I am trying to evaluate the association between the velocity a biomarker rises in the blood and the length of stay in the hospital.
1) Value #1 at time 0
2) Value #2 at time #2
3) Length of stay in days.
*** Time #2 is different between the subject (between 4-24 hrs after time 0)
Velocity = (Value2-Value1) / (time2-0)
The velocity is dependent on the value #1 (i.e., a patient arriving with a very high value will have a low velocity even though his condition is bad, because the biomarker has a maximal value. On the other hand, a patient with a very low value #1 may have a high velocity even though his condition is good, because the marker can be produced but will never get to very high levels).
How do I estimate the association between velocity and my outcome variable ?
One of the options I thought of was to divide the value #1 into categories and then evaluate the velocity in each category (maybe with a Linear Mixed Model) however, I am not sure how to create the categories (there is no reason to assume that clinical meaningful cutpoints are equally distributed in the data).