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```					          CHAPTER 26
Discriminant Analysis

Tables, Figures, and Equations

From: McCune, B. & J. B. Grace. 2002. Analysis of
Ecological Communities. MjM Software Design,
Gleneden Beach, Oregon http://www.pcord.com
Purposes:
1. Summarizing the differences between groups (often used as a
follow-up to clustering, to help describe the groups);
"descriptive discriminant analysis." With community data, you
could use indicator species analysis as a nonparametric
alternative.
Purposes:
1. Summarizing the differences between groups (often used as a
follow-up to clustering, to help describe the groups);
"descriptive discriminant analysis." With community data, you
could use indicator species analysis as a nonparametric
alternative.
2. Multivariate testing of whether or not two or more groups
differ significantly from each other. For ecological community
data this is better done with MRPP, thus avoiding the
assumptions listed below.
Purposes:
1. Summarizing the differences between groups (often used as a
follow-up to clustering, to help describe the groups);
"descriptive discriminant analysis." With community data, you
could use indicator species analysis as a nonparametric
alternative.
2. Multivariate testing of whether or not two or more groups
differ significantly from each other. For ecological community
data this is better done with MRPP, thus avoiding the
assumptions listed below.
3. Determining the dimensionality of group differences.
Purposes:
1. Summarizing the differences between groups (often used as a
follow-up to clustering, to help describe the groups);
"descriptive discriminant analysis." With community data, you
could use indicator species analysis as a nonparametric
alternative.
2. Multivariate testing of whether or not two or more groups
differ significantly from each other. For ecological community
data this is better done with MRPP, thus avoiding the
assumptions listed below.
3. Determining the dimensionality of group differences.
4. Checking for misclassified items.
Purposes (cont.):
5. Predicting group membership or classifying new cases
("predictive discriminant analysis").
Purposes (cont.):
5. Predicting group membership or classifying new cases
("predictive discriminant analysis").
6. Comparing occupied vs. unoccupied habitat to determine the
habitat characteristics that allow or prevent a species' existence.
DA has been widely used for this purpose in wildlife studies
and rare plant studies.
Assumptions
1. Homogeneous within-group variances

2. Multivariate normality within groups.

3. Linearity among all pairs of variables.

4. Prior probabilities.
How it works
The "direct" procedure is described below.

1. Calculate variance/covariance matrix for each group.
How it works
The "direct" procedure is described below.

1. Calculate variance/covariance matrix for each group.

2. Calculate pooled variance/covariance matrix (Sp) from the
above matrices.
How it works
The "direct" procedure is described below.

1. Calculate variance/covariance matrix for each group.

2. Calculate pooled variance/covariance matrix (Sp) from the
above matrices.

3. Calculate between group variance (Sg) for each variable.
4. Maximize the F-ratio:

y  Sg y
F =
y  Sp y

where the y is an the eigenvector associated with a
particular discriminant function.
We seek y to maximize F.
Maximize this ratio by finding the partial derivatives with
a characteristic equation:

| S Sg -  I| = 0
-1
p

The number of roots is g-1, where g is number of groups.
In other words, the number of functions (axes) derived is
one less than the number of groups.
The eigenvalues thus express the percent of variance
among groups explained by those axes.
6. Solve for each eigenvector y (also known as the
"canonical variates" or "discriminant functions").

[S Sg -  I ]y = 0
-1
p
7. Locate points (sample units) on each axis.

X = AY
X = scores (coordinates) for n rows (sample units) on m
dimensions, where m = g-1.
A = original data matrix of n rows by p columns

Each eigenvector is known as a discriminant function.
These unstandardized discriminant functions Y can be used as
(linear) prediction equations, assigning scores to unclassified
items.

Standardized discriminant function coefficients standardize to
unit variance. The absolute value of these coefficients indicate
the relative importance of the individual variables in
contributing to the discriminant function.
8. Classification phase.

a. Derive a classification equation for each group, one term
in the equation for each variable, plus a constant.
b. Insert data values for a given SU to calculate a
classification score for each group for that SU.
c. The SU is assigned to the group in which it had the
highest score.

The coefficients in the equation are derived from:
p  p within-group variance-covariance matrix (Sp) and
p  1 vector of the means for each variable in group k, Mk.

First, calculate W by dividing each term of Sp by the
within-group degrees of freedom. Then:
8. Classification phase, cont.

The coefficients in the equation are derived from:
p  p within-group variance-covariance matrix (Sp)
p  1 vector of the means for each variable in group k, Mk.

First, calculate W by dividing each term of Sp by the within-
group degrees of freedom.

Then:       Ck = W-1 M k
1
The constant is derived as:      constant k   = - Ck M k
2

The constant and the coefficients in Ck define a linear
equation of the usual form, one equation for each group k.
Summary statistics
 Wilk's lambda (). Wilk's  is the error sum of squares
divided by the sum of the effect sum of squares and the
error sum of squares. Thus, it is the variance among the
objects not explained by the discriminant functions. It
ranges from zero (perfect separation of groups) to one
(no separation of groups).

Statistical significance of lambda is tested with a chi-
square approximation.

 Chi-square (derived from Wilk’s lambda).

 Variance explained.
Figure 26.1. Comparison of DA and PCA. Groups are tighter in DA than in
PCA because DA maximizes group separation while PCA maximizes the
representation of variance among individual points. Groups were
superimposed on an ordination of pine species in ecological trait space (after
McCune 1988). Pinus resinosa was not assigned to a group, so it does not
appear in the DA ordination.
Table 26.1. Predictions of goshawk nesting sites from DA
compared to actual results, in one case using equal prior
probabilities, in the other case using prior probabilities based
on the occupancy rate of landscape cells. The first value of
0.83 means that 83% of the sites that were predicted by DA to
be nesting sites actually were nesting sites.

Predicted with             Predicted with
EQUAL PRIORS              UNEQUAL PRIORS
Nest       Not nest        Nest      Not nest
Actual   Nest       0.83        0.17           0.48        0.52
Not nest   0.17        0.83           0.02        0.98
Table 26.1. Predictions of goshawk nesting sites from DA
compared to actual results, in one case using equal prior
probabilities, in the other case using prior probabilities based
on the occupancy rate of landscape cells. The first value of
0.83 means that 83% of the sites that were predicted by DA to
be nesting sites actually were nesting sites.

Predicted with               Predicted with
EQUAL PRIORS                UNEQUAL PRIORS
Nest       Not nest          Nest      Not nest
Actual   Nest     0.5    0.83        0.17      0.07   0.48       0.52
Not nest 0.5    0.17        0.83      0.93   0.02       0.98

priors                          priors
EQUAL priors:
No. non-nests predicted nests = p(predicted nest but not nest) 
number of non-nests
= 0.17  93      False
= 15.8           positives

No. nests predicted non-nests = p(predicted not nest but nest) 
number of nests
= 0.17  7       False
= 1.2            negatives
Total number of errors         = 15.8 + 1.2
= 17
UNEQUAL priors:
No. non-nests predicted nests = p(predicted nest but not nest) 
number of non-nests
= 0.02  93      False
= 1.9            positives
No. nests predicted non-nests = p(predicted not nest but nest) 
number of nests
= 0.52  7
= 3.6            False
negatives
Total number of errors    = 1.9 + 3.6
= 5.5

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