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Fit Analyses

The Likelihood Function and Maximum-Likelihood Estimation

The log-likelihood function
l(\theta , \phi ; y) = \log f(y ; \theta, \phi) =
 \frac{y \theta-b(\theta)}{a(\phi)} + c(y , \phi)
can be expressed in terms of the mean \muand the dispersion parameter \phi:
{l(\mu , \phi ; y) = -\frac{1}2 \log(\phi)
-\frac{1}{2\phi} (y-\mu)^2} \hfil {for -\infty\lt y\lt\infty}


Inverse Gaussian
{l(\mu , \phi ; y) = - \log( y^3 \phi)
-\frac{(y-\mu)^2}{2 y \mu^2 \phi} }  {for y\gt}


{l(\mu , \phi ; y) = -\log(y {\Gamma}(\frac{1}{\phi})) +
 -\frac{y}{\mu\phi}}  {for y\gt}


{l(\mu , \phi ; y) = y \log(\mu)-\mu }   for y = 0, 1, 2, ...


{l(\mu , \phi ; y) = r \log(\mu) + (m-r) \log(1-\mu) }

for y=r/m, r=0, 1, 2,..., m


Some terms in the density function have been dropped in the log-likelihood function since they do not affect the estimation of the mean and scale parameters.

SAS/INSIGHT software uses a ridge stabilized Newton-Raphson algorithm to maximize the log-likelihood function l(\mu , \phi ; y) with respect to the regression parameters. On the rth iteration, the algorithm updates the parameter vector b by
b(r) = b(r-1) - H-1(r-1) u(r-1)
where H is the Hessian matrix and u is the gradient vector, both evaluated at {{\beta}= b_{(r-1)}}.
H = ( h_{jk} ) =
 ( \frac{\partial^2l}{\partial \beta_{j}
 \partial \beta_{k}} )
u = ( u_{j} ) = ( 
 \frac{\partial l}{\partial \beta_{j} } ).

The Hessian matrix H can be expressed as
H = - X' Wo X
where X is the design matrix, Wo is a diagonal matrix with ith diagonal element
w_{oi} = w_{ei} +
 ( y_{i}- \mu_{i})
 \frac{V_{i}{g_{i}''} +
 {V_{i}'} {g_{i}'}}{V^2_{i}
 a_{i}(\phi) }
w_{ei} = E( w_{oi}) =
 \frac{1}{a_{i}(\phi) V_{i}
 ({g_{i}'})^2 }
where gi is the link function, Vi is the variance function, and the primes denote derivatives of g and V with respect to \mu.All values are evaluated at the current mean estimate { \mu_{i}}.{ a_{i}(\phi) = \phi / w_{i}},where wi is the prior weight for the ith observation.

SAS/INSIGHT software uses either the full Hessian matrix H = - X' Wo X or the Fisher's scoring method in the maximum-likelihood estimation. In the Fisher's scoring method, Wo is replaced by its expected value We with ith element wei.
H = X' We X

The estimated variance-covariance matrix of the parameter estimates is
\hat{{{\Sigma}}} = - H^{-1}
where H is the Hessian matrix evaluated at the model parameter estimates. The estimated correlation matrix of the parameter estimates is derived by scaling the estimated variance-covariance matrix to 1 on the diagonal.

A warning message appears when the specified model fails to converge. The output tables, graphs, and variables are based on the results from the last iteration.

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.