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%\VignetteIndexEntry{Comparison of methods in the metap package}
\title{Comparison of methods in the \pkg{metap} package}
\author{Michael Dewey}
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\begin{document}
\maketitle

\section{Introduction}

\subsection{What is this document for?}

This document describes some methods for the meta--analysis of
$p$--values
(significance values)
contained in the package \pkg{metap}
and contains comments on
the performance of the various algorithms
under a small number of different
scenarios with
hints on the choice of method.


\subsection{Notation}

The $k$ studies give rise to $p$--values,
$p_i,\;i = 1, \dots, k$.
These are assumed to be independent.
We shall also need the ordered $p$--values:
$p_{[1]} \le p_{[2]}, \dots, \le p_{[k]}$
and weights
$w_i,\;i = 1, \dots, k$.
Logarithms are natural.
A function for combining $p$--values is denoted $g$.
The size of the test is $\alpha$.
We may also need $k$ degrees of freedom, $\nu_i$.

The methods are referred to by the name of the
function in \func{metap}.
Table \ref{funcs} shows other
descriptions of each method.

\begin{table}[htbp]
\begin{tabular}{lll}
Function name & \multicolumn{2}{c}{Description(s)} \\[1ex]
 & \multicolumn{1}{c}{Eponym} \\
\func{invchisq} & Lancaster's method & Inverse chi square \\
\func{invt} & & Inverse t \\
\func{logitp} & & Logistic\\
\func{meanp} \\
\func{meanz} \\
\func{maximump} \\
\func{minimump} & Tippett's method \\
\func{sumlog} & Fisher's method & Chi square (2 df)\\
\func{sump} & Edgington's method & Uniform\\
\func{sumz} & Stouffer's method & Normal\\
\func{truncated} & Truncated Fisher\\
\func{truncated} & & rank--truncated\\
\func{votep} \\
\func{wilkinsonp} & Wilkinson's method \\
\end{tabular}
\caption{Methods considered in this document}
\label{funcs}
\end{table}

\section{Theoretical results}

There have been various attempts to clarify the
problem and to discuss optimality
of the methods.
A detailed account was provided by
\citet{liptak58}.

\citet{birnbaum54} considered the property of
admissibility.
A method is admissible if when it rejects $H_0$ for
a set of $p_i$ it will also reject $H_0$ for
$P^*_i$ where $p^*_i \le p_i$ for all $i$.
He considered that Fisher's and Tippett's method were admissible.
See also \citet{owen09}.

He also points out the problem
is poorly specified.
This may
account for the number of methods available
and their differing behaviour.
The null hypothesis $H_0$ is well defined,
that all $p_i$ have a uniform distribution on the unit interval.
There are
two classes of alternative hypothesis
\begin{itemize}
\item
$H_A$: all $p_i$ have the same (unknown)
non--uniform, non--increasing density,
\item
$H_B$:
at least one $p_i$ has an (unknown)
non--uniform, non--increasing density.
\end{itemize}

If all the tests being combined come from
what are basically replicates then $H_A$ is appropriate
whereas if they are of different kinds
of test or different conditions
then $H_B$ is appropriate.
Note that Birnbaum specifically considers the
possibility that the tests being combined may be
very different 
for instance some tests of means, some of variances,
and so on.


\section{The methods}

\subsection{Comparison scenarios}

To provide a standard of comparison
we shall use the following two situations.
Some authors have also used the case of exactly two
$p_i$.

\begin{description}
%\subsubsection{What if all $p_i = p$?\label{twopisection}}
\item[What if all $p_i = p$?]\label{twopisection}
Perhaps surprisingly there are substantial differences here
as we shall see when we look at each method.
We shall describe how the returned value
varies with $p$ and $k$.
%\subsubsection{Cancellation}
\item[Cancellation]
When the collection of primary studies
contains a number of values significant in both directions
the methods can give very different results.
If the intention of the synthesis is to examine a directional
hypothesis one would want a method where these cancelled out.
The decision between methods should be made on theoretical
grounds of course.
We shall use the following four values as our
example.
\end{description}

