Goodness of Fit

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, d_i, 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 m_i 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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