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\usepackage[pdftex,
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pdfauthor={Bradley Efron, Brit B. Turnbull and Balasubramanian Narasimhan},
pdftitle={locfdr Vignette}]
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\title{locfdr Vignette\\
Complete Help Documentation\\ Including Usage Tips and Simulation Example}
\author{Bradley Efron, Brit B. Turnbull and Balasubramanian Narasimhan\\
  Department of Statistics\\
  Stanford University\\ 
  Stanford, CA 94305}

\date{\today}

\SweaveOpts{echo=TRUE, pdf=TRUE, eps=FALSE}

\begin{document}
%\VignetteIndexEntry{Local FDR}
%\VignetteKeywords{FDR, local FDR}
%\VignettePackage{locfdr}
\maketitle

This vignette includes locfdr's complete help documentation, including
usage tips, which could not fit in the R help file.  It also
demonstrates usage of locfdr through an example using the
simulated data included in the package.

\section{Description and Usage}

locfdr computes local false discovery rates, following the definitions and
  description in the references listed below.

\begin{verbatim}
locfdr(zz, bre=120, df=7, pct=0, pct0=1/4, nulltype=1, type=0, plot=1,
        mult, mlests, main=" ", sw=0)
\end{verbatim}

\section{Arguments}

\subsection{zz}
  zz is a vector of summary statistics, one for each case under
  consideration.  In a microarray experiment, there would be one
  element of zz for each gene, perhaps a $t$-statistic comparing
  gene expression levels under two different conditions.  The
  calculations  assume a large number of cases, say
  \texttt{length(zz)} exceeding 200.

Results may be improved by transforming
  zz so that its elements are theoretically distributed
  as $N(0,1)$ under the null hypothesis.
  For example, when using $t$-statistics, transform them by
  \texttt{zz = qnorm(pt(t,df)).}  Recentering and rescaling zz may be
  necessary if its central histogram looks very far removed from mean 0 and variance 1.

  When using a permutation null
  distribution with sample zperm, transform the original
  statistics zorig by 
  \texttt{zz = qnorm(ecdf(zperm)(zorig)).}
  Such transformation is especially
  important when the theoretical null option is invoked (see nulltype below).


\subsection{bre}

bre is the number of breaks in the discretization of the $z$-score axis,
        or a vector of breakpoints fully describing the
        discretization.  If \texttt{length(zz)} is small, such as when the
        number of cases is less than about 1000, set bre to a number
        less than the default of 120.

\subsection{df}

df is the degrees of freedom for fitting the estimated
        density $f(z)$ (see type below). Larger values of df may be required if
        $f(z)$ has sharp bends or other irregularities. A warning
        is issued if the fitted curve does not adequately match the
        histogram counts. It is a good idea to use the plot option to
        view the histogram and fitted curve.

\subsection{pct}

pct is the excluded tail proportions of zz's when fitting
           $f$. The default \texttt{pct=0} includes the full range of
           zz's. pct can also be a 2-vector, describing the fitting range.

\subsection{pct0}

pct0 is the proportion of the zz distribution used in fitting the
            null density $f_0$ by central matching.  If it is a 2-vector,
            e.g. \texttt{pct0=c(0.25,0.60)}, the range \texttt{[pct0[1],
            pct0[2]]} is used.  If a scalar, \texttt{[pct0, 1-pct0]} is used.

\subsection{nulltype}

nulltype is the type of null hypothesis assumed in estimating $f_0$,
        for use in the fdr calculations.
\begin{itemize}
\item 0 is the
        theoretical null $N(0,1)$, which assumes that zz
        has been scaled to have a $N(0,1)$ distribution under
        the null hypothesis.
\item 1 (the default) is the empirical null
        with parameters estimated by maximum likelihood.
\item 2 is the empirical null
        with parameters estimated by central matching (see
        \cite{size}).
\item 3 is a ``split normal'' version of 2, in
        which $f_0(z)$ is allowed to have different
        scales on the two sides of the maximum.
\end{itemize}

Unless sw is set to 2 or 3, the theoretical, maximum
        likelihood, and central matching estimates all will be output in
        the matrix fp0, and both the theoretical and the specified
        nulltype will be used in the calculations output in mat, but
        only the specified nulltype is used in the calculation of the
        output fdr (local fdr estimates for every case).

\subsection{type}
type is the type of fitting used for $f$.
\begin{itemize}
\item 0 is a natural spline.
\item 1 is a polynomial.
\end{itemize}
In either case, $f$ is fit with degrees of freedom df (so
        total degrees of freedom including the intercept is $\mbox{df}+1$).

