To　assess　the　effectiveness　of　a　treatment　in　phase　II　clinical　trial　for　cancer　study,　an　adaptive　multi-stage　design,　especially　a　two-stage　one,　is　commonly　used.　This　type　of　design　allows　an　on-going　study　to　end　early　for　the　participant＇s　safety　and　design＇s　efficiency.　Since　a　large　value　of　the　response　rate　p　for　the　treatment　is　wanted,　lower　one-sided　confidence　intervals　for　p　are　of　interest.　Due　to　the　limited　sample　size,　exact　intervals　with　a　guaranteed　confidence　level　are　derived　using　a　rank　function　that　is　based　on　the　Clopper-Pearson　lower　confidence　limit.　When　the　sample　size　in　stage　2　is　a　constant,　two　kinds　of　smallest　intervals　are　constructed　with　or　without　using　the　sufficiency　principle.　The　proposed　intervals　outperform　the　existing　exact　intervals,　and　the　intervals　not　based　on　minimal　sufficient　statistic　are　recommended　for　practice　due　to　their　small　expected　lengths.　When　the　sample　size　in　stage　2　varies,　the　smallest　interval　is　also　proposed.