Predicting Individual Responses Using Multinomial Logit Analysis

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Predicting Individual ResponsesUsing Multinomial
Logit Analysis

• Modeling an individuals response to marketing
  effort
• The BookBinders Book Club case

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The Logit Model

• The objective of the model is to predict the
  probabilities that an individual will choose each
  of several choice alternatives (e.g., buy versus
  not buy Select from among three brands A, B, and
  C). The model has the following properties
• The probabilities lie between 0 and 1, and sum
  to 1.
• The model is consistent with the proposition
  that customers pick the choice alternative that
  offer them the highest utility on a purchase
  occasion, but the utility has a random
  component that varies from one purchase
  occasion to the next.
• The model has the proportional draw property --
  each choice alternative draws from other choice
  alternatives in proportion to their utility.

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Technical Specification of the Multinomial Logit
Model

• Individual is probability of choosing brand
  1(Pi1) is given by
• where Aij is the attractiveness of alternative
  j to customer i å wk bijk
• k
• bijk is the value (observed or measured) of
  variable k (e.g., price) for alternative j when
  customer i made a purchase.
• Wk is the importance weight associated with
  variable k (estimated by the model)
• Similar equations can be specified for the
  probabilities that customer i will choose other
  alternatives.

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Technical Specification ofthe Multinomial Logit
Model

• On each purchase occasion, the (unobserved)
  utility that customer i gets from alternative j
  is given by
• where ?ij is an error term. Notice that utility
  is the sum of an observable term (Aij) and an
  unobservable term (?ij ).

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Example Choosing Among Three Brands
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Example Computations

• (a) (b) (c) (d) (e)
• Share ShareBrand Aij wk bijk
  estimate estimate Draw without with
  (c)(d) new brand new brand
• A 4.70 109.9 0.512 0.407 0.105
• B 3.30 27.1 0.126 0.100 0.026
• C 4.35 77.5 0.362 0.287 0.075
• D 4.02 55.7 0.206

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An Important Logit Model Implication
High
Marginal Impact of a Marketing Action (
)
Low
0.0
0.5
1.0
Probability of Choosing Alternative 1 ( )
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Quote for the Day

• You will lose money sending a terrific piece of
  mail to a lousy list, but make money sending a
  lousy piece of mail to a terrific list!
• -- Direct mail lore

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MNL Model of Response to Direct Mail

• Probability of function of (past response
  behavior,
• responding to marketing effort,
• direct mail characteristics of
• solicitation customers)

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BookBinders Book Club Case

• Predict response to a mailing for the Art
  History of Florence based on the following
  variables
• Gender
• Amount Purchased
• Months since first purchase
• Months since last purchase
• Frequency of purchase
• Past purchases of art books
• Past purchases of childrens books
• Past purchases of cook books
• Past purchases of DIY books
• Past purchases of youth books

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Scoring Using Current Industry Practice

• Dominant Scoring Rule used in the industry is
  the RFM (Recency, Frequency, and Monetary) model
• Recency
• Last purchased in the past 3 months 25 points
• Last purchased in the past 3 - 6 months 20
• Last purchased in the past 6 - 9 months 10
• Last purchased in the past 12 - 18 months 5
• Last purchased in the past 18 months 0
• Come up with similar scoring rules for
  Frequency and Monetary.
• For each customer, add up his/her score on each
  of the components (recency, frequency, and
  monetary) to compute an overall score.

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Scoring Based on Regression

• Regression Model
• Pij wo ?wkbijk ?ij
• where Pij is the probability that individual i
  will choose alternative j, wk are the regression
  coefficients and bijk are the independent
  variables described earlier. Note that Pij
  computed this way need not necessarily lie
  between 0 and 1.

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Scoring Model using Artificial Neural Networks

• What is a neural network?
• Determinants of network properties
• Description of feed-forward network with back
  propagation
• Potential value of neural networks

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Artificial Neural Networks

• An artificial neural network is a general
  response model that relates inputs (e.g.,
  advertising) to outputs (e.g., product
  awareness). The modeler need not specify the
  functional form of this relationship.
• A neural net attempts to mimic how the human
  brain processes input information and consists of
  a richly interlinked set of simple processing
  mechanisms (nodes).

