Brian S. Everitt A Handbook of Statistical Analyses using SPSS

z 30. 463
(12.2)
The coefficients defining Fisher’s linear discriminant function in equation
(12.1) are proportional to the unstandardized coefficients given in
the “Canonical Discriminant Function Coefficients” table which is produced
when Unstandardized is checked in the Statistics sub-dialogue box (see
Display 12.3). The latter (scaled) discriminant function is used by SPSS to
calculate discriminant scores for each object (skulls). The discriminant
scores are centered so that they have a sample mean zero. These scores
can be compared with the average of their group means (shown in the
“Functions at Group Centroids” table) to allocate skulls into groups. Here
the threshold against which a skull’s discriminant score is evaluated is
0. 0585 12 0. 877 0. 994
(12.3)
Thus new skulls with discriminant scores above 0.0585 would be assigned
to the Lhasa site (type B); otherwise, they would be classified as type A.
When variables are measured on different scales, the magnitude of an
unstandardized coefficient provides little indication of the relative contribution
of the variable to the overall discrimination. The “Standardized
Canonical Discriminant Function Coefficients” listed attempt to overcome
this problem by rescaling of the variables to unit standard deviation. For
our data, such standardization is not necessary since all skull measurements
were in millimeters. Standardization should, however, not matter much
since the within-group standard deviations were similar across different
skull measures (see Display 12.4). According to the standardized coefficients,
skull height (meas3) seems to contribute little to discriminating
between the two types of skulls.
A question of some importance about a discriminant function is: how
well does it perform? One possible method of evaluating performance is
to apply the derived classification rule to the data set and calculate the
misclassification rate. This is known as the resubstitution estimate and the
corresponding results are shown in the “Original” part of the “Classification
Results” table in Display 12.7. According to this estimate, 81.3% of skulls
can be correctly classified as type A or type B on the basis of the
discriminant rule. However, estimating misclassification rates in this way
is known to be overly optimistic and several alternatives for estimating
misclassification rates in discriminant analysis have been suggested (Hand,
1997). One of the most commonly used of these alternatives is the socalled
leaving one out method, in which the discriminant function is first
derived from only n – 1 sample members, and then used to classify the
observation left out. The procedure is repeated n times, each time omitting
© 2004 by Chapman & Hall/CRC Press LLC