[Gsdesign-commits] r167 - pkg/tex/Sections

[noreply at r-forge.r-project.org]

Author: keaven
Date: 2009-05-22 23:42:04 +0200 (Fri, 22 May 2009)
New Revision: 167

Added:
   pkg/tex/Sections/ConditionalPower.tex
Removed:
   pkg/tex/Sections/ConditionalPower.tex.undo
Log:
v 2 manual completion

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+\section{Conditional power and B-values\label{sec:CPB}}
+\subsection{Group sequential test statistics as sums of independent increments\label{sec:increments}}
+In some cases, rather than working with $Z_{1}$, $Z_{2}$,...,$Z_{k}$ as in Section \ref{sec:gsProbability}, it is desirable to consider
+variables representing incremental sets of observations between analyses.
+This approach will be applied here to define conditional power and B-values, two common measures of interim results and boundaries.
+Letting $I_{0}=n_{0}=0$ we define $Y_{i}=\sum_{j=n_{i-1}+1}^{n_{i}}X_{j}%
+/\sqrt{I_{i}-I_{i-1}}$ for $i=1,2,\ldots,k$. This implies $Y_{1}%
+,Y_{2},...,Y_{k}$ are independent and normally distributed with
+\begin{equation}
+Y_{i}\sim N(\sqrt{I_{i}-I_{i-1}}\theta,1),\text{ }%
+i=1,2,...,k.\label{Yi joint dist}%
+\end{equation}
+For $i=1,2,...,k$ if we let $w_{i}=\sqrt{I_{i}-I_{i-1}}$, note that
+\begin{equation}
+Z_{i}=\frac{\sum_{j=1}^{i}\sqrt{I_{j}-I_{j-1}}Y_{j}}{\sqrt{I_{i}}}=\frac
+{\sum_{j=1}^{i}w_{j}Y_{j}}{\sqrt{\sum_{j=1}^{i}w_{j}^{2}}}%
+.\label{Z sum of ind Y}%
+\end{equation}
+Finally, we define notation for independent increments between arbitrary
+analyses. Select $i$ and $j$ with $1\leq i<j\leq k$ and let $Z_{i,j}%
+=\sum_{m=n_{i}+1}^{n_{j}}X_{m}/\sqrt{I_{_{j}}-I_{i}}$. Thus, $Z_{i,j}\sim
+N(\sqrt{I_{j}-I_{i}}\theta,1)$ is independent of $Z_{i}$\ and
+\begin{equation}
+Z_{j}=\frac{\sqrt{I_{i}}Z_{i}+\sqrt{I_{j}-I_{i}}Z_{i,j}}{\sqrt{I_{j}}%
+}.\label{Zj as ind inc}
+\end{equation}
+By definition, for $i=2,3,...k$,
+\begin{equation}
+Y_{i}=Z_{i-1,i}.\label{Yi=Zi-1,i}%
+\end{equation}
+\bigskip
+
+For the more general canonical form not defined using $X_{1},X_{2},...$ we
+define $Y_{1}=Z_{1}$ and for $1\leq j<i\leq k$
+\begin{equation}
+Z_{j,i}=\frac{\sqrt{I_{i}}Z_{i}-\sqrt{I_{j}}Z_{j}}{\sqrt{I_{i}-I_{j}}%
+}.\label{Zij implicit}%
+\end{equation}
+\bigskip The variables $Z_{j,i}$ and $Z_{j}$ are independent, as before, for
+any $1\leq j<i\leq k$. We use (\ref{Yi=Zi-1,i}) to define $Y_{i}$,
+$i=2,3,...,k.$ As before $Y_{i}\symbol{126}N(\sqrt{I_{i}-I_{i-1}}\theta,1)$,
+$1<i\leq k,$ and these random variables are independent of each other.
+
+\subsection{Conditional power\label{sec:CP}}
+
