I need some explanation of the results I got my multi-state Cox model. I looked at the absolute risk or the proportion of people in a state at a particular age for a time-dependent covariate which is whether the patients having long-stay hospital admissions are more at risk to be institutionalised? The curves from the Ps(t) using Aalen-Johansen shows that there was a greater proportion of people with Long-stay admission in the institutionalised state than people without long-stay admission. But the results from the multi-state Cox model i.e, the HR are lowered for patients with long-stay admission than without long-stay admission.

Does these results make any sense? I thought that if the proportion in a state is higher then the HR should also be higher.

You did not provide much information on your analysis. Did you include any other covariates/variables in the model(s)? If so, the association of these other variables with your outcome could well explain your results. (That is, if some of the other covariates for example work as confounders of the relation between length of stay and hospitalization, including them would be expected to change the estimate of the association between length of stay and hospitalization).