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\pubyear{2009}
\volume{0}
%\volumetitle{IMS Collections}
\firstpage{326}
\lastpage{332}
\doi{0017/0000.0000}
\arxiv{math.PR/0000017}



\begin{document}
\begin{frontmatter}
\title{Bayesian Decision Theory for Multiple Comparisons}
%\protect\thanksref{T1}}
\runtitle{Bayesian Decision Theory for Multiple Comparisons}

\begin{aug}
\author{\fnms{Charles}
\snm{Lewis}\thanksref{a1}\ead[label=e1]{}}
\and
\author{\fnms{Dorothy T.}
\snm{Thayer}\thanksref{a2}\ead[label=e2]{}}
\address[a1]{Fordham University and Educational Testing Service}
\address[a2]{Educational Testing Service}

\runauthor{Lewis and Thayer}
\contributor{Lewis, C.}{Fordham University and Educational Testing Service}
\contributor{Thayer, D.T.}{Educational Testing Service}


\affiliation{Fordham University and Educational Testing Service}

\end{aug}

\begin{abstract}
Applying a decision theoretic approach to multiple comparisons very similar
to that described by Lehmann (1950, 1957a, 1957b), we introduce a loss
function based on the concept of the false discovery rate (\textit{FDR}). We derive a
Bayes rule for this loss function and show that it is very closely related
to a Bayesian version of the original multiple comparisons procedure
proposed by Benjamini and Hochberg (1995) to control the sampling theory
\textit{FDR}. We provide the results of a Monte Carlo simulation that illustrates the
very similar sampling behavior of our Bayes rule and Benjamini and
Hochberg's procedure when applied to making all pair-wise comparisons in a
one-way fixed effects analysis of variance setup with 10 and with 20 means.
\end{abstract}

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%\kwd{cervical cancer}
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\tableofcontents
\end{frontmatter}

\section{Introduction}
A previous paper by the authors (Lewis {\&} Thayer, 2004) considered the
application of Bayesian decision theory to the multiple comparisons problem
for random effects designs, following the earlier work of Shaffer (1999),
Duncan (1965), and Waller and Duncan (1969). In our paper, we demonstrated
that the Bayes rule for a per-comparison ``0-1'' loss function controls a
random effects version of the false discovery rate (\textit{FDR}), thus supporting and
extending Shaffer's (1999) results.

A recent paper by Sarkar and Zhou (2008) adopts a random effects setup very
similar to that of our earlier paper. Rather than considering Bayes rules,
they introduce a procedure that controls the random effects \textit{FDR} discussed by us
while maximizing the random effects per-comparison power rate that we had
considered. This approach produces substantial power gains over other
procedures (including ours), but it ``declares even small differences
significant when $\tau $ [the between-groups standard deviation] is large,
thereby achieving [even] greater power than the unadjusted (per-comparison)
procedure for large values of $\tau $'' (Sarkar {\&} Zhou, 2008, p. 692). We
view this as a weakness, rather than a strength, of their method, as it
seems to ignore the basic principle behind multiple comparisons procedures,
namely that making multiple inferences calls for increased conservatism
relative to making a single inference.

The present study considers a more general setting for making multiple
comparisons and introduces a new loss function that is more directly tied to
the \textit{FDR}. We derive a Bayes rule for this loss function and show that it is very
closely related to a Bayesian version of the original multiple comparisons
procedure proposed by Benjamini and Hochberg (1995) to control the sampling
theory \textit{FDR}. We provide the results of a Monte Carlo simulation that illustrates
the very similar sampling behavior of our Bayes rule and Benjamini and
Hochberg's procedure when applied to testing all pairwise comparisons for a
one-way fixed effects analysis of variance setup with 10 and with 20 means.

