The reason for this study is to see if the study drug baricitinib is safe and effective in participants from 1 year to less than 18 years old with systemic juvenile idiopathic arthritis (sJIA). Participants are assigned to 1 of 2 cohorts. In cohort 1, participants will receive baricitinib or tocilizumab reference. In cohort 2, participants will receive baricitinib.