📄 pa 765 discriminant function analysis.mht
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Subject: PA 765: Discriminant Function Analysis
Date: Sun, 20 Aug 2000 20:38:08 +0800
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<H1>Discriminant Function Analysis</H1></CENTER>
<P><BR>
<H2>Overview</H2>Discriminant function analysis, a.k.a. discriminant =
analysis or=20
DA, is used to classify cases into the values of a categorical =
dependent,=20
usually a dichotomy.If discriminant function analysis is effective for a =
set of=20
data, the classification table of correct and incorrect estimates will =
yield a=20
high percentage correct. There are several purposes for DA:=20
<P>
<UL>
<LI>To investigate differences between groups.=20
<LI>To determine the most parsimonious way to distinguish between =
groups.=20
<LI>To discard variables which are little related to group =
distinctions.=20
<LI>To classify cases into groups.=20
<LI>To test theory by observing whether cases are classified as =
predicted.=20
</LI></UL>
<P>Discriminant analysis shares all the usual assumptions of =
correlation,=20
requiring linear and homoscedastic relationships, and untruncated =
interval or=20
near interval data. Like multiple regression, it also assumes proper =
model=20
specification (inclusion of all important independents and exclusion of=20
extraneous variables). DA also assumes the dependent variable is a true=20
dichotomy since data which are forced into dichotomous coding are =
truncated,=20
attenuating correlation.=20
<P>DA is an earlier alternative to <A=20
href=3D"http://www2.chass.ncsu.edu/garson/pa765/logistic.htm">logistic=20
regression</A>, which is now frequently used in place of DA as it =
usually=20
involves fewer violations of assumptions, is robust, and has =
coefficients which=20
many find easier to interpret.. See also the separate topic on <A=20
href=3D"http://www2.chass.ncsu.edu/garson/pa765/mda.htm">multiple =
discriminant=20
function analysis</A> (MDA) for dependents with more than two =
categories.=20
<P>
<P><BR>
<H2>Key Terms and Concepts</H2>
<UL>
<P><A name=3Ddav></A>
<LI><B>Discriminating variables: </B>These are the independent =
variables, also=20
called <I>predictors</I>.=20
<P></P>
<LI><B>The criterion variable</B>. This is the dependent variable, =
which is=20
the object of classification efforts.=20
<P><A name=3Ddf></A></P>
<LI><B>Discriminant function: </B>A discriminant function, also called =
a=20
<I>canonical root</I>, is a latent variable which is created as a =
linear=20
combination of discriminating (independent) variables, such that L =3D =
b<SUB>1</SUB>x<SUB>1</SUB> + b<SUB>2</SUB>x<SUB>2</SUB> + ... +=20
b<SUB>n</SUB>x<SUB>n</SUB> + c, where the b's are discriminant =
coefficients,=20
the x's are discriminating variables, and c is a <A=20
=
href=3D"http://www2.chass.ncsu.edu/garson/pa765/discrim.htm#constant">con=
stant</A>.=20
This is analogous to multiple regression, but the b's are discriminant =
coefficients which maximize the distance between the means of the =
criterion=20
(dependent) variable. Note that the foregoing assumes the discriminant =
function is estimated using ordinary least-squares, the traditional =
method,=20
but there is also a version involving <A=20
=
href=3D"http://www2.chass.ncsu.edu/garson/pa765/discrim.htm#mle">maximum =
likelihood estimation</A>.=20
<P>
<UL>
<P><A name=3Dds></A>
<LI>The <B>discriminant score</B>, also called the DA score, is the =
value=20
resulting from applying a discriminant function formula to the data =
for a=20
given case. The <I>Z score</I> is the discriminant score for =
standardized=20
data.=20
<P></P>
<LI><B>Cutoff: </B>If the discriminant score of the function is less =
than or=20
equal to the cutoff, the case is classed as 0, or if above it is =
classed as=20
1. When group sizes are equal, the cutoff is the mean of the two =
centroids=20
(for two-group DA). If the groups are unequal, the cutoff is the =
weighted=20
mean.=20
<P><A name=3Dcoeff></A></P>
<LI><B>Unstandardized discriminant coefficients</B> are used in the =
formula=20
for making the classifications in DA, much as b coefficients are =
used in=20
regression in making predictions. The product of the unstandardized=20
coefficients with the observations yields the discriminant scores.=20
<P><A name=3Dcoeff2></A></P>
<LI><B>Standardized discriminant coefficients</B> are used to =
compare the=20
relative importance of the independent variables, much as beta =
weights are=20
used in regression.=20
<P></P>
<LI>The <B>group centroid</B> is the mean value for the discriminant =
scores=20
for a given category of the dependent. Two-group discriminant =
analysis has=20
two centroids, one for each group.=20
<P></P>
<LI><B>Number of discriminant functions</B>. There is one =
discriminant=20
function for 2-group discriminant analysis, but for higher order DA, =
the=20
number of functions (each with its own cut-off value) is the lesser =
of (g -=20
1), where g is the number of groups, or p,the number of =
discriminating=20
(independent) variables. Each discriminant function is orthogonal to =
the=20
others. See the section on <A=20
href=3D"http://www2.chass.ncsu.edu/garson/pa765/mda.htm">multiple =
discriminant=20
analysis</A>. </LI></UL>
<P><A name=3Dsignif></A></P>
<LI><B>Tests of significance</B>=20
<P>
<UL><A name=3Dlambda></A>
<LI><B>Wilks's lambda</B> is used in an <B>ANOVA (F) test of mean=20
differences</B> in DA, such that the <U>smaller</U> the lambda for =
an=20
independent variable, the <U>more</U> that variable contributes to =
the=20
discriminant function. Lambda varies from 0 to 1, with 0 meaning =
group means=20
differ (thus the more the variable differentiates the groups), and 1 =
meaning=20
all group means are the same. The F test of Wilks's lambda shows =
which=20
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