+\bigskip
+As an alternative to $\beta$-spending, stopping
+rules for futility are interpreted by considering the conditional power of a
+positive trial given the value of a test statistic at an interim analysis.
+Thus, we consider the conditional probabities of boundary crossing for a group
+sequential design given an interim result. Assume $1\leq$ $i<m\leq j\leq k$
+and let $z_{i}$\ be any real value. Define%
+\begin{equation}
+u_{m,j}(z_{i})=\frac{u_{j}\sqrt{I_{j}}-z_{i}\sqrt{I_{m}}}{\sqrt{(I_{j}-I_{m}%
+)}}\label{umj}%
+\end{equation}
+and
+\begin{equation}
+l_{m,j}(z_{i})=\frac{l_{j}\sqrt{I_{j}}-z_{i}\sqrt{I_{m}}}{\sqrt{(I_{j}-I_{m}%
+)}}.\label{lmj}%
+\end{equation}
+Recall (\ref{Zj as ind inc}) and consider the conditional probabilities%
+\begin{align}
+\alpha_{i,j}(\theta|z_{i})  & =P_{\theta}\{\{Z_{j}\geqslant u_{j}%
+\}\cap_{m=i+1}^{j-1}\{l_{m}<Z_{m}<u_{m}\}|Z_{i}=z_{i}%
+\}\label{Cond lower bound prob}\\
+& =P_{\theta}\left\{  \left\{  \frac{\sqrt{I_{i}}z_{i}+\sqrt{I_{j}-I_{i}%
+}Z_{i,j}}{\sqrt{I_{j}}}\geqslant u_{j}\right\}  \cap_{m=i+1}^{j-1}\left\{
+l_{m}<\frac{\sqrt{I_{i}}z_{i}+\sqrt{I_{j}-I_{i}}Z_{i,m}}{\sqrt{I_{j}}%
+}\right\}  <u_{m}\right\} \nonumber\\
+& =P_{\theta}\{\{Z_{i,j}\geqslant u_{i,j}(z_{i})\}\cap_{m=i+1}^{j-1}%
+\{l_{m,j}(z_{i})<Z_{m,j}<u_{m,j}(z_{i})\}\}.\nonumber
+\end{align}
+This last line is of the same general form as $\alpha_{i}(\theta)$ and can
+thus be computed in a similar fashion. For a non-binding bound, the same logic
+applied ignoring the lower bound yields%
+
+\begin{align}
+\alpha_{i,j}^{+}(\theta|z_{i})  & =P_{\theta}\{\{Z_{j}\geqslant u_{j}%
+\}\cap_{m=i+1}^{j-1}\{Z_{m}<u_{m}\}|Z_{i}=z_{i}\}\label{alphaij+}\\
+& =P_{\theta}\{\{Z_{i,j}\geqslant u_{i,j}(z_{i})\}\cap_{m=i+1}^{j-1}%
+\{Z_{m,j}<u_{m,j}(z_{i})\}\}.\nonumber
+\end{align}
+Finally, the conditional probability of crossing a lower bound at analysis $j$ given a test statistic $z_{i}$ at analysis $i$ is denoted by
+\begin{align}
+\beta_{i,j}(\theta|z_{i})  & =P_{\theta}\{\{Z_{j}\leq l_{j}\}\cap
+_{m=i+1}^{j-1}\{l_{m}<Z_{m}<u_{m}\}|Z_{i}=z_{i}\}\label{Conditional power}\\
+& =P_{\theta}\{\{Z_{i,j}\leq l_{i,j}(z_{i})\}\cap_{m=i+1}^{j-1}\{l_{m,j}
+(z_{i})<Z_{m,j}<u_{m,j}(z_{i})\}\}.\nonumber
+\end{align}
+Since $\alpha_{i,j}^{+}(\theta|z_{i})$ and $\beta_{i,j}(\theta|z_{i})$\ are of
+the same general form as $\alpha_{i}^{+}(\theta)$ and $\beta_{i}(\theta)$,
+respectively, they can be computed using the same tools.
+
+\subsection{B-values}
+Proschan, Lan and Wittes \cite{PLWBook}.

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