Summary: As C. R. Nelson and R. Startz [“The distribution of the instrumental variable estimator and its t ratio when the instrument is a poor one”, J. Bus. 63, No. 1, 125–140 (1990), http://www.jstor.org/stable/2353264; Econometrica 58, No. 4, 967–976 (1990; Zbl 0727.62109)] dramatically demonstrated, standard hypothesis tests and confidence intervals in instrumental variables regression are invalid when instruments are weak. Recent work on hypothesis tests for the coefficient on a single included endogenous regressor when instruments may be weak has focused on similar tests. This paper extends that work to nonsimilar tests, of which similar tests are a subset. The power envelope for two-sided invariant (to rotations of the instruments) nonsimilar tests is characterized theoretically, then evaluated numerically for five IVs. The power envelopes for similar and nonsimilar tests differ theoretically, but are found to be very close numerically. The nonsimilar test power envelope is effectively achieved by the M. J. Moreira [Econometrica 71, No. 4, 1027–1048 (2003; Zbl 1151.62367)] conditional likelihood ratio test, so that test is effectively uniformly most powerful invariant (UMPI). We also provide a new nonsimilar test, \(P^*\), which has \(\chi^2_1\) critical values, is asymptotically efficient under strong instruments, involves only elementary functions, and is very nearly UMPI.