{\codefont
<<>>=
cancel <- c(0.001, 0.001, 0.999, 0.999)
@
}

<<echo = FALSE>>=
library(metap)
data(dat.metap)
validity <- dat.metap$validity$p
genvec <- function(pvals, kvals, fun, name) {
   ps <- length(pvals)
   ks <- length(kvals)
   temp <- matrix(-1, nrow = ps, ncol = ks)
   for(i in 1:ps)
   for(j in 1:ks) {
      temp[i, j] <- fun(rep(pvals[i], kvals[j]))$p
   }
   temp2 <- as.vector(temp)
   res <- data.frame(method = rep(name, length(temp2)),
      p = rep(pvals, ks),
      k = rep(kvals, each = ps),
      g = temp2
   )
   res
}
@


\subsection{Methods using transformation of the $p$--values}

One class of methods relies on transforming the $p$--values
first.

\begin{table}[htbp]
\begin{tabular}{lll}
Function name & Definition & Critical value \\[1ex]
\func{invchisq} & $\sum_{i=1}^k \chi^2_{\nu_i}(p_i)$ & $\chi^2_{\sum{\nu_i}}(\alpha)$ \\[1ex]
\func{invt} & $\frac{\sum_{i=1}^k t_{\nu_i}(p_i)}%
{\sqrt{\sum_{i=1}^k \frac{\nu_i}{\nu_i - 2}}}$ & $z(\alpha)$ \\[1ex]
\func{logitp} & $\frac{\sum_{i=1}^k \log\frac{p}{1 - p}}{C}$ & $t_{5k+4}$ \\
 & where $C = \sqrt\frac{k \pi^2 (5 k + 2)}{3(5 k + 4)}$ & \\[1ex]
\func{meanz} & $\frac{\bar{z}}{s_{\bar{z}}}$ & $t_{k-1}(\alpha)$ \\
 & where $\bar{z} = \sum_{i=1}^k \frac{z(p_i)}{k}$ \\
 & and $s_{\bar{z}} = \frac{s_z}{\sqrt{k}}$ & \\[1ex]
\func{sumlog} & $\sum_{i=1}^{k} - 2 \log p_i$ & $\chi_{2k}(\alpha)$ \\[1ex]
\func{sumz} & $\frac{\sum_{i=1}^k z(p_i)}{\sqrt{k}}$ & $z(\alpha)$\\
\end{tabular}
\caption{Definitions of methods using transformation of the $p$ values}
\label{transdefs}
\end{table}

<<echo = FALSE>>=
   kvals <- c(4, 5, 6, 8, 10, 15, 20)
   pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6)
   dat <- rbind(
      genvec(pvals, kvals, logitp, "logitp"),
      genvec(pvals, kvals, meanz, "meanz"),
      genvec(pvals, kvals, sumlog, "sumlog"),
      genvec(pvals, kvals, sumz, "sumz")
   )
@

<<fig=TRUE,label=transeqp,include=FALSE,echo=FALSE>>=
   lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat,
      auto.key = list(space = "left", lines = TRUE, title = "p"),
      ylab = "g(p)"
   )
@

\subsubsection{The method of summation of logs, Fisher's method}

See Table \ref{transdefs} for the definition.
This works because $- 2 \log p_i$ is a $\chi^2_2$
and the sum of $\chi^2$ is itself a $\chi^2$
with degrees of freedom equal to the sum of the degrees
of freedom of the
individual $\chi^2$.
Of course the sum of the log of the $p_i$
is also the log of the product of the $p_i$.
Fisher's method \citep{fisher25} is provided in \func{sumlog}.

<<echo = FALSE>>=
set.seed(18122019)
temp <- matrix(runif(10000), nrow = 100)
fisher <- apply(temp, 1, function(x) sumlog(x)$p)
lanc4 <- apply(temp, 1, function(x) invchisq(x, 4)$p)
lanc16 <- apply(temp, 1, function(x) invchisq(x, 16)$p)
lanc256 <- apply(temp, 1, function(x) invchisq(x, 256)$p)
banda <- function(x, y) {
   res <- data.frame(sum = x + y, diff = (x - y))
   res
}
dat <- data.frame(rbind(banda(fisher, lanc4),
   banda(fisher, lanc16),
   banda(fisher, lanc256),
   banda(lanc4, lanc16),
   banda(lanc4, lanc256),
   banda(lanc16, lanc256)
   ),
   name = factor(c(rep("FL4", 100), rep("FL16", 100),
      rep("FL256", 100), rep("L4L16", 100),
      rep("L4L256", 100), rep("L16L256", 100)),
      levels = c("FL4", "FL16", "FL256", "L4L16", "L4L256", "L16L256")
   )
)
@

