Using case-control, cohort, and case-crossover analysis, the investigators found that tramadol increased the risk of hospitalization for hypoglycemia by more than three-fold, with the risk particularly elevated in the first 30 days of treatment.
In my opinion, "with" is just a preposition. And after preposition we only need a noun or noun phrase. But in this sentence, it seems that "with" has the function of conjunction. Is it true? would you explain it for me, please!