\section{Setup}

We start with a general likelihood $p\left( {{\rm {\bf y}}\left| {\rm {\bf
\theta }} \right.} \right)$, prior $p\left( {\rm {\bf \theta }} \right)$,
and resulting posterior $p\left( {{\rm {\bf \theta }}\left| {\rm {\bf y}}
\right.} \right)$. Let ${\rm {\bf \psi }}={\rm {\bf f}}\left( {\rm {\bf
\theta }} \right)$ be a vector of $m$ ``contrasts'' among the elements of ${\rm
{\bf \theta }}$. Suppose our goal is to identify the sign of each of the
elements of ${\rm {\bf \psi }}$, given \textbf{y}. In the language of
decision theory, for each $\psi _i ,\mbox{ }i=1,\cdots ,m,$ we will take
action $a_i $, with $a_i =+1$ used to indicate that we declare $\psi _i $ to
be positive, $a_i =-1$ indicating that we declare $\psi _i $ to be negative,
and $a_i =0$ used to indicate that we are unable to determine sign of $\psi
_i $. Although directly inspired by Williams, Jones and Tukey (1999), and
Jones and Tukey (2000), this approach to (multiple) hypothesis testing has
its origins in the much earlier work of Lehmann (1950, 1957a, 1957b).

To continue, we introduce two component loss functions: $L_1 \left( {\psi _i
,a_i } \right)=1$ if the signs of $\psi _i $ and $a_i $ disagree, and $L_1
=0$ otherwise (used to count the number of incorrect sign declarations);
$L_2 \left( {\psi _i ,a_i } \right)=1$ if $a_i =0$, and $L_2 =0$ otherwise
(used to count the signs not declared). These actions and losses are very
similar to those given by Lehmann (1957b, p. 549). They differ from
conventional treatments of hypothesis testing in the sense that they focus
on identifying the sign of each contrast and do not formally consider the
possibility that the value of the contrast could be (exactly) 0. The
reasonableness of this approach, compared with conventional point hypothesis
testing is emphasized by Jones and Tukey (2000), among others.

We now propose a loss function that combines $L_1 $ and $L_2 $ as follows:
\begin{equation}
\label{eq1}
L_{DFDR} \left( {{\rm {\bf \psi }},{\rm {\bf a}}}
\right)=\frac{\sum\limits_{i=1}^m {L_1 \left( {\psi _i ,a_i } \right)}
}{\max \left\{ {1,m-\sum\limits_{i=1}^m {L_2 \left( {\psi _i ,a_i } \right)}
} \right\}}+\left( {\frac{\alpha }{2}} \right)\frac{\sum\limits_{i=1}^m {L_2
\left( {\psi _i ,a_i } \right)} }{m},
\end{equation}
for a fixed choice of $0<\alpha <1$ (such as $\alpha =0.05)$. Here, \textit{DFDR} (in
notation introduced by Shaffer, 2002) stands for Directional False Discovery
Rate. The first term in Equation 1 is the sample value of the \textit{DFDR} for a given
${\rm {\bf \psi }}$ and a vector of actions \textbf{a}, namely the number of
incorrect sign declarations, divided by the total number of signs declared
(or divided by 1 if no signs are declared by \textbf{a}).

The second term in Equation 1 is $\alpha \mathord{\left/ {\vphantom {\alpha
2}} \right. \kern-\nulldelimiterspace} 2$ times the sample proportion of
signs not declared. This term may be interpreted as a sample per-comparison
Type II error rate, weighted by a relative importance factor of $\alpha
\mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-\nulldelimiterspace}
2$. Using a per-comparison formulation, as well as assigning this loss
component a small weight, serves to emphasize that failure to declare a sign
is considered to be much less serious than declaring that sign incorrectly.
This emphasis is in keeping with the concern about controlling Type I errors
(at the expense of making Type II errors) in traditional treatments of the
multiple comparisons problem. The Bayes decision rule for the complete loss
function in Equation 1 minimizes its posterior expected value, in this sense
balancing the two types of losses against each other, with the major focus
being on reducing the \textit{DFDR}.

