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stat400lec37 - Statistics 400 Section 8.5 Contingency...

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Statistics 400 Section 8.5 Contingency Tables h×k Contingency Table Group 1 2 k 1 Y11 Y21 Yk1 2 Y12 Y22 Yk2 h Y1h Y2h Ykh 2×k contingency table Group 1 2 k 1 Y11 Y21 Yk1 2 Y12 Y22 Yk2 Test whether two multinomial distributions are equal Two sets of independent random variables have the following multinomial distributions. The joint pmf of 11 21 1 , ,..., k Y Y Y is 1 11 21 1 11 21 1 11 21 1 11 21 1 ! ( , ,..., ) ... ! !... ! k y y y k k k n f y y y p p p y y y = The joint pmf of 12 22 2 , ,..., k Y Y Y is 2 12 22 2 12 22 2 12 22 2 12 22 2 ! ( , ,..., ) ... ! !... ! k y y y k k k n f y y y p p p y y y = 2 2 2 11 1 11 21 1 21 1 1 1 1 11 1 21 1 1 ( ) ( ) ( ) ... k k k Y n p Y n p Y n p n p n p n p - - - + + + + 2 2 2 12 2 12 22 2 22 2 2 2 2 12 2 22 2 2 ( ) ( ) ( ) ... k k k Y n p Y n p Y n p n p n p n p - - - + + + ~ Ping Ma Lecture 37 Fall 2005 - 1 -
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Test the hypothesis H0: p i1 =p i2 i=1,2,…,k Ha: at least one of p i1 ≠p i2 Test statistic 2 2 2 11 1 11 12 1 2 21 1 21 22 1 2 1 1 1 2 1 2 1 1 11 12 1 2 1 21 22 1 2 1 1 2 1 2 [ ( ) /( )] [ ( ) /( )] [ ( ) /( )] ... ( ) /( ) ( ) /( ) ( ) /( ) k k k k k k Y n Y Y n n Y n Y Y n n Y n Y Y n n Q n Y Y n n n Y Y n n n Y Y n n - - + + - + + - + + = + + + + + + + + + 2 2 2 12 1 11 12 1 2 22 1 21 22 1 2 2 1 1 2 1 2 1 11 12 1 2 1 21
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