Used in multilevel modelling, it is a statistic that follows the chi-square distribution, which is used to compare the fit of two models after one or more parameters have been added or deleted. The lower deviance, the better the fit. Many models can fit the data, and so the usual tactic is to obtain deviance for the full model and for a nested model, excluding some effects. A chi-square difference test can then be used to check whether the full model differs significantly from the fit of the nested model. If there is no difference, then the nested model is preferred as it is more parsimonious.
See Multilevel modelling (MLM)