\section{Bayes Decision Rule}

To identify the actions that minimize the posterior expected value of the
loss function given in Equation 1, we begin by introducing some notation.\\ 
If
$\Pr \left( {\psi _i >0\left| {\rm {\bf y}} \right.} \right)>0.5$, define
$a_i^\ast =+1$ and $p_i =\Pr \left( {\psi _i <0\left| {\rm {\bf y}} \right.}
\right)$; if $\Pr \left( {\psi _i >0\left| {\rm {\bf y}} \right.}
\right)\leqslant 0.5$, define $a_i^\ast =-1$ and $p_i =\Pr \left( {\psi _i
>0\left| {\rm {\bf y}} \right.} \right)$. Note that $a_i^\ast $ and $p_i $
are related by the following result: $E_{{\rm {\bf \theta }}\left| {\rm {\bf
y}} \right.} \left[ {L_1 \left( {\psi _i ,a_i^\ast } \right)\left| {\rm {\bf
y}} \right.} \right]=p_i $. Now order the $p_i $ so that $p_{\left( 1
\right)} \leqslant \cdots \leqslant p_{\left( m \right)} $. Define ${\rm
{\bf a}}^{\left( k \right)}$ for $k=1,\cdots ,m$ as $a_{\left( i
\right)}^{\left( k \right)} =a_{\left( i \right)}^\ast ,\mbox{ for
}i=1,\cdots ,k,$ and $a_{\left( i \right)}^{\left( k \right)} =0,\mbox{ for
}i=k+1,\cdots ,m$. For $k=0$, take $a_{\left( i \right)}^{\left( 0 \right)}
=0,\mbox{ for }i=1,\cdots ,m$.

The posterior expected loss for ${\rm {\bf a}}^{\left( k \right)}$ is given
by
\begin{equation}
\label{eq2}
E_{{\rm {\bf \theta }}\left| {\rm {\bf y}} \right.} \left[ {L_{DFDR} \left(
{{\rm {\bf \psi }},{\rm {\bf a}}^{\left( k \right)}} \right)\left| {\rm {\bf
y}} \right.} \right]=\frac{\sum\limits_{i=1}^k {p_{\left( i \right)} }
}{\max \left\{ {1,k} \right\}}+\left( {\frac{\alpha }{2}} \right)\left(
{1-\frac{k}{m}} \right).
\end{equation}
Clearly, ${\rm {\bf a}}^{\left( k \right)}$ minimizes the posterior expected
loss among all action vectors \textbf{a} that declare exactly $k$ signs. Let
$k_{DFDR} $ be the value of $k$ for which the posterior expected loss given in
Equation 2 is minimized. (The value of $k_{DFDR} $ in a given setting would
normally be determined by an exhaustive search over all values of
$k=0,\cdots ,m$.) The Bayes decision rule for this problem is given by ${\rm
{\bf \delta }}_{DFDR} \left( {\rm {\bf y}} \right)={\rm {\bf a}}^{\left(
{k_{DFDR} } \right)}$, and the corresponding Bayes risk is
\begin{equation}
\label{eq3}
\begin{array}{l}
 r\left( {{\rm {\bf \delta }}_{DFDR} } \right)=E_{{\rm {\bf \theta }},{\rm
{\bf y}}} \left[ {L_{DFDR} \left( {{\rm {\bf \psi }},{\rm {\bf \delta
}}_{DFDR} \left( {\rm {\bf y}} \right)} \right)} \right] \\
 \mbox{ }=E_{\rm {\bf y}} \left[ {\frac{\sum\limits_{i=1}^{k_{DFDR} }
{p_{\left( i \right)} } }{\max \left\{ {1,k_{DFDR} } \right\}}+\left(
{\frac{\alpha }{2}} \right)\left( {1-\frac{k_{DFDR} }{m}} \right)} \right].
\\
 \end{array}
\end{equation}
The latter expectation in Equation 3 is taken with respect to the predictive
distribution of \textbf{y}, and it should be noted that the $p_{\left( i
\right)} $ and, consequently, $k_{DFDR} $ all depend on \textbf{y}.

