## Goodness of Fit

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

where is the log-likelihood under the model
and is
the log-likelihood under the maximum achievable (saturated) model.
For generalized linear models,
the scaled deviance can be expressed as

where is the residual deviance
for the model and is the sum of individual deviance contributions.

The forms of the individual deviance contributions,
*d*_{i}, are

**Normal**

**Inverse Gaussian**

**Gamma**

**Poisson**

**Binomial**

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

For a binomial distribution with *m*_{i}
trials in the *i*th observation, the Pearson
statistic is

For other distributions, the Pearson
statistic is

The scaled Pearson statistic is / .Either the mean deviance or
the mean Pearson statistic can be used to estimate the dispersion parameter .The approximation is usually quite accurate for
the differences of deviances for nested models (McCullagh and Nelder 1989).

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