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help for ^williams^                                       
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Logistic Regression Using William's Procedure
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 ^williams^ numerator denominator [varlist] [^if^ exp] [^in^ range] 
       [^, ef^orm ^le^vel(#) ^lt^olera ^nolo^g ^res^id

 eform         : provides estimates in terms of odds ratios

 level         : allows specification of confidence interval other than
                   the default of 95

 ltolera       : allows change in default dispersion convergence of .0001
 
 nolog         : suppresses display of iteration

 resid         : the following residuals are created:           
                   _mu     : mu, fitted value 
                   _lp     : eta, linear predictor
                   _Pear   : Pearson residuals
                   _Dev    : Deviance residuals
                   _Hat    : hat matrix diagonal
                   _Anscom : Anscombe residuals
                   _StandP : Standardized Pearson residuals
                   _StandD : Standardized deviance residuals

      Be certain that you have sufficient variable space to add 8 
      additional variables to the data in memory. If not, then use
      ^set maxvar=x^ command to make adjustment. 
      
The final weighted logistic regression displays p-values based on
t-statistics per recommendation of D. Collett (1991), Modeling Binary Data,
Chapman & Hall, p 196.

The William's procedure iteratively reduces the Chi2-based dispersion to 
approximately 1.0. An extra parameter, called ^phi^, is used to scale the  
variance function. It changes with each iteration. The value of phi which 
results in a chi2-based dispersion of 1.0 is then used to weight a standard
grouped logistic regression. The weighting formula is: 1/(1+(m-1)*phi) where
m is the binomial denominator.

CHI2 = -2(LL0 - LLf): p-value evaluates the null hypothesis that all
coefficients in the model, except the intercept, equal 0. Degree of freedom
is the number of predictors. CHI2 is to be distinguished from the traditional
GLM Chi2 which is defined as SUM{(y-mu)^^2/V} where V=variance.

Pseudo R2 = 1-LLf/LL0: This is standard Stata output; however there are 
several alternative R2 formulations in common use.

For additional information contact: 
Joseph Hilbe, email: hilbe@@asu.edu