Since $E_{{\rm {\bf \theta }}\left| {\rm {\bf y}} \right.} \left[ {L_{DFDR}
\left( {{\rm {\bf \psi }},{\rm {\bf a}}^{\left( 0 \right)}} \right)\left|
{\rm {\bf y}} \right.} \right]=\alpha \mathord{\left/ {\vphantom {\alpha 2}}
\right. \kern-\nulldelimiterspace} 2$ for all \textbf{y}, it follows that
$r\left( {{\rm {\bf \delta }}_{DFDR} } \right)\leqslant \alpha
\mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-\nulldelimiterspace}
2$. Consequently,
\begin{equation}
\label{eq4}
E_{{\rm {\bf \theta }},{\rm {\bf y}}} \left[ {\frac{\sum\limits_{i=1}^m {L_1
\left( {\psi _i ,\delta _{DFDR,i} \left( {\rm {\bf y}} \right)} \right)}
}{\max \left\{ {1,m-\sum\limits_{i=1}^m {L_2 \left( {\psi _i ,\delta
_{DFDR,i} \left( {\rm {\bf y}} \right)} \right)} } \right\}}}
\right]\leqslant \frac{\alpha }{2}.
\end{equation}
Equation 4 says that a Bayesian version of the \textit{DFDR} is bounded by $\alpha
\mathord{\left/ {\vphantom {\alpha 2}} \right. \kern-\nulldelimiterspace} 2$
when the Bayes rule ${\rm {\bf \delta }}_{DFDR} $ is used. It may be worth
observing that these results apply to a very general class of multiple
comparison problems. Essentially the only restriction is that the set of
contrasts be finite. Indeed, these do not even have to be contrasts in the
usual sense of that term. They could also, for example, be a set of
independent parameters that formed a family of interest.

\section{A Bayesian Version of Benjamini {\&} Hochberg's Procedure}

Next, we consider the multiple comparisons procedure proposed by Benjamini
and Hochberg (1995) and modified for directional testing by Williams, Jones
and Tukey (1999).

However, we will translate the procedure into our Bayesian framework. Define
$k_{DB\And H} $ to be the largest value of $k=1,\cdots ,m,$ such that
\[
p_{\left( k \right)} \leqslant \left( {\frac{\alpha }{2}} \right)\left(
{\frac{k}{m}} \right),
\]
with $k_{DB\And H} =0$ if no such value of $k$ exists. Define ${\rm {\bf \delta
}}_{DB\And H} \left( {\rm {\bf y}} \right)={\rm {\bf a}}^{\left( {k_{DB\And
H} } \right)}$.

If $k_{DB\And H} =0$, then the posterior expected loss for ${\rm {\bf
a}}^{\left( {k_{DB\And H} } \right)}$ is equal to $\alpha \mathord{\left/
{\vphantom {\alpha 2}} \right. \kern-\nulldelimiterspace} 2$. If $k_{DB\And
H} >0$, the posterior expected loss for ${\rm {\bf a}}^{\left( {k_{DB\And H}
} \right)}$ is given by Equation 2 as
\begin{equation}
\label{eq5}
E_{{\rm {\bf \theta }}\left| {\rm {\bf y}} \right.} \left[ {L_{DFDR} \left(
{{\rm {\bf \psi }},{\rm {\bf a}}^{\left( {k_{DB\And H} } \right)}}
\right)\left| {\rm {\bf y}} \right.}
\right]=\frac{\sum\limits_{i=1}^{k_{DB\And H} } {p_{\left( i \right)} }
}{k_{DB\And H} }+\left( {\frac{\alpha }{2}} \right)\left( {1-\frac{k_{DB\And
H} }{m}} \right).
\end{equation}
From the definition of $k_{DB\And H} $, it follows that
\begin{equation}
\label{eq6}
p_{\left( i \right)} \leqslant \left( {\frac{\alpha }{2}} \right)\left(
{\frac{k_{DB\And H} }{m}} \right)\mbox{ for }i=1,\cdots ,k_{DB\And H} .
\end{equation}
Consequently, applying Inequality 6 to Equation 5, it follows that
\[
E_{{\rm {\bf \theta }}\left| {\rm {\bf y}} \right.} \left[ {L_{DFDR} \left(
{{\rm {\bf \psi }},{\rm {\bf a}}^{\left( {k_{DB\And H} } \right)}}
\right)\left| {\rm {\bf y}} \right.} \right]\leqslant \left( {\frac{\alpha
}{2}} \right)\left( {\frac{k_{DB\And H} }{m}} \right)+\left( {\frac{\alpha
}{2}} \right)\left( {1-\frac{k_{DB\And H} }{m}} \right)=\frac{\alpha }{2}.
\]
Since this inequality holds for all \textbf{y}, it implies that $r\left(
{{\rm {\bf \delta }}_{DB\And H} } \right)\leqslant \alpha \mathord{\left/
{\vphantom {\alpha 2}} \right. \kern-\nulldelimiterspace} 2$, and so, just
as with ${\rm {\bf \delta }}_{DFDR} $,
\begin{equation}
\label{eq7}
E_{{\rm {\bf \theta }},{\rm {\bf y}}} \left[ {\frac{\sum\limits_{i=1}^m {L_1
\left( {\psi _i ,\delta _{DB\And H,i} \left( {\rm {\bf y}} \right)} \right)}
}{\max \left\{ {1,m-\sum\limits_{i=1}^m {L_2 \left( {\psi _i ,\delta
_{DB\And H,i} \left( {\rm {\bf y}} \right)} \right)} } \right\}}}
\right]\leqslant \frac{\alpha }{2}.
\end{equation}
Equation 7 says that ${\rm {\bf \delta }}_{DB\And H} $ also controls our
Bayesian \textit{DFDR}.