\subsection{plot}

plot specifies the plots desired.
\begin{itemize}
\item 0 gives no plots.
\item 1 (the default) gives a single
            plot showing the histogram of zz and fitted
            mixture density $f$ (green solid curve) and null
            subdensity $p_0f_0$ (blue dashed curve).
            Colored histogram
            bars indicate estimated non-null counts.  Yellow triangles
            on the zz-axis indicate threshold values for $fdr(z) \leq
            0.2$, if such cases exist.
\item 2 also gives plot of fdr, and the right and
            left tail area Fdr curves.
\item 3 gives instead the $f_1$ cdf
            of the estimated fdr curve, as in Figure 4 of \cite{size}.
\item 4 gives all three plots.
\end{itemize}

We recommend setting plot to 1 or greater, to check the fit of
              $p_0f_0$ to the histogram.  (If the fit is poor, try
              a different
              nulltype or a different value of the mlests argument.)

\subsection{mult}

mult is an optional scalar multiple (or vector of multiples) of the
            sample size for calculation of the corresponding
            hypothetical Efdr value(s).

\subsection{mlests}

mlests is an optional vector of initial values for $(\delta_0, \sigma_0)$ in
  the maximum likelihood iteration.  In addition, these are used to
  determine the interval over which the maximum likelihood estimation is
performed.  If, for example, zz was transformed quantile-wise from F
statistics, most of zz's elements corresponding to interesting features will
be positive.  To shift the interval away from such elements,
specify a negative initial value for $\delta_0$, the first element of
mlests.  If the default results in a poor fit of $p_0f_0$ to the
histogram in the first plot, try setting mlests to move the estimates
toward the values suggested by the histogram.


\subsection{main}
main is the main heading for the histogram plot.

\subsection{sw}
sw determines the type of output desired.
\begin{itemize}
\item 2 gives a list
          consisting of the last 5 values listed under Value below.
\item 3 gives
          the square matrix of dimension bre-1 representing the influence
          function of $\log(fdr)$.  The
          $(i,j)$ entry of the matrix is the derivative of
          $\log(fdr)$ at the midpoint of bin $i$ with respect to the
          count value of bin $j$.
\item Any other value
          of sw returns a list consisting of the first 7 (8 if mult is
          supplied) values listed below.
\end{itemize}

\section{Value}

\subsection{fdr}
fdr is the estimated local false discovery rate for each case,
        using the selected type and nulltype.

\subsection{fp0}
fp0 is a matrix containing the estimated parameters delta (mean of
$f_0$), sigma (standard deviation of $f_0$), and p0 (proportion of
tests that are null), along with their estimated standard errors.  If \texttt{nulltype<3}, fp0 is a $5\times 3$ matrix,
        with columns representing delta, sigma, and p0 and rows
        representing nulltypes and estimate vs. standard
        error.  If \texttt{nulltype==3}, the second column corresponds
        to the estimate of sigma for the left side of $f_0$, and a
        fourth column corresponds to the sigma estimate for the right.

\subsection{Efdr}
Efdr is the expected local false discovery rate for the non-null cases,
        a measure of the experiment's power as described in Section 3
        of \cite{size}. Large values of Efdr, say \texttt{Efdr>0.4},
        indicate low power. Overall Efdr and right and left values are
        given, both for the specified nulltype and for nulltype 0.  (If
        \texttt{nulltype==0}, values are given for nulltypes 1 and 0.)

\subsection{cdf1}
cdf1 is a $99\times 2$ matrix giving the estimated cdf of fdr under the
        non-null distribution $f_1$. Large values of the cdf for small fdr
        values indicate good power.  See Section 3 of \cite{size}.
        Set plot to 3 or 4 to see the plot of cdf1.

\subsection{mat}

mat is a $(\mbox{bre}-1)\times 11$ matrix, convenient for
        comparisons and plotting.  Each row corresponds to a
bin of the zz histogram, and the columns contain the following:
\begin{enumerate}
\item x: the midpoint of the bin.
\item fdr: the estimated local false discovery rate at $x$, calculated based on
  the specified type and nulltype (using \texttt{nulltype=1} if
  \texttt{nulltype=0} is specified).
\item Fdrleft: the left tail false discovery rate at $x$.
\item Fdrright: the right tail false discovery rate at $x$.
\item f: the mixture density estimate at $x$, calculated based on the
  specified type, df, and pct, scaled to sum to \texttt{length(zz)}.
\item f0: the null density estimate at $x$, calculated based on the
  specified nulltype (using \texttt{nulltype=1} if \texttt{nulltype=0}
  is specified) and pct0 and scaled to sum to \texttt{length(zz)}.
\item f0theo: the null density estimate at $x$, calculated using the
  theoretical null $N(0,1)$ and scaled to sum to \texttt{length(zz)}.
\item fdrtheo: the local false discovery rate at $x$, calculated based on
  the specified type and \texttt{nulltype=0}.
\item counts: the number of elements of zz in the bin.
\item lfdrse: the delta-method estimate of the standard error of the
  log of the local false discovery rate for the specified nulltype.  
  This estimate assumes independence of the
        zz values and should usually be considered as a lower bound on
        the true standard errors. See \cite{size}.
\item p1f1: the estimated subdensity of the zz elements that come from
  non-null tests.  p1f1 is scaled to sum to approximately (1-p0) times
  \texttt{length(z)}, i.e. the estimated number of non-null tests.
\end{enumerate}

\subsection{z.2}

z.2 is the interval along the zz-axis outside of which $fdr(z)<0.2$,
the locations of the yellow triangles in the histogram plot.  If no
elements of zz on the left or right satisfy the criterion, the
corresponding element of z.2 is NA, and the corresponding triangle
does not appear.

