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 Fit Analyses

## 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.