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