I've read some works on the Hosmer-Lemeshow test (HLT) but still I'm uncertain what is the exact null hypothesis for the HLT. I'd like to formulate a nullhypothesis as precisely as possible.
I think the fitted model is significant says everything and nothing.
The observed frequencies equal the estimated frequencies is more acccurate for a pearson chi-square test and misses the logistic model.
I prefer this one:
$$\hat{\pi}_{k_j} = \pi_{{k_j}_0} \; (j =1, \ldots, c_k) (k=1, \ldots, g)$$
where
$\hat{\pi}_{k_j}$ is the from the logistic model estimated probability
$\pi_{{k_j}_0}$ is the real but unknown probalbility
$c_k$ is the number of covariate patterns in the $k$th grouop
$g$ is the number of groups
What are your suggestions?
Edit
As I think now, the proper null hypothesis is in fact The model fits and the alternative is the model does not fit. (Hosmer, Lemeshow: Applied logistic Regression, Kuß: Globale Anpassungstests im logistischen Regressionsmodell)
That means the HL-test evaluates the fitted model.