\subsection{call}

call is the function call.

\subsection{mult}

If the argument mult was supplied, the value mult is the vector of the
ratios of the hypothetical Efdr for the supplied multiples of the
sample size to the Efdr for the actual sample size.

\subsection{pds}

pds is the vector of estimates of p0, delta, and sigma.

\subsection{x}

x is the vector of bin midpoints.

\subsection{f}

f is the vector of estimated values of $f(x)$ at the bin midpoints.

\subsection{pds.}

pds. is the derivative of the estimates of p0, delta, and sigma with
            respect to the bin counts.

\subsection{stdev}
stdev is the vector of delta-method estimates of the standard deviations of
             the p0, delta, and sigma estimates.


\section{Simulation Example}

This simulation example involves 2000 ``genes'', each of which has
yielded a test statistic $z_i$, with $z_i \sim N(\mu_i, 1)$,
independently for $i=1,2,\ldots,2000$.

Here $\mu_i$ is the ``true score'' of gene $i$, which we observe only
noisily. 1800 (90\%) of the $\mu_i$ values are zero; the remaining 200
(10\%) are from a $N(3,1)$ distribution. The data are contained in the
dataset \texttt{lfdrsim}, where the $z_i$ are the column \texttt{zex}.  

<<Preliminaries>>=
library(locfdr)
data(lfdrsim)
zex <- lfdrsim[, 2]
@ 

If we are confident that the null $z_i$'s are distributed as $N(0,1)$,
we run \texttt{locfdr} with \texttt{nulltype=0}.  Otherwise, we use
the default \texttt{nulltype=1}, which uses empirical estimates of the
null density parameters.

<<Compute-local-fdr, fig=TRUE, echo=TRUE>>=
w <- locfdr(zex)
@ 

In the figure, the green solid line is the spline-based estimate of
the mixture density $f$.  The blue dashed line is the null
subdensity $p_0f_0$, estimated by default by maximum likelihood
(nulltype=1).  Whichever nulltype is specified, \texttt{locfdr}
returns a matrix \texttt{fp0} containing parameters of all three
nulltypes and corresponding estimates of the proportion $p_0$ of cases
that are null, along with standard errors.  In this example, the null
distribution is $N(0,1)$, and both the MLE and central matching
estimates come close to this.


<<Show fp0>>=
w$fp0
@ 

The output mat contains estimates of the local false discovery rates
and other functions for each bin midpoint $x$.

<<Show mat>>=
w$mat[1:5,]
@ 

The output fdr contains the local false discovery rate estimate
for each $z_i$.  One might use this vector to create a list of
Interesting cases.

<<list interesting cases>>=
which(w$fdr<.2)
@ 

Here $0.2$ is a rule-of-thumb cut-off.  In the simulated data, the
first 200 cases have nonzero $\mu_i$.  So we can find the observed tail
false discovery proportion.

<<tail FDP>>=
sum(which(w$fdr<.2)>200)/sum(w$fdr<.2)
@ 

The estimated tail FDR can be found from the \texttt{mat} output.

<<tail FDR>>=
w$mat[which(w$mat[,"fdr"]<.2)[1],"Fdrright"]
@ 

The tail FDR is the mean local fdr over the entire tail and is
therefore smaller than the local fdr cutoff.

\begin{thebibliography}{99}
\bibitem{choice} Efron, B. (2004) ``Large-scale simultaneous hypothesis
  testing: the choice of a null hypothesis,'' \textit{JASA}, \textbf{99},
  pp. 96--104.
\bibitem{locfdr} Efron, B. (2005) ``Local False Discovery Rates,''\\
\texttt{http://www-stat.stanford.edu/}$\tilde{\mbox{\,\,\,\,\,}}$\texttt{brad/papers/False.pdf}
\bibitem{size} Efron, B. (2006) ``Size, Power, and False Discovery Rates,''\\
\texttt{http://www-stat.stanford.edu/}$\tilde{\mbox{\,\,\,\,\,}}$\texttt{brad/papers/Size.pdf}
\bibitem{cor} Efron, B. (2006) ``Correlation and Large-Scale Simultaneous
  Significance Testing,'' \textit{JASA}, \textbf{102}, pp. 93--103.
\end{thebibliography}

\end{document}