<<fig=TRUE,label=fishlanc,include=FALSE,echo=FALSE>>=
   lattice::xyplot(diff ~ sum | name, data = dat,
   panel = function(x, y, ...) {
      lattice::panel.xyplot(x, y, ...)
      lattice::panel.abline(h = mean(y), lty = 2)
      lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3)
      lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3)
   }
   )
@

<<echo = FALSE>>=
stouff <- apply(temp, 1, function(x) sumz(x)$p)
invt4 <- apply(temp, 1, function(x) invt(x, 4)$p)
invt16 <- apply(temp, 1, function(x) invt(x, 16)$p)
invt256 <- apply(temp, 1, function(x) invt(x, 256)$p)
banda <- function(x, y) {
   res <- data.frame(sum = x + y, diff = (x - y))
   res
}
dat <- data.frame(rbind(banda(stouff, invt4),
   banda(stouff, invt16),
   banda(stouff, invt256),
   banda(invt4, invt16),
   banda(invt4, invt256),
   banda(invt16, invt256)
   ),
   name = factor(c(rep("St4", 100), rep("St16", 100),
      rep("St256", 100), rep("t4t16", 100),
      rep("t4t256", 100), rep("t16t256", 100)),
      levels = c("St4", "St16", "St256", "t4t16", "t4t256", "t16t256")
   )
)
@

<<fig=TRUE,label=stouffinvt,include=FALSE,echo=FALSE>>=
lattice::xyplot(diff ~ sum | name, data = dat,
   panel = function(x, y, ...) {
      lattice::panel.xyplot(x, y, ...)
      lattice::panel.abline(h = mean(y), lty = 2)
      lattice::panel.abline(h = mean(y) + 1.96 * sd(y), lty = 3)
      lattice::panel.abline(h = mean(y) - 1.96 * sd(y), lty = 3)
   }
)
@


As can be seen in Figure \ref{equalp}
when all the $p_i=p$
\func{sumlog} returns
a value which
decreases with $k$
when $p<0.32$, increases with $k$
when $p>0.37$, and in between
increases with $k$ and then
decreases.
Some detailed algebra provided in a post
to https://stats.stackexchange.com/questions/243003 by Christoph Hanck
suggests that the breakpoint is $e^{-1} = 0.3679$.
Where the $p_i$ are less than this then for a sufficiently
large $k$ (several
hundred) the result will be significant and not if above that.
Over the range of $k$ we are plotting this bound is not
yet closely approached.

\mygraph{transeqp}{Behaviour of the methods using transformed $p$ values for $k$ values of $p=p_i$}{equalp}

\subsubsection{Inverse $\chi^2$ Lancaster's method}

It would of course be possible to generalise
Fisher's method to use
transformation to $\chi^2$ with any other number
of degrees of freedom rather than 2.
\citet{lancaster61} suggests that this is highly correlated
with \func{sumlog}.
Lancaster's method is provided in \func{invchisq}.
In fact the resemblance to \func{sumlog} becomes
less as the number of degrees of freedom increases.
Figure \ref{fishlanc} shows for a small number of selected
degrees of freedom how it compares to Fisher's method.

\twograph{fishlanc}{Fisher's method and Lancaster's method}{fishlanc}{stouffinvt}{Stouffer's method and inverse $t$}{stouffinvt}{Sum and difference plots of Fisher v Lancaster and Stouffer v inverse $t$}{compfishstouff}

\subsubsection{The method of summation of $z$ values, Stouffer's method}

The method of summation of $z$
values is provided in \func{sumz} \citep{stouffer49}.
See Table \ref{transdefs} for the definition.
As can be seen in Figure \ref{equalp}
it returns a value for our $p_i=p$
example which
decreases with $k$ when $p$ below 0.5
and increases above.

A weighted version of Stouffer's method is available
%\begin{equation}
$\frac{\sum_{i=1}^k w_i z(p_i)}{\sqrt {\sum_{i=1}^k w_i ^ 2}}$
%\end{equation}
where $w_i$ are the weights.
In the absence of effect sizes (in which case a method
using effect sizes would be more appropriate anyway)
best results are believed to be obtained with weights
proportional to the square root of the sample sizes
\citep{zaykin11} following \citet{liptak58}.


\subsubsection{Mean of normals method}

There is also a method closely related to Stouffer's using
the mean of normals provided in \func{meanz}
also defined in Table \ref{transdefs}
which has very similar properties except that
when all the $p_i$ are equal it either gives 0 or 1
as can be seen in Figure \ref{equalp}.

