# Probability and Statistics for Engineering and the Sciences (with CD-ROM and InfoTrac )

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Stat 312: Lecture 26 Contingency Tables Moo K. Chung [email protected] May 6, 2003 Concepts 1. Two-way contingency table: suppose there are I populations and each population is classified into J categories. Let n ij be the number of observed elements in population i which fall into category j . We denote n j = I i =1 n ij and n i = J j =1 n ij . 2. Testing for homogeneity . Let p ij be the pro- portion of the elements in population i which fall into category j . Note that J j =1 p ij = 1 . We want to test if the proportions in the differ- ent categories are the same for all populations, i.e. H 0 : p 1 j = p 2 j = · · · = p Ij for all j. 3. The expected number of element N ij in pop- ulation i which falls into category j . N ij = n i p ij . Under the null hypothesis, p ij = · · · = p Ij = p j . So N ij = n i p j . We estimate p j by pooling I samples together. ˆ p j = n j /n . 4. Test statistic: χ 2 = X i,j ( n ij - N ij ) 2 N ij χ 2 ( I - 1)( J - 1) .