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Discriminant Analysis with Misclassification Costs: Prior Probabilities Adjustment - Prof., Study notes of Introduction to Business Management

How to use misclassification costs in discriminant analysis for two groups using spss. The author discusses the concept of misclassification costs (c(1|2) and c(2|1)) and how to adjust prior probabilities (p1 and p2) using a constant k. The document also provides instructions on how to set up the analysis in spss and interpret the results.

Typology: Study notes

Pre 2010

Uploaded on 02/12/2009

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Discriminant Analysis for 2 Groups
A Note on Using Misclassification Costs
R.L. Andrews
Consider two groups labeled 1 and 2. Let p1 denote the proportion of the total in
group 1 and p2 denote the proportion of the total in group 2. p1 and p2 are the prior
probabilities of an item being in the respective groups. C(2|1) is the cost of
misclassification for saying that an item from group 1 is in group 2. C(1|2) is the
cost of misclassification for saying that an item from group 2 is in group 1
Define a constant K such that
KC
C
( )
( )
2 1
1 2
.
SPSS uses equal misclassification costs for the two groups but one can use K to
adjust the prior probabilities to account for uneaqual misclassificataion costs.
Therefore after the adjustment for misclassification costs,
1
the input prior for
group 1 is
p K
p K p
1
1 2
and the input prior for group 2 is
p
p K p
2
1 2
. However, if
one prefers to use
C and C( ) ( )2 1 1 2
rather than K, then the input prior for group 1
is
p C
p C p C
1
1 2
2 1
2 1 1 2
( )
( ) ( )
and the input prior for group 2 is
p C
p C p C
2
1 2
1 2
2 1 1 2
( )
( ) ( )
.
To enter these values in SPSS, set up the analysis using the menus (see the next
page) for Statistics and Classify.
The menu for Classify button provides only two options for specifying prior
probabilities as is shown on the next page.
pf2

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Discriminant Analysis for 2 Groups

A Note on Using Misclassification Costs

R.L. Andrews Consider two groups labeled 1 and 2. Let p 1 denote the proportion of the total in group 1 and p 2 denote the proportion of the total in group 2. p 1 and p 2 are the prior probabilities of an item being in the respective groups. C(2|1) is the cost of misclassification for saying that an item from group 1 is in group 2. C(1|2) is the cost of misclassification for saying that an item from group 2 is in group 1 Define a constant K such that K^

C

C

SPSS uses equal misclassification costs for the two groups but one can use K to adjust the prior probabilities to account for uneaqual misclassificataion costs. Therefore after the adjustment for misclassification costs, 1 the input prior for group 1 is p K p K p 1 1 2

and the input prior for group 2 is p p K p 2 1 ^ ^2

. However, if one prefers to use C^ (^2 1 )^ and^ C(^1 2 )rather than K, then the input prior for group 1 is p C p C p C 1 1 2

and the input prior for group 2 is p C p C p C 2 1 2

To enter these values in SPSS, set up the analysis using the menus (see the next page) for Statistics and Classify. The menu for Classify button provides only two options for specifying prior probabilities as is shown on the next page.

Clicking on Summary Table gives a summary of classification results (hit rate results) for the data in this set. Clicking on Leave-one-ouit classification gives a summary of jackknife (see page 308 of Stevens) classification results for the data in this set. Clicking on Fisher’s gives coefficients that can be used to classify a data point into one of the categories. Once these are set up click on the Paste button in the menu on the previous page. For this situation I want to use p1=.2 and p2=.8. I have typed these values in after PRIORS replacing either EQUAL or SIZE. Now click on Run and Current to obtain the analysis output for this Discriminant Analysis.