Quasi-Likelihood Functions

Fit Analyses

Quasi-Likelihood Functions

For binomial and Poisson distributions, the scale parameter has a value of 1. The variance of Y is ${\rm{Var}(y) = \mu(1-\mu) / m}$ for the binomial distribution and ${\rm{Var}(y) = \mu}$ for the Poisson distribution. Overdispersion occurs when the variance of Y exceeds the Var(y) above. That is, the variance of Y is ${ {\sigma}^2 V(\mu)}$ , where ${\sigma}$ >1.

With overdispersion, methods based on quasi-likelihood can be used to estimate the parameters $\beta$ and ${\sigma}$ .A quasi-likelihood function

$Q(\mu ; y) = \int_{y}^{\mu}{\frac{y-t}{{\sigma}^2 V(t)} dt}$

is specified by its associated variance function.

SAS/INSIGHT software includes the quasi-likelihoods associated with the variance functions $V(\mu) = 1$ , $\mu$ , $\mu^2$ , $\mu^3$ , and $\mu(1-\mu$ ). The associated distributions (with the same variance function), the quasi-likelihoods ${Q(\mu ; y)}$ , the canonical links ${g(\mu)}$ , and the scale parameters ${\sigma}$ and ${\nu}$ for these variance functions are

$V(\mu) = 1$: Normal

${ {\sigma}^2 Q(\mu ; y) = -\frac{1}2 (y-\mu)^2} {for -\infty\lt y\lt\infty}$

$g(\mu) = \mu$

${{\sigma} = \sqrt{\phi}}$
${V(\mu) = \mu}$: Poisson

${ {\sigma}^2 Q(\mu ; y) = y \log(\mu)-\mu} {for \mu\gt, y\ge 0}$

${g(\mu) = \log \mu}$

${{\sigma} = \sqrt{\phi}}$
${V(\mu) = \mu^2}$: Gamma

${ {\sigma}^2 Q(\mu ; y) = - y / \mu - \log(\mu)} {for \mu\gt, y\ge 0}$

$g(\mu) = \mu^{-1}$

${{\nu} = \phi^{-1}}$
$V(\mu) = \mu^3$: Inverse Gaussian

${ {\sigma}^2 Q(\mu ; y) = - y / (2 \mu^2) + 1 / \mu} {for \mu\gt, y\ge 0}$

$g(\mu) = \mu^{-2}$

${{\sigma} = \sqrt{\phi}}$
${V(\mu) = \mu (1-\mu)}$: Binomial

${ {\sigma}^2 Q(\mu ; y) = r \log(\mu) + (m-r) \log(1-\mu) }$

for $0\lt\mu\lt 1$ , y= r/m, r = 0, 1, 2, ... , m

$g(\mu) = \log(\frac{\mu}{1-\mu})$

${{\sigma} = \sqrt{\phi}}$

SAS/INSIGHT software uses the mean deviance, the mean Pearson $\chi^2$ , or the value in the Constant entry field to estimate the dispersion parameter $\phi$ .The conventional estimate of $\phi$ is the mean Pearson $\chi^2$ 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 $\phi$ , but $\phi$ 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.