<<>>=
meanz(c(0.3, 0.31))$p
meanz(c(0.1, 0.2))$p
@

The method of \func{meanz} also
has the unusual property
that a set of $p$--values which are all less than those in another
set can still give rise to a larger overall $p$.
See example above.
This is the only method considered here which has this
property so if it is a desirable one then that is the only method to consider.

\subsubsection{The inverse $t$ method}

A closely related method is the inverse $t$ method.
See Table \ref{transdefs} for the definition.
This method is provided in \func{invt}.
As is clear from the definition this method tends to
Stouffer's method as $\nu_i \to \infty$.
Figure \ref{stouffinvt} shows this for selected degrees of
freedom.

\subsubsection{The method of summation of logits}

See Table \ref{transdefs} for the definition.
This method is provided in \func{logitp}.
The constant $C$ was arrived at by equating
skewness and kurtosis with that of the $t$--distribution
\citep{loughin04}.
As can be seen in Figure \ref{equalp}
this method returns a value for our $p_i=p$
example which
decreases with $k$ when $p$ below 0.5
and increases above.

\subsubsection{Examples for methods using transformations of the $p$ values}

<<echo=FALSE>>=
log10p <- function(x) {
   res <- round(-log(x, base = 10), 2)
   res
}
@
\begin{table}[htbp]
\begin{tabular}{lll}
Function name & \multicolumn{1}{c}{validity}  & \multicolumn{1}{c}{cancel} \\
 & \multicolumn{1}{c}{value expressed} \\
 & \multicolumn{1}{c}{as $-\log_{10}p$}  \\[1ex]
\func{logitp} & \Sexpr{log10p(logitp(validity)$p)} & \Sexpr{logitp(cancel)$p} \\
\func{meanz} & \Sexpr{log10p(meanz(validity)$p)} & \Sexpr{meanz(cancel)$p} \\
\func{sumlog} & \Sexpr{log10p(sumlog(validity)$p)} & \Sexpr{round(sumlog(cancel)$p, 5)} \\
\func{sumz} & \Sexpr{log10p(sumz(validity)$p)} & \Sexpr{sumz(cancel)$p}\\
\end{tabular}
\caption{Examples of methods using transformation of the $p$ values}
\label{transexamples}
\end{table}

Using the same example dataset which we have already plotted and our cancellation
dataset we have the values in Table \ref{transexamples}.
As can be seen all the methods cancel except for \func{sumlog}.
The agreement for the validity dataset is close except
for \func{meanz} whoch gives a value several orders of magnitude
greater than the other three.
Lancaster's method and inverse $t$ are not shown as they are both
infinite families of possible methods
and in any event are similar to Fisher's method
and Stouffer's method respectively.

\subsection{Methods using untransformed $p$--values}

\begin{table}[htbp]
\begin{tabular}{lll}
Function name & Definition & Critical value \\[1ex]
\func{meanp} & $\bar p = \frac{\sum_{i=1}^k p_i}{k}$ \\
 & $z = (0.5 - \bar{p}) \sqrt{12k}$ & $z(\alpha)$ \\
\func{minimump} & $p_{[1]}$ & $1 - (1 - \alpha)^{\frac{1}{k}}$ \\
\func{maximump} & $p_{[k]}$ & $\alpha^k$ \\
\func{wilkinsonp} & $p_{[r]}$ & $\sum_{s=r}^k {k \choose s}\alpha^s (1 - \alpha)^{k-s}$\\[1ex]
\func{sump} & $\frac{(S)^k}{k!}%
- {k \choose 1}\frac{(S - 1)^k}{k!}%
+ {k \choose 2}\frac{(S - 2)^k}{k!} - \dots$ & $\alpha$ \\
 & where $S = \sum_{i=1}^k p_i$ \\
\end{tabular}
\caption{Definitions of methods not using transformation of the $p$ values, %
the series for \func{sump} continues
until the term in in the numerator $(S-i)$
becomes negative}
\label{untransdefs}
\end{table}