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Characteristics of Biological Neural Networks

• Massively parallel
• Distributed representation and computation
• Learning ability
• Generalization ability
• Adaptivity
• Inherent contextual information
• Fault tolerance
• Low energy consumption

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An Example Artificial Neural Network
Neurons
Inputs In humanssensory data. In
4Thoughtadvertising, selling effort, price, etc.
Outputs In humansmuscular reflexes. In
4Thoughtsales model.
Synapses
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Determinants of the Behavior of Artificial Neural
Network

• Network properties (depends on whether network is
  feedforward or feedback number of nodes, number
  of layers in the network, and order of
  connections between nodes).
• Node properties (threshold, activation range,
  transfer function).
• System dynamics (initial weights, learning rule).

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Processing Mechanism of Individual Neurons

• Each neuron converts input signals into an
  overall signal value by weighting and summing the
  incoming signals.
• Z å Wi Xi
• i
• It transforms the overall signal value into an
  output signal (Y) using a transfer function.

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Transfer Function Formulations

• Hard limiter (Y 1 if Z T else 0)
• Sigmoidal (0 Y 1)
• 1
• Y g(Z)
• 1 e(ZT)
• Tanh (1 Y 1)
• Y g(Z) tanh (Z T)

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Role of Hidden Unit in a Two-Dimensional Input
Space
Exclusive or Problem
Classes with meshed regions
General region shapes
Description of decision regions
Structure
Half plane bounded by hyperplane
Single layer
Arbitrary (complexity limited by number of hidden
units)
Two layer
Arbitrary (complexity limited by number of hidden
units)
Three layer
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System Dynamics(Learning Mechanism)

• Supervised learning using back propagation of
  errors. Goal of this process is to reduce the
  total error at output nodes
• EP å (tPk OPk)2
• k
• where
• EP error to be minimized
• tPk target value associated with the kth
  input values to the output nodes
• OPk Output of neural net as calculated from
  the current set of weights.

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Error Propagation

• The error is calculated at each node for each
  input set k
• The error at the output node is equal to
• diL g (ZiL)tiL YiL
• where
• TiL Target value on the i-th output node
  (layer L of network)
• diL Error to be back propagated from node i
  in layer L
• g gradient of transfer function.

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Error Propagation

• Error is propagated back as follows
• dil g (Zil) å wijl1 djl1
• j
• for l (L1), . . . 1. (Lth layer is output)
• The weights are then adjusted using an
  optimality rule (in conjunction with a learning
  rate) to minimize overall error EP.

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So, Whats the Big Deal?

• With a sigmoidal transfer function and back
  propagation, the neural network can learn to
  represent any sampled function to any required
  degree of accuracy with a sufficient number of
  nodes and hidden layers.
• This allows us to capture underlying
  relationships without knowing the form of the
  relationship.

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Some Successful Applications

• Recognizing handwritten characters (e.g., zip
  codes)
• Recognizing speech (e.g., Dragons Naturally
  Speaking software)
• Estimating response to direct mail operations

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Predictions of Probability of Purchase

• RFM Model Use computed score as a measure of
  probability of purchase.
• Regression
• MNL
• RFM and Regression models can be implemented in
  Excel.
• Also, all three scoring procedures for
  probability of
• purchase can be implemented in Excel.

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Predictions of Probability of Purchase

• Neural Net Use the 4Thought software to compute
  choice probability. Note, as in regression,
  these predictions need not necessarily lie
  between 0 and 1. Follow the tutorial closely in
  doing this exercise.

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Scoring Customers for their Potential
Profitability

• A B C D Average Cus
  tomer Purchase Purchase ScoreCustomer Probabi
  lity Volume Margin A B C
• 1 30 31.00 0.70 6.51 2 2 143.00 0.60
  1.72 3 10 54.00 0.67 3.62 4 5 88.00
  0.62 2.73 5 60 20.00 0.58 6.96 6 22 6
  0.00 0.47 6.20 7 11 77.00 0.38 3.22 8 1
  3 39.00 0.66 3.35 9 1 184.00 0.56 1.03
  10 4 72.00 0.65 1.87
• Average Expected Score per customer 3.72

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Develop Tables such as the Following (Example
Shown for Mailing to the Top 60
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Summary of Coefficients
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Economics of Mailings
Note If we mailed to everyone on the list, we
can expect a response rate of 8.9.