## Quasi-Likelihood Functions

For binomial and Poisson distributions,
the scale parameter has a value of 1.
The variance of **Y** is
for the binomial distribution and
for the Poisson distribution.
*Overdispersion* occurs when the variance
of **Y** exceeds the Var(y) above.
That is, the variance of **Y** is
, where >1.
With overdispersion, methods based on quasi-likelihood
can be used to estimate the parameters and .A quasi-likelihood function

is specified by its associated variance function.

SAS/INSIGHT software includes the quasi-likelihoods
associated with the variance functions
, , , , and ).
The associated distributions (with the same variance function),
the quasi-likelihoods , the canonical links
, and the scale parameters and for these variance functions are

**Normal**

**Poisson**

**Gamma**

**Inverse Gaussian**

**Binomial**

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

SAS/INSIGHT software uses the mean deviance, the mean
Pearson , or the value in the **Constant**
entry field to estimate the dispersion parameter .The conventional estimate of is the
mean Pearson statistic.
Maximum quasi-likelihood estimation is similar to ordinary
maximum-likelihood estimation and has the same parameter
estimates as the distribution with the same variance function.
These estimates are not affected by the dispersion
parameter , but is used in the
variance-covariance matrix of the parameter estimates.
However, the likelihood-ratio based statistics, such as
**Type I (LR)**, **Type III (LR)**, and **C.I.(LR) for Parameters**
tables, are not produced in the analysis.

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