<<echo = FALSE>>=
   kvals <- c(4, 5, 6, 8, 10, 15, 20)
   pvals <- c(0.2, 0.3, 0.3679, 0.4, 0.5, 0.6)
   dat <- rbind(
      genvec(pvals, kvals, meanp, "meanp"),
      genvec(pvals, kvals, maximump, "maximump"),
      genvec(pvals, kvals, minimump, "minimump"),
      genvec(pvals, kvals, sump, "sump"),
      genvec(pvals, kvals, votep, "votep")
   )
@

<<fig=TRUE,label=untranseqp,include=FALSE,echo=FALSE>>=
   lattice::xyplot(g ~ k | method, groups = p, type = "l", data = dat,
      auto.key = list(space = "left", lines = TRUE, title = "p"),
      ylab = "g(p)"
   )
@

\mygraph{untranseqp}{Behaviour of the methods using untransformed $p$ values for $k$ values of $p=p_i$}{unequalp}


\subsubsection{The method of minimum $p$, maximum $p$, and Wilkinson's method}

The methods of minimum $p$ \citep{tippett31}, maximum $p$ and Wilkinson \citep{wilkinson51}
are defined in Table \ref{untransdefs}.
Wilkinson's method depends on which value
(the $r$th) of $p_{[i]}$ is
selected.
% p is pbeta(p[r], r, k+1-r)
% critical p is qbeta(alpha, r, k+1-r)
Wilkinson's method is provided in \func{wilkinsonp} and a
convenience function \func{minimump}
with its own \code{print} method is provided for
the minimum $p$ method ($r=1$).
It is also possible to use the method for the
maximum $p$ (that is $r=k$) and a convenience function \func{maximump}
is provided for that purpose.

As can be seen in Figure \ref{unequalp}
these methods return a value for our $p_i=p$
example which
always increases with $k$
which is true for
\func{minimump}
and
which always decreases with $k$
which is true for
\func{maximump}

\subsubsection{The method of summation of $p$--values, Edgington's method\label{sump}}

Defined in Table \ref{untransdefs}
\citep{edgington72a}.
This method is provided in \func{sump}.
As can be seen in Figure \ref{unequalp}
this method returns a value for our $p_i=p$
example which
decreases with $k$ when $p$ below 0.5
and increases above.

Some authors use a simpler version, $\frac{(\sum p)^k}{k!}$,
for instance \citet{rosenthal78} in the text
although compare his Table 4.
This can be very conservative when
$\sum p > 1$
There seems no particular need to use this method but
it is returned by \func{sump}
as the value of \code{conservativep}
for use in checking published values.

Note also that there can be numerical problems for extreme values
of $S$ and in that case recourse might be made to
\func{meanp} which has similar
properties.

\subsubsection{The mean $p$ method}

Defined in Table \ref{untransdefs}.
Although this method is attributed to Edgington \citep{edgington72b}
when the phrase Edgington's method is used
it refers to the method of summation of $p$--values
described above in Section \ref{sump}.
As can be seen in Figure \ref{unequalp}
this method returns a value for our $p_i=p$
example which
decreases with $k$ when $p$ below 0.5
and increases above.

Not surprisingly this method gives very similar results
to Edington's other method implemented
in \func{sump} and since it does not have the
numerical problems of that method it might perhaps be preferred.

\subsubsection{Examples for methods using untransformed $p$--values}

Using the same example dataset which we have already plotted and our cancellation
dataset we have the values in Table \ref{untransexamples}.
As can be seen \func{meanp} and \func{sump} cancel but the
other two do not.
Agreement here is not so good especially for the maximum p method.
Wilkinson's method not shown as it depends on the value of $r$.

\begin{table}[htbp]
\begin{tabular}{lrl}
Function name & \multicolumn{1}{c}{validity}  & \multicolumn{1}{c}{cancel} \\
 & \multicolumn{1}{c}{value expressed} \\
 & \multicolumn{1}{c}{as $-\log_{10}p$}  \\[1ex]
 \\[1ex]
\func{minimump} & \Sexpr{log10p(minimump(validity)$p)} & \Sexpr{round(minimump(cancel)$p, 5)} \\
\func{maximump} & \Sexpr{log10p(maximump(validity)$p)} & \Sexpr{round(maximump(cancel)$p, 5)} \\
\func{meanp} & \Sexpr{log10p(meanp(validity)$p)} & \Sexpr{meanp(cancel)$p}\\
\func{sump} & \Sexpr{log10p(sump(validity)$p)} & \Sexpr{sump(cancel)$p} \\
\end{tabular}
\caption{Examples for methods using the untransformed $p$ values}
\label{untransexamples}
\end{table}

\subsection{Other methods}

\subsubsection{The method of vote--counting}

A simple way of looking at the problem is
vote counting.
Strictly speaking this is not a method which combines
$p$--values in the same sense as the other methods.
If most of the studies have produced results in favour of the
alternative hypothesis irrespective of whether any of them is
individually significant then that might be regarded as evidence
for that alternative.
The numbers for and against may be compared with what
would be expected under the null using the binomial distribution.
A variation on this would allow for a neutral zone of studies
which are considered neither for nor against.
For instance one might only
count studies which have reached some conventional level of
statistical significance in the two different directions.

