lecture9-BW.pdf - Biostatistics 695 Generalized Linear Models Lecture 9 CDA Chap 4 Lecture 9 \u2013 p.1\/32 Components of a GLM 1 A random component \u2022

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Biostatistics 695 Generalized Linear Models Lecture 9, CDA Chap 4 Lecture 9 – p.1/32
Components of a GLM 1. A random component consists of a response variable Y w/independent observations ( y 1 , . . . , y n ) from a distribution in the exponential family of distributions. 2. A systematic component specifies explanatory variables used in a linear predictor function. 3. A link function is a function of E( Y ) that equates to the linear function of explanatory variables it links the random component, through a function of its mean, to the systematic component. Lecture 9 – p.2/32
The Random Component of a GLM Suppose we have n independent observations ( y 1 , . . . , y n ) with pmf or density function for y i given by f ( y i ; θ i , φ ) = exp y i θ i b ( θ i ) a ( φ ) + c ( y i , φ ) (1) This is called the exponential dispersion family of distributions. φ is called the dispersion parameter . θ i is called the natural parameter . Note that only independence in observations is required, the observations need not be identically distributed...each observation can have its own natural parameter, θ i . Lecture 9 – p.3/32
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Lecture 9 – p.6/32
Exponential Family of Distributions The binomial and Poisson distributions do not have a dispersion parameter φ . True of all one parameter family members of the exponential family. An alternative characterization for one parameter members is the natural exponential family . f ( y i ; θ i ) = a ( θ i ) b ( y i ) exp [ y i Q ( θ i )] (2) The correspondence between (2) and (1) is Q ( θ ) = θ/a ( φ ) a ( θ ) = exp [ b ( θ ) /a ( φ )] b ( y ) = exp [ c ( y ; φ )] Q ( θ ) called the natural parameter. Lecture 9 – p.7/32
The Systematic Component of a GLM Let x i = ( x i 1 , x i 2 , . . . , x ip ) T be a column vector of observations for explanatory variables from observation i . Let β = ( β 1 , . . . , β p ) T denote the column vector of model parameters. The systematic component of a GLM relates parameters, η = ( η 1 , . . . , η n ) T to x i through a linear function: η i = x T i β , i = 1 , . . . , n. or in matrix notation: η = X β where X = ( x 1 , . . . , x n ) T is an n × p matrix of observations of the explanatory variables.