This seems like an appropriate place to note that what has just been
established (namely the fact that ${\rm {\bf \delta }}_{DB\And H} $ controls
the \textit{DFDR} for an arbitrary set of contrasts) is a Bayesian, rather than a
sampling theory result. Indeed, Benjamini and Hochberg's (1995) sampling
theory procedure has only been shown to control the sampling theory \textit{FDR} in
special circumstances, such as the case of independent tests. In particular,
it has not been shown to provide sampling theory control of the \textit{FDR} when making
all pairwise comparisons among a set of means in a one-way, fixed effects
analysis of variance setup.

Since ${\rm {\bf \delta }}_{DFDR} $ is a Bayes decision rule, it must be the
case that $r\left( {{\rm {\bf \delta }}_{DFDR} } \right)\leqslant r\left(
{{\rm {\bf \delta }}_{DB\And H} } \right)$. Moreover, it is also possible to
show that $k_{DFDR} \geqslant k_{DB\And H} $ for all \textbf{y}. To see
this, suppose the contrary: $k_{DB\And H} =k_{DFDR} +d$ with $d>0$. By the
definition of $k_{DFDR} $, we must have
\[
\frac{\sum\limits_{i=1}^{k_{DFDR} } {p_{\left( i \right)} } }{\max \left\{
{1,k_{DFDR} } \right\}}+\left( {\frac{\alpha }{2}} \right)\left(
{1-\frac{k_{DFDR} }{m}} \right)<\frac{\sum\limits_{i=1}^{k_{DB\And H} }
{p_{\left( i \right)} } }{k_{DB\And H} }+\left( {\frac{\alpha }{2}}
\right)\left( {1-\frac{k_{DB\And H} }{m}} \right)
\]
or
\begin{align}
\label{eq8}
\frac{\sum\limits_{i=1}^{k_{DFDR} } {p_{\left( i \right)} } }{\max \left\{
{1,k_{DFDR} } \right\}}+\left( {\frac{\alpha }{2}} \right)\left(
{1-\frac{k_{DFDR} }{m}} \right) &< \frac{\sum\limits_{i=1}^{k_{DFDR} }
{p_{\left( i \right)} } +\sum\limits_{i=k_{DFDR} +1}^{k_{DFDR+d} }
{p_{\left( i \right)} } }{k_{DFDR} +d}\nonumber\\
&+\left( {\frac{\alpha }{2}}
\right)\left( {1-\frac{k_{DFDR} +d}{m}} \right).
\end{align}
Note that a strict inequality has been used here, implying that, in case of
ties, $k_{DFDR} $ would be chosen to be the largest value of $k$ that minimizes
the posterior expected loss. To continue, using the definition of $k_{DB\And
H} $,
\begin{equation}
\label{eq9}
\sum\limits_{i=k_{DFDR} +1}^{k_{DFDR} +d} {p_{\left( i \right)} } \leqslant
d\left( {\frac{\alpha }{2}} \right)\left( {\frac{k_{DFDR} +d}{m}} \right).
\end{equation}
Combining Inequalities 8 and 9 gives
\begin{align*}
\frac{\sum\limits_{i=1}^{k_{DFDR} } {p_{\left( i \right)} } }{\max \left\{
{1,k_{DFDR} } \right\}}+\left( {\frac{\alpha }{2}} \right)\left(
{1-\frac{k_{DFDR} }{m}} \right) &<\frac{\sum\limits_{i=1}^{k_{DFDR} }
{p_{\left( i \right)} } }{k_{DFDR} +d}\\
&+\left( {\frac{\alpha }{2}}
\right)\left( {\frac{d}{m}} \right)+\left( {\frac{\alpha }{2}} \right)\left(
{1-\frac{k_{DFDR} +d}{m}} \right)
\end{align*}
or
\begin{equation}
\label{eq10}
\frac{\sum\limits_{i=1}^{k_{DFDR} } {p_{\left( i \right)} } }{\max \left\{
{1,k_{DFDR} } \right\}}<\frac{\sum\limits_{i=1}^{k_{DFDR} } {p_{\left( i
\right)} } }{k_{DFDR} +d},
\end{equation}
If $k_{DFDR} >0$, Inequality 10 implies that $d=0$, contrary to our initial
assumption. If $k_{DFDR} =0$, both sides of Inequality 10 would be 0,
contradicting the strict inequality. Thus, we have demonstrated that
$k_{DFDR} \geqslant k_{DB\And H} $ for all \textbf{y}. In other words, the
Bayes rule ${\rm {\bf \delta }}_{DFDR} $ will always declare at least as
many signs as ${\rm {\bf \delta }}_{DB\And H} $.