This method returns a value for our $p_i=p$
example which
is 1 for $p$
values above 0.5 and otherwise invariant with $p$
but decreases with $k$.
This method does cancel significant values in both
directions.

\begin{table}[htbp]
\begin{tabular}{lll}
Function name & validity  & cancel \\[1ex]
\func{votep} & \Sexpr{round(votep(validity)$p, 6)} &
 \Sexpr{round(votep(cancel)$p, 5)} \\
\end{tabular}
\caption{Examples for vote counting}
\label{votepexamples}
\end{table}

\subsubsection{Methods not using all $p$--values}

If there is a hypothesis that the signal will be
concentrated in only a few $p$--values then alternative methods
are available in \func{truncated}.
This is a wrapper to two packages available on
CRAN: \pkg{TFisher} which provides the
truncated Fisher method \citep{zaykin07,zhang18}
and \pkg{mutoss} which provides
the rank--truncated Fisher
method \citep{dudbridge03}.
Note that Table \ref{truncatedexamples} only shows
results for the validity data--set as, since the
methods explicitly only consider results in one direction
the cancellation issue does not arise.

\begin{table}[htbp]
\begin{tabular}{lll}
Function name & truncated at $p$ = 0.5  & truncated at rank = 5 \\[1ex]
\func{truncated} & \Sexpr{log10p(truncated(validity, ptrunc = 0.5)$p)}
 & \Sexpr{log10p(truncated(validity, rtrunc = 5)$p)} \\
\end{tabular}
\caption{Examples for truncated using the validity data--set
 expressed as $-\log_{10}p$}
\label{truncatedexamples}
\end{table}

\citet{dudbridge03} compare these two methods.
They comment that in a meta--analysis the method
using truncated Fisher may be preferred particularly
if reporting bias is suspected.
When the interest is in a small set of signals
in the presence of much noise as occurs in
genome wide association scanning then the rank
truncation method is recommended.
Their article provides more details about
the methods.

\section{Loughin's recommendations}

In his simulation study \citet{loughin04} carried out extensive comparisons.
Note that he did not consider all the methods implemented here.
These omissions are not too important for our purposes.
The methods implemented here as \func{invchisq}, \func{invt},
\func{meanp} and \func{meanz} are all very similar to ones which he did
study.
The truncation methods appeared about the same time as
his work but in any case are fundamentally different.
Vote counting is arguably not a method of the same sort.

As Loughin points out
the first thing to consider is whether large $p$--values
should cancel small ones.
If this is not desired then the only methods to
consider are those in \func{sumlog} (Fisher),
\func{minimump} (Tippett) and \func{maximump}.

\mypdf{struct}{Loughin's recommendations based on structure}{struct}{[height=6cm,width=8cm]}

He bases his recommendations on criteria of structure and the
arrangement of evidence against $H_0$.
Figure \ref{struct} shows a summary of his recommendations
about the structure of the evidence.

\mypdf{strength}{Loughin's recommendations based on where the strength of the evidence is located}{strength}{[height=8cm,width=12cm]}

Figure \ref{strength} summarise his recommendations
about the arrangement of evidence.

Overall he considered the choice to lie between
Stouffer's method, Fisher's method
and the logistic method
implemented in \func{logitp}.
As has already been mentioned Fisher's method cancels
whereas the other two do not so if the weak evidence
in a small number of $p$--values is not to be
over--whelmed by the others then Fisher is the
best choice.
However where the evidence is more evenly spread
Stouffer's method may be preferred.
The logistic method represents a compromise between
them and is perhaps best suited where the pattern
of evidence is not clear in advance.
The other methods are not universally ruled out
and may be helpful in the specific circumstance
outlined in his summaries.


\bibliography{metap}
\bibliographystyle{plainnat}
\end{document}

