Methods and formulas for the goodness-of-fit statistics in Fit Poisson Model

Deviance measures the discrepancy between the current model and the full model. The full model is the model that has n parameters, one parameter per observation. The full model maximizes the log-likelihood function. The full model provides a point of comparison for models with fewer than n parameters. Comparisons to the full model use the scaled deviance.

The contribution to the scaled deviance from each individual data point depends on the model.

Model	Deviance
Binomial
Poisson

The degrees of freedom for the test depend on the sample size and the number of terms in the model:

Notation

Term	Description
Lf	the log-likelihood for the full model
Lc	the log-likelihood of the model with a subset of terms from the full model
yi	the number of events for the ith row in the data
	the estimated mean response for the ith row in the data
mi	the number of trials for the ith row in the data
n	the number of rows in the data
p	the regression degrees of freedom

The generalized Pearson chi-square statistic assesses the relative difference between the observed and fitted values.

The degrees of freedom for the test depend on the sample size and the number of terms in the model. The Pearson statistic has an exact chi-square distribution for normal data. For non-normal data, like the binomial distribution and the Poisson distribution, the statistic approaches the distribution asymptotically.

Notation

Term	Description
n	the number of rows in the data
p	the regression degrees of freedom
yi	the response value for the ith factor/covariate pattern
	the estimated mean response of the ith row
V(·)	the variance function for the model, defined below

The variance function depends on the model:

Model	Variance function
Binomial
Poisson