\section{Simulation Results}

It is important to recall that the procedure actually proposed by Benjamini
and Hochberg (1995) uses sampling theory $p$-values (one-tailed values in
Williams, Jones and Tukey's 1999 version), rather than posterior tail
probabilities and controls the sampling theory version of the \textit{FDR} (or \textit{DFDR} in
Willams et al.'s version). Now consider a standard multiple comparisons
problem: the one-way, fixed effects analysis of variance setup, with the
$\psi _i $ chosen to be all pair-wise differences among the group means. In
this case, the relevant sampling theory and Bayesian (based on a vague prior
for all parameters) $p$-values are identical tail probabilities from the
appropriate Student's $t$-distribution (see, for instance, Box {\&} Tiao, 1973,
p. 140).

Tables 1 and 2 give the results of sampling theory simulations (based on
25,000 replications for each condition) of one-way, fixed effects ANOVA
setups, considering all pair-wise differences for 10 evenly spaced means,
and for 25 evenly spaced means. In these simulations, the within-group
variance was set at 3.0 with $n=3$ sample observations per group (so the
sampling variances of the sample means are all equal to 1.0 and the
within-group degrees of freedom equals 20 in the first case and 50 in the
second case). In addition, we chose $\alpha =0.05$, with the intention of
controlling the sampling theory \textit{DFDR} at $\alpha \mathord{\left/ {\vphantom
{\alpha 2}} \right. \kern-\nulldelimiterspace} 2=0.025$, although we
emphasize again that there is no theory to support that control for
pair-wise comparisons.

The \textit{DFDR} values in Table 1 are sampling theory averages of the sample \textit{DFDR} used as
the first term of our loss function for the two rules with two numbers of
means, and a range of spacings among the means. To make the results
comparable across the two setups, we used the population standard deviation
(denoted here by $\tau )$ to index the spread of the means. The Average
Power values in Table 2 are sampling theory averages of the sample
per-comparison correct sign declaration rate. All four quantities for a
given number of means are computed from the same 25,000 replications at each
spread of the means. Note that the spread is effectively given in units
equal to the standard errors of the sample means. For the spread labeled
``0.00+'' all population mean values were set equal, and an arbitrary
ordering was chosen to evaluate the ``wrong sign'' errors. Both procedures
conservatively control the \textit{DFDR} for all conditions considered. The Bayes rule
procedure provides slightly greater per-comparison power than that of
Benjamini and Hochberg in these conditions, but the actual differences are
trivial.

