The purpose of this phase III, randomized, open-label, comparative, multicenter, international study is to compare the efficacy and safety of brigatinib to that of crizotinib in anaplastic lymphoma kinase (ALK)-positive locally advanced or metastatic non-small cell lung cancer (NSCLC) patients who have not previously been treated with an ALK inhibitor. Participants will be randomized in a 1:1 ratio to receive either brigatinib, 90 milligrams (mg) orally once daily (QD) for 7 days, then a 180 mg orally QD, or crizotinib, 250 mg orally twice daily (BID). Participants will receive treatment until disease progression, intolerable toxicity, consent withdrawal, or death. The total estimated duration of the study is at least 5 years, including 2 years to accrue patients, with at least 3 years for treatment and follow-up.