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Home Page Title Page Contents Page 1 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit STAT231: STATISTICS Paul Marriott [email protected] July 8, 2008

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Home Page Title Page Contents Page 2 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Chapter 5: Model Assessment In this chapter we will look at the problem of fitting models to data Define the likelihood function and the method of maximum likelihood estimation After fitting we use the data to evaluate how good the model is for the Problem
Home Page Title Page Contents Page 3 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Association across categories One assumption that is often used in modelling is that data is independent. If the data is counts of categories then there is a simple χ 2 -test which we can use to test the hypothesis of independence. For simplicity suppose we are comparing two binary variates X and Y . Independence means that knowing the value of x for a ran- domly selected unit in the population does not affect the prob- ability that y equals 1 . In terms of probabilities we would want to check P ( Y = 1 | X = 1) = P ( Y = 1) .

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Home Page Title Page Contents Page 4 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Association across categories Y/X X = 0 X = 1 Total Y = 0 a b a + b Y = 1 c d c + d Total a + c b + d n = a + b + c + d Table 1: Observed Frequencies for paired categorical variables If X and Y were independent we would expect that the num- ber of counts of units where X = 0 & Y = 0 would equal n × P ( X = 0) × P ( Y = 0) . Could be estimated via ( a + b + c + d ) a + b ( a + b + c + d ) a + c ( a + b + c + d ) = ( a + b )( a + c ) a + b + c + d
Home Page Title Page Contents Page 5 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Association across categories Y/X X = 0 X = 1 Total Y = 0 ( a + b )( a + c ) n ( a + b )( b + d ) n a + b Y = 1 ( c + d )( a + c ) n ( c + d )( b + d ) n c + d Total a + c b + d a + b + c + d Table 2: Expected Frequencies for paired categorical variables when independence is assumed

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Home Page Title Page Contents Page 6 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Association across categories Denote the four cell counts in observed table by o i Denote the four counts in expected table by e i Calculate the following statistic. s = i ( o i - e i ) 2 e i . Under independence the random variable version of this has an approximate χ 2 1 distribution. Thus we can calculate a p -value for the hypothesis of inde- pendence between X and Y via P ( S s ) where S χ 2 1 .
Home Page Title Page Contents Page 7 of 16 Go Back Full Screen Close Quit First Prev Next Last Go Back Full Screen Close Quit Association across categories

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