1. I believe that judging the whole model-data fit through a statistical index is relatively meaningless for users because it does not provide any information of how to refine their measurement.
2. Are these indices included in the study independent with data size? If yes, then reporting a result of hypothesis testing without its effect size is insufficient.
3. Only the case of dichotomous responses was discussed at present. If one want to develop a more general index to cover polytomous outcomes, an unimaginable multivariate distribution is needed to derive mathematically.