MODIFICATION OF ALGORITHM 



The original algorithm was modified to deal with unequal weighting of observations. 

 An extension of the work of Bartlett (1951) demonstrates the inverse of the expression 

 (X'X + mxx') may be written as 



(X ' X + mxx ' )" 1 = (X'X)" 1 - (X'X)' 1 ^xx' (X'X)- 1 (4) 



1 + mx 1 (X'X) _1 x 



where m is the weight or blowup factor associated with each observation, x. This rela- 

 tionship may be verified by showing that the product of (X'X + mxx') and the right-hand 

 side of (4) yields the identity matrix. 



The first degree Taylor Series expansion of the logistic function around a guessed 

 value of the parameter vector, 8, may be expressed as 



y* = X' S/w + e 



a n n n n 



where for the nth observation in the population 



V* = P n ~ U + exp(x n '8 )) _1 + x n ' 1/w n (5) 



P n = observed value of dependent variable (either or 1) 



w = l/P Q 



n n n 



v n J 



Var (e ) = a 2 P Q 



n n in 



* 



Let Y = WY so that the weighted Normal Equations may be written as 



X'W" 1 X3 = X'W -1 Y 



which implies that the least squares estimator of 6 is 



b = (X'W^Xr^'W^Y 



- 1 * 

 Since W is diagonal we can define W such that 



WW = W (i.e., w = l/P Q and w* = Jw ) 



n n n n n 



Then let X* = W*X and 



* * 



b = (X 'X )- 1 X'W" 1 Y (6) 



8 



