<!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>Regression Analysis</TITLE>
<META NAME="description" CONTENT="Regression Analysis">
<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="node29.html">
<LINK REL="previous" HREF="node27.html">
<LINK REL="up" HREF="node27.html">
<LINK REL="next" HREF="node29.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html1785"
HREF="node29.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html1782"
HREF="node27.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html1776"
HREF="node27.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1784"
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="tex2html1786"
HREF="node29.html">Statistical Tests</A>
<B> Up:</B> <A NAME="tex2html1783"
HREF="node27.html">Analysis of Results</A>
<B> Previous:</B> <A NAME="tex2html1777"
HREF="node27.html">Analysis of Results</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION00551000000000000000">
Regression Analysis</A>
</H2>
The degree of correlation between two parameters can visually be
estimated by plotting them. Numerically this is expressed by the
correlation coefficient :
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
\rho = \frac{\sum(x-\bar{x})(y-\bar{y})}
{\sqrt{\sum (x-\bar{x})^2 \sum (y-\bar{y})^2}}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:corr-coef"> </A><IMG
WIDTH="263" HEIGHT="63"
SRC="img121.gif"
ALT="\begin{displaymath}\rho = \frac{\sum(x-\bar{x})(y-\bar{y})}
{\sqrt{\sum (x-\bar{x})^2 \sum (y-\bar{y})^2}}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.32)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
for the two variables <I>x</I> and <I>y</I>. This expression assumes that the
quantities have normal distribution. When the distribution is
unknown, the correlation can be given by the Spearman rank correlation
coefficient :
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{equation}
r_s = 1 - \frac{6\sum D^2}{n(n^2-1)}
\end{equation} -->
<TABLE WIDTH="100%" ALIGN="CENTER">
<TR VALIGN="MIDDLE"><TD ALIGN="CENTER" NOWRAP><A NAME="eq:sr-corr-coef"> </A><IMG
WIDTH="177" HEIGHT="63"
SRC="img122.gif"
ALT="\begin{displaymath}r_s = 1 - \frac{6\sum D^2}{n(n^2-1)}
\end{displaymath}"></TD>
<TD WIDTH=10 ALIGN="RIGHT">
(2.33)</TD></TR>
</TABLE>
</DIV>
<BR CLEAR="ALL"><P></P>
where <I>n</I> is the number of pairs and <I>D</I> is the difference of the rank
between <I>x</I> and <I>y</I> in a pair.
<P>
When the correlation coefficient indicates a significant correlation,
the functional relation is given by the regression line which is
computed by least squares methods for normal distributions or maximum
likelihood procedures (see Section <A HREF="node14.html#estim">2.1.3</A>). If non-linear
relations are expected, the variables are transformed into a linear
domain by means of standard mathematical functions. Such
transformation may also be used to make the variance of a variable
homogenous.
<BR>
<DIV ALIGN="CENTER"><A NAME="fig:regr-var"> </A><A NAME="854"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 2.13:</STRONG>
Correlation between two measures of the inclination angle of
galaxies: (A) with angle <I>i</I> as variable, and (B) with
<IMG
WIDTH="60" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
SRC="img124.gif"
ALT="$\cos(i)$">.</CAPTION>
<TR><TD><IMG
WIDTH="1105" HEIGHT="531"
SRC="img125.gif"
ALT="\begin{figure}\psfig{figure=fig13_correlation.eps,width=15cm,clip=} \end{figure}"></TD></TR>
</TABLE>
</DIV>
<BR>
In Figure <A HREF="node28.html#fig:regr-var">2.13</A>, a correlation between two different
measures of the inclination angle <I>i</I> of galaxies is shown. Since the
angle is obtained from the axial ratio, a cosine transformation is
used to achieve a homogenous variance. Unfortunately, it is not
always possible to obtain both a linear relation and homogeneity of
variance with the same transformation.
<P>
All methods assume that the single data points are unbiased and
uncorrelated. Great care must be taken to assure that this is
fulfilled. If this is not the case, significant systematic errors may
be introduced in the estimates. A standard example is magnitude
limited samples which may be affected by the Malmquist bias.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html1785"
HREF="node29.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="icons.gif/next_motif.gif"></A>
<A NAME="tex2html1782"
HREF="node27.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="icons.gif/up_motif.gif"></A>
<A NAME="tex2html1776"
HREF="node27.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1784"
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="tex2html1786"
HREF="node29.html">Statistical Tests</A>
<B> Up:</B> <A NAME="tex2html1783"
HREF="node27.html">Analysis of Results</A>
<B> Previous:</B> <A NAME="tex2html1777"
HREF="node27.html">Analysis of Results</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
</ADDRESS>
</BODY>
</HTML>
