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Unformatted text preview: Home Page Title Page Contents JJ II J I 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 Home Page Title Page Contents JJ II J I 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 JJ II J I 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) . Home Page Title Page Contents JJ II J I 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 JJ II J I 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 Home Page Title Page Contents JJ II J I 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....
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This note was uploaded on 04/18/2010 for the course STAT 231 taught by Professor Cantremember during the Spring '08 term at Waterloo.

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