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A Generalized Linear Model for Bernoulli Response Data

A Generalized Linear Model forBernoulli Response DataCopyrightc 2017 Dan Nettleton (Iowa State University)Statistics 5101 / 46 Consider the Gauss-Markov Linear Model with normalerrors:y=X + , N(0, 2I).Another way to write this Model is8i=1,..,n,yi N( i, 2), i=x0i ,andy1,..,ynare 2017 Dan Nettleton (Iowa State University)Statistics 5102 / 46 This is a special case of what is known as ageneralized Linear is another special case:8i=1,..,n,yi Bernoulli ( i), i=exp(x0i )1+exp(x0i ),andy1,..,ynare 2017 Dan Nettleton (Iowa State University)Statistics 5103 / 46In each example, all responses are independent, andeach Response is a draw from one type of distributionwhose parameters may depend on explanatoryvariables through a Linear predictorx0i .Copyrightc 2017 Dan Nettleton (Iowa State University)Statistics 5104 / 46 The second Model , for the case of a binary Response ,is often called a logistic regression responses are common (success/failure,survive/die, good customer/bad customer, win/lose,etc.)

For Generalized Linear Models, Fisher’s Scoring Method is typically used to obtain an MLE for , denoted as ˆ. Fisher’s Scoring Method is a variation of the Newton-Raphson algorithm in which the Hessian matrix (matrix of second partial derivatives) is replaced by its expected value (-Fisher Information matrix).

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