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Fit Analyses

Goodness of Fit

The log-likelihood can be expressed in terms of the mean parameter \mu and the log-likelihood-ratio statistic is the scaled deviance

D^{{\ast}}(y; \hat{\mu}) = -2 (l(\hat{\mu}; y) - l( \hat{\mu}_{max}; y))
where {l(\hat{\mu}; y)} is the log-likelihood under the model and {l( \hat{\mu}_{max}; y)} is the log-likelihood under the maximum achievable (saturated) model.

For generalized linear models, the scaled deviance can be expressed as

D^{{\ast}}(y ; \hat{ \mu}) = \frac{1}{\phi} D (y ; \hat{ \mu})
where {D (y ; \hat{ \mu})} is the residual deviance for the model and is the sum of individual deviance contributions.

The forms of the individual deviance contributions, di, are

Normal
{ (y-\hat{\mu})^2}

Inverse Gaussian
{ (y-\hat{\mu})^2 / ( \hat{\mu}^2 y)}

Gamma
{-2 \log(y/\hat{\mu}) + 2 (y-\hat{\mu})/\hat{\mu}}

Poisson
{2 y \log(y/\hat{\mu}) - 2 (y-\hat{\mu})}

Binomial
{2 ( r \log(y/\hat{\mu}) + (m-r) \log((1-y)/(1-\hat{\mu}))}

where y=r/m, r is the number of successes in m trials.


For a binomial distribution with mi trials in the ith observation, the Pearson \chi^2 statistic is
\chi^2 = \sum_{i=1}^n{m_{i} \frac{( y_{i}- \mu_{i})^2}{V( \mu_{i}) }}

For other distributions, the Pearson \chi^2 statistic is
\chi^2 = \sum_{i=1}^n{\frac{( y_{i}- \mu_{i})^2}{V( \mu_{i}) }}

The scaled Pearson \chi^2 statistic is \chi^2 / \phi.Either the mean deviance {D (y; \hat{\mu}) / (n-p)} or the mean Pearson \chi^2 statistic { \chi^2 / (n-p)}can be used to estimate the dispersion parameter \phi.The \chi^2 approximation is usually quite accurate for the differences of deviances for nested models (McCullagh and Nelder 1989).

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