\begin{table}[htbp]
\label{tab1} Table 1. {Sampling Theory $DFDR$ for All Pair-wise
Comparisons Made Using Our Bayes Rule and Benjamini \& Hochberg's
Procedures.}
\begin{center}
\begin{tabular}{ccccc}
$\tau :$ Spread & \multicolumn{ 2}{c}{10 Means} & \multicolumn{ 2}{c}{25 Means} \\\cline{2-5}
  of Means &        ${\rm {\bf \delta }}_{DFDR}$ &       ${\rm {\bf \delta }}_{DB\And H} $ &        ${\rm {\bf \delta }}_{DFDR} $ &        ${\rm {\bf \delta }}_{DB\And H} $ \\ 
\hline
     0.00+ &      0.0204 &      0.0171 &      0.0206 &      0.0176 \\

     0.721 &     0.0062 &     0.0044 &     0.0067 &     0.0046 \\

     3.606 &     0.0005 &     0.0005 &     0.0013 &     0.0012 \\

     5.408 &     0.0001 &     0.0001 &     0.0006 &     0.0006 \\

     7.211 &     0.0000 &     0.0000 &     0.0003 &     0.0003 \\

    14.422 &     0.0000 &     0.0000 &     0.0000 &     0.0000 \\
\hline
\end{tabular}  
  
\end{center}

\end{table}

\begin{table}[htbp]
\label{tab2} Table 2. Sampling Theory Average Power for All Pair-wise
Comparisons Made Using Our Bayes Rule and Benjamini {\&} Hochberg's
Procedure.
\begin{center}
\begin{tabular}{ccccc}
$\tau :$ Spread & \multicolumn{ 2}{c}{10 Means} & \multicolumn{ 2}{c}{25 Means} \\\cline{2-5}
  of Means &        ${\rm {\bf \delta }}_{DFDR}$ &       ${\rm {\bf \delta }}_{DB\And H} $ &        ${\rm {\bf \delta }}_{DFDR} $ &        ${\rm {\bf \delta }}_{DB\And H} $ \\ 
\hline
     0.00+ &      0.002 &      0.002 &      0.001 &      0.000 \\

     0.721 &     0.022 &     0.016 &     0.012 &     0.007 \\

     3.606 &     0.634 &     0.621 &     0.604 &     0.594 \\

     5.408 &     0.783 &     0.778 &     0.741 &     0.737 \\

     7.211 &     0.860 &     0.857 &     0.813 &     0.811 \\

    14.422 &     0.984 &     0.984 &     0.924 &     0.924 \\
\hline
\end{tabular}  
  
\end{center}

\end{table}






Note: 25,000 replications for each condition, $n=3$ observations per
group, within degrees of freedom $\nu =20$ for 10 means and $\nu
=50$ for 25 means, within variance $\sigma ^2=3.0$, $\alpha
\mathord{\left/ {\vphantom {\alpha 2}} \right.
\kern-\nulldelimiterspace} 2=.025$.

\section{Conclusions}

The decision rule ${\rm {\bf \delta }}_{DFDR} $ has been shown to be optimal
(from a Bayesian perspective) relative to the loss function $L_{DFDR} $ for
a wide class of multiple comparison problems involving sign declarations. It
has also been shown to control a Bayesian version of the directional false
discovery rate (\textit{DFDR}), as has a Bayesian version of the procedure proposed by
Benjamini and Hochberg (${\rm {\bf \delta }}_{DB\And H} )$. There is no
guarantee that ${\rm {\bf \delta }}_{DFDR} $ or ${\rm {\bf \delta }}_{DB\And
H} $ will control a sampling theory \textit{DFDR} for the case of pair-wise
comparisons$,$ although that appears to occur in the ANOVA examples given,
where the two rules behave very similarly.

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\end{document}