Discriminant Analysis

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Example 8.8

• Discriminant Analysis

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DISCRIM.XLS

• This file contains the annual income and size of
  investment portfolio (both in thousands of
  dollars) for 84 people.
• It also indicates whether each of these people
  subscribes or does not subscribe to the Wall
  Street Journal.
• Using income and size of investment portfolio,
  determine a classification rule that maximizes
  the number of people correctly classified as
  subscribers or nonsubscribers.

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Solution

• The model is actually simpler for cluster
  analysis.
• Using appropriate weights, we create a
  discriminant score for each of the 84 customers.
• Then based on a cutoff score, we classify each
  customer as a subscriber or nonsubscriber, and we
  tally the number of correct classifications.

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Developing the Model

• The model appears on the next slide. It can be
  formed as follows.
• Customer data. Enter the customer data in the
  shaded range. This includes the data on the
  variable used for classification (income and
  investment amount), as well as an indication of
  which group each customer is in. These 84
  customers represent the training sample, so we
  know which group each of them is in.
• Decision variables. The decision variables are
  the weights used to form discriminant scores and
  the cutoff value for classification. Enter any
  values for these in the Weights and Cutoff ranges.

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Developing the Model -- continued

• Discriminant variables. Each discriminant score
  is a weighted combination of the persons income
  and investment amount. To calculate these in
  column E, enter the formula SUMPRODUCT(Weights,B1
  2C12) in cell E12 and copy it down.
• Classifications. We will classify a person as a
  nonsubscriber if the persons discriminant score
  is below the cutoff value and as a subscriber
  otherwise. Therefore, enter the formula
  IF(E12ltCutoff,No,Yes) in cell F12 and copy
  it down.
• Correct? Check whether each classification is
  correct by entering the formula
  IF(D12F12,Yes,No) in cell G12 and copying
  it down.

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Developing the Model -- continued

• Tallies. It is customary to tally the
  classifications in a classification matrix. Do
  this in the range B99C100 by entering the
  formulas COUNTIF(YesGroup,Yes) and
  COUNTA(YesGroup)-B99 in cells B99 and C99. Note
  that the YesGroup range contains the
  classifications for all subscribers. Then enter
  similar formulas in cells B100 and C100. These
  are based on the NoGroup range. Finally,
  calculate the objective, the percentage of all 84
  people classified correctly, in the PctCorrect
  cell with the formula SUM(B99,C100)/SUM(B99C100)
  .

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Using the Evolutionary Solver

• First, note that the Evolutionary Solver is
  required because of the IF (and COUNIF and
  COUNTA) functions used to make (and tally) the
  classifications.
• The completed Solver dialog box appears here.

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Using the Evolutionary Solver -- continued

• It is straightforward except for the lower and
  upper limits on the changing cells.
• There are no natural weights or cutoff values to
  use. However, we can always constrain the weights
  to be between 1 and 1.
• To obtain lower and upper limits on the cutoff
  value, we first calculated the maximum sum of
  income and investment amount for any customer,
  which is slightly less than 160.

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Using the Evolutionary Solver -- continued

• This means that the largest discriminant score,
  using weights of 1, is no less than 160.
• Therefore, there is no need to consider cutoff
  values below 160 or above 160.

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Solution

• The solution shown is certainly not unique.
• There are many other sets of weights and cutoffs
  values that obtain 98.6 correct classification
  rate, and you will probably obtain a different
  solution from ours.
• Note that only 6 of the 84 people are
  misclassified 4 subscribers are classified as
  nonsubscribers and 2 nonsubscribers are
  classified as subscribers.
• Also, we see from the weights that the
  classification is based primarily on the
  investment amount, people with large investments
  amounts are classified as subscribers.

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Solution -- continued

• Therefore, a subscriber such as person 81 is
  misclassified because his investment amount is
  abnormally small relative to other subscribers.
• On the other hand, a nonsubscriber is
  misclassified if his investment amount is
  abnormally large relative to other
  nonsubscribers.
• In a real application, we would use this analysis
  for people other than the 84 in the training
  sample.

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Solution -- continued

• That is, we would calculate a discriminant score
  for each such person and classify each as a
  nonsubscriber if her discriminant score is less
  than the cutoff value.
• However, the percentage correctly classified
  would typically be less maybe even considerably
  less than the 92.86 rate we obtained in the
  training sample.