Under the existence of nuisance parameters, we consider a class of tests $\mathscr{S}$ which contains the likelihood ratio, Wald and Rao's score tests as special cases. To investigate the influence of nuisance parameters, we derive the second order asymptotic expansion of the distribution of $T \in \mathscr{S}$ under a sequence of local alternatives. This result and concrete examples illuminate some interesting features of effects due to nuisance parameters. Optimum properties for a modified likelihood ratio test proposed in Mukerjee [\ref{M-1994}] are shown under the criteria of second order local maximinity.