Chapter Contents Previous Next
 Fit Analyses

## The Likelihood Function and Maximum-Likelihood Estimation

The log-likelihood function
can be expressed in terms of the mean and the dispersion parameter :
Normal

Inverse Gaussian

Gamma

Poisson
for y = 0, 1, 2, ...

Binomial

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

 Note 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( , ; 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 .

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
where gi is the link function, Vi is the variance function, and the primes denote derivatives of g and V with respect to .All values are evaluated at the current mean estimate .,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
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.

 Note 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.

 Chapter Contents Previous Next Top