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\begin{document}

Examples of simple tables ``in text.''

     Monte Carlo experiments are often the simplest way to
estimate and compare the power functions of complex test statistics.
Unfortunately, at the end of such studies one may find results such
as the following:
  \begin{center}
{\bf Table 1}
  \end{center}
  \begin{center}
\begin{tabular}{ccccc}
       &$\delta=0$ &$\delta=1.5$ &$\delta=3.0$ &$\delta=4.5$  \\
Test 1:&  .08  &.24   &.37  & .74  \\
Test 2:&  .03  &.20   &.30  & .62
\end{tabular}
  \end{center}
 where $\delta=0$ corresponds to the null hypothesis, .05 is the nominal
level of the tests, and the entries
such as .08 are the proportion of test rejections in $N$ Monte Carlo replications.

\vspace*{1in}

Of course the adjusted power at the null is just $\alpha$ because of
the way $\hat{C}_\alpha$ is chosen.  If we carried out these
calculations for the two statistics in Table 1, we might replace
Table 1 by

  \begin{center}
{\bf Table 2}
  \end{center}
  \begin{center}
\begin{tabular}{ccccc}
       &$\delta=0$ &$\delta=1.5$ &$\delta=3.0$ &$\delta=4.5$  \\
Test 1:&  .08 (.05) &.24 (.19) &.37 (.30) & .74 (.62)  \\
Test 2:&  .03 (.05) &.20 (.22) &.30 (.33) & .62 (.68)
\end{tabular}
  \end{center}

The adjusted powers in parentheses are then comparable.  On the
other hand we have left the original power estimates in Table 2
because they show what kind of power one would obtain by using
the standard critical value $C_\alpha^*$ even though such a test
does not have level $\alpha$.

\end{document}