The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors (i.e. maximizing power for a fixed significance level).
ðɪ ænˈoʊvə ˈɛftˈɛst ɪz nˈoʊn təbi nˌɪɹli ˈɑːptɪməl ɪnðə sˈɛns ʌv mˈɪnɪmˌaɪzɪŋ fˈɔls nˈɛɡətˌɪv ˈɛɹɚz fɚɹə fˈɪkst ɹˈeɪt ʌv fˈɔls pˈɑːzᵻtˌɪv ˈɛɹɚz ˈaɪ.ˈiː. mˈæksɪmˌaɪzɪŋ pˈaʊɚ fɚɹə fˈɪkst sɪɡnˈɪfɪkəns lˈɛvəl.