I'm working with an organization that is using JMP in their analysis, and I can't tell from the description in JMP's help files if the test for goodness of fit in their logistic regression is the Hosmer-Lemeshow test. If it makes a difference, my data set has only one predictor variable, so we aren't considering complex models.
The next questions that JMP addresses are whether there is enough information using the variables in the current model or whether more complex terms need to be added. The Lack of Fit test, sometimes called a Goodness of Fit test, provides this information. It calculates a pure-error negative log-likelihood by constructing categories for every combination of the regressor values in the data (Saturated line in the Lack Of Fit table), and it tests whether this log-likelihood is significantly better than the Fitted model.
The Saturated degrees of freedom is m–1, where m is the number of unique populations. The Fitted degrees of freedom is the number of parameters not including the intercept. For the Ingots example, these are 18 and 2 DF, respectively. The Lack of Fit DF is the difference between the Saturated and Fitted models, in this case 18–2=16.
The Lack of Fit table lists the negative log-likelihood for error due to Lack of Fit, error in a Saturated model (pure error), and the total error in the Fitted model. Chi-square statistics test for lack of fit.
(I'm an engineer, not a statistician, the saturated line parts are confusing me).