<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998)
originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>Statistical Tests</TITLE>
<META NAME="description" CONTENT="Statistical Tests">
<META NAME="keywords" CONTENT="vol2">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="vol2.css">
<LINK REL="next" HREF="node30.html">
<LINK REL="previous" HREF="node28.html">
<LINK REL="up" HREF="node27.html">
<LINK REL="next" HREF="node30.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html1796"
HREF="node30.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html1793"
HREF="node27.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html1787"
HREF="node28.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1795"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1797"
HREF="node30.html">Multivariate Statistical Methods</A>
<B> Up:</B> <A NAME="tex2html1794"
HREF="node27.html">Analysis of Results</A>
<B> Previous:</B> <A NAME="tex2html1788"
HREF="node28.html">Regression Analysis</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION00552000000000000000">
Statistical Tests</A>
</H2>
Often, the computed estimators have to be compared with other values
or model predictions. Different statistical tests are used to compute
the probability of a given hypotheses being correct. A typical null
hypotheses is that two quantities or samples are taken from the same
population. For single quantities, a confidence interval is estimated
for the desired significance level. The null hypotheses is then
accepted at this level of significance if the value is within the
interval. When the underlying distribution is normal, the
``Student's'' <I>t</I> and the <IMG
WIDTH="30" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
SRC="img126.gif"
ALT="$\chi^2$">
distributions are used to estimate
the confidence intervals for the mean and standard deviation,
respectively.
<P>
It is also possible to test if a set of observed values are taken from
a given distribution.
For this purpose the test variable
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
\hat{\chi}^2 = \sum \frac{(O-E)^2}{E}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:chi2-test"> </A><IMG
WIDTH="175" HEIGHT="57"
SRC="img127.gif"
ALT="\begin{displaymath}\hat{\chi}^2 = \sum \frac{(O-E)^2}{E}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.34)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
is used where <I>O</I> and <I>E</I> are the observed and expected frequencies,
respectively. The level of significance is derived from
<!-- MATH: $\hat{\chi}^2$ -->
<IMG
WIDTH="30" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
SRC="img128.gif"
ALT="$\hat{\chi}^2$">
which is <IMG
WIDTH="29" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
SRC="img129.gif"
ALT="$\chi^2$">
distributed. The bins must be so
large that <I>E</I> is larger than 5 for all intervals.
<P>
When the underlying distribution is unknown, two independant samples can
be compared using the Kolmogoroff and Smirnoff test.
It uses the test variable
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
D = \max \left| \left(\frac{F_1}{n_1} - \frac{F_2}{n_2} \right)_i \right|
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:ks-test"> </A><IMG
WIDTH="221" HEIGHT="58"
SRC="img130.gif"
ALT="\begin{displaymath}D = \max \left\vert \left(\frac{F_1}{n_1} - \frac{F_2}{n_2} \right)_i \right\vert
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.35)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <I>F</I><SUB><I>j</I></SUB> is the cumulative frequency in the <I>i</I><SUP><I>th</I></SUP> interval with
<I>n</I><SUB><I>j</I></SUB> values for the two samples <I>j</I>=1,2. The intervals must be of
equal size and have the same limits for both samples. Special tables
give the confidence interval for this test variable. Several other
tests are available for comparing independent samples such as the
U-test of Wilcoxon, Mann and Whitney which uses the rank in its test
variable.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html1796"
HREF="node30.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html1793"
HREF="node27.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html1787"
HREF="node28.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1795"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1797"
HREF="node30.html">Multivariate Statistical Methods</A>
<B> Up:</B> <A NAME="tex2html1794"
HREF="node27.html">Analysis of Results</A>
<B> Previous:</B> <A NAME="tex2html1788"
HREF="node28.html">Regression Analysis</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
