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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!

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  • "With" is a preposition and preps freely take clausal complements, both finite and non-finite (like yours). Here, the PP is an adjunct (grammatically optional).
    – user36764
    Commented Aug 11, 2016 at 11:39

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Here, "with" means "accompanied by". There is an implied "being" between "risk" and "particularly elevated". That is, the sentence is equivalent to:

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, accompanied by the risk being particularly elevated in the first 30 days of treatment.

Though, of course, the original is more idiomatic and flows better.

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