27GLM2(1) - Logistic Regr. Model for Binomial Count Data I...

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Unformatted text preview: Logistic Regr. Model for Binomial Count Data I Bernoulli model appropriate for 0/1 response on an individual I What if data are # events out of # trials per subject? I Example: Toxicology study of the carcenogenicity of aflatoxicol. I (from Ramsey and Schaefer, The Statistical Sleuth , p 641) I Tank of trout randomly assigned to dose of aflatoxicol I 5 doses. CRD. 4 replicate tanks per dose I 86-90 trout per tank I Response is # trout with liver tumor I Could use Bernoulli model for each individual fish I But, all fish in a tank have the same covariate values (dose) I easier to analyze data in summarized form (# with tumor, # in tank) c 2011 Dept. Statistics (Iowa State University) Stat 511 section 27 1 / 23 0.05 0.10 0.15 0.20 0.25 0.0 0.2 0.4 0.6 0.8 Dose (ppm) # tumor/# in tank c 2011 Dept. Statistics (Iowa State University) Stat 511 section 27 2 / 23 I Each response is # events out of # trials. I y i Binomial ( m i , i ) , i = 1 , ..., n , where m i is a known number of trials for observation i. I i = exp ( x i ) 1 + exp ( x i ) I y 1 , ..., y n are independent. I Note: two levels of independence assumed in this model I each response, y i , is independent I trials within each each response are independent | i I y i Binomial ( m i , i ) when I y i = m i j = 1 Z ij , where Z ij Bernoulli ( i ) I And Z ij independent I May be an issue for both the Challenger and trout data sets I Binomial model assumes no flight (tank) effects I Data are same as 360 fish raised individually, or the same as one tank with 360 fish c 2011 Dept. Statistics (Iowa State University) Stat 511 section 27 3 / 23 I For now, assume Binomial model reasonable I Facts about Binomial distributions: If y i Binomial ( m i , i ) I E ( y i ) = m i i I Var ( y i ) = m i i ( 1- i ) I f ( y i ) = m i y i y i i ( 1- i ) m i- y i for y i { , ..., m i } I l ( | y ) = n i = 1 [ y i log ( i 1- i ) + m i log ( 1- i )] + const = n i = 1 [ y i x i - m i log ( 1 + exp {- x i } )] + const . I l ( | y , m , x ) for ( y 1 , m 1 , x 1 ) , ( y 2 , m 2 , x 2 ) , . .. ( y n , m n , x n ) same (apart from constant) as Bernoulli lnL: l ( | z ) for ( z ij , x ij ) = ( , x 1 ) , ( , x 1 ) , .. . ( 1 , x 1 ) , .. . , ( , x 2 ) , .. . ( 1 , x 2 ) , .. . , , ( , x n ) , .. . , ( 1 , x n ) , .. . I MLEs obtained by numerically maximizing l ( | y ) over IR p c 2011 Dept. Statistics (Iowa State University) Stat 511 section 27 4 / 23 I Results for trout data: Coefficient Estimate se z p Intercept-0.867 0.076-11.30 < 0.0001 dose 14.33 0.937 15.30 < 0.0001 I Looks impressive, but ......
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27GLM2(1) - Logistic Regr. Model for Binomial Count Data I...

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