Transcript BookBinders Book Club

Predicting Individual Responses
Using Multinomial Logit Analysis
Modeling an individual’s
response to marketing
effort
The BookBinders Book
Club case
Marketing Engineering, Spring 1999
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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 i’s probability of choosing brand 1(Pi1) is
given by:
Pi1 
e A i1
e
A ij
j
where Aij is the “attractiveness” of alternative j to customer i =  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 of
the Multinomial Logit Model
On each purchase occasion, the (unobserved)
utility that customer i gets from alternative j is
given by:
U ij  A ij   ij
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
bijk
Brand Performance Quality Variety
Value
A
0.7
0.5
0.7
0.7
B
0.3
0.4
0.2
0.
C
0.6
0.8
0.7
0.4
D (new)
0.6
0.4
0.8
0.5
Estimated
Importance
Weight (wk) 2.0
1.7
1.3
2.2
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Example Computations
(a)
Brand Aij = wk bijk
(b)
e
A ij
(c)
Share
estimate
without
new brand
(d)
Share
estimate
with
new brand
(e)
Draw
(c)–(d)
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
dPil
 w k Pil (1  Pil )
db ijk
High
Marginal Impact
of a Marketing
Action ( dPil )
db ijk
Low
0.0
0.5
1.0
Probability of Choosing Alternative 1 ( Pi1 )
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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
responding to =
direct mail
solicitation
function of (past response behavior,
marketing effort,
characteristics of
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 children’s 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
Inputs
Neurons
Outputs
In humans:
sensory
data.
In humans:
muscular
reflexes.
In 4Thought:
advertising,
selling
effort, price,
etc.
In 4Thought:
sales 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 =  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–(Z–T)
 Tanh (–1 Y  1)
Y = g(Z) = tanh (Z – T)
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Role of Hidden Unit in a TwoDimensional Input Space
Structure
Description of
decision regions
Exclusive or
Problem
Classes with
meshed regions
General
region shapes
Half plane
bounded by
hyperplane
Single layer
Two layer
Three layer
Arbitrary
(complexity
limited by number
of hidden units)
Arbitrary
(complexity limited
by number of
hidden units)
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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 =  (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)[  wijl+1 djl+1]
j
for l = (L–1), . . . 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, What’s 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., Dragon’s 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: Score ( for respondent i )  w
 0  w
 k b ijk
k

MNL:
i' s probability of purchase 
e
 0  w
 kb ijk
w
1 e
 0  w
 kb ijk
w
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
Customer
Purchase
Probability
B
Average
Purchase
Volume
C
Margin
D
Customer
Score
=ABC
1
30%
$31.00
0.70
6.51
2
3
4
5
6
7
8
9
10
2%
10%
5%
60%
22%
11%
13%
1%
4%
$143.00
$54.00
$88.00
$20.00
$60.00
$77.00
$39.00
$184.00
$72.00
0.60
0.67
0.62
0.58
0.47
0.38
0.66
0.56
0.65
1.72
3.62
2.73
6.96
6.20
3.22
3.35
1.03
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%
Model
Number of hits
(favorable responses at
60th percentile of
ordered scores)
Expected response
rate by mailing the
top 60% of customers
in the ordered list
% of favorable
respondents
recovered at
60th percentile
RFM
Regression
MNL
Neural Net
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Summary of Coefficients
Coefficient
Gender
Amount Purchased
Months since first purchase
Months since last purchase
Frequency of purchase
Purchase of art books
Purchase of children’s books
Purchase of Cook books
Purchase of DIY books
Purchase of Youth books
Regression
Model
