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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 {\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.


Related
Reading
Logistic Regression, Chapter 16.

Related
Reading
Poisson Regression, Chapter 17.

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