When　establishing　a　treatment　in　clinical　trials,　it　is　important　to　evaluate　both　effectiveness　and　toxicity.　In　phase　II　clinical　trials,　multinomial　data　are　collected　in　m-stage　designs,　especially　in　two-stage　(m　=　2)　design.　Exact　tests　on　two　proportions,　p(r)　for　the　response　rate　and　p(t)　for　the　nontoxicity　rate,　should　be　employed　due　to　limited　sample　sizes.　However,　existing　tests　use　certain　parameter　configurations　at　the　boundary　of　null　hypothesis　space　to　determine　rejection　regions　without　showing　that　the　maximum　Type　I　error　rate　is　achieved　at　the　boundary　of　null　hypothesis.　In　this　paper,　we　show　that　the　power　function　for　each　test　in　a　large　family　of　tests　is　nondecreasing　in　both　p(r)　and　p(t);　identify　the　parameter　configurations　at　which　the　maximum　Type　I　error　rate　and　the　minimum　power　are　achieved　and　derive　level-alpha　tests;　provide　optimal　two-stage　designs　with　the　least　expected　total　sample　size　and　the　optimization　algorithm;　and　extend　the　results　to　the　case　of　m　＞　2.　Some　R-codes　are　given　in　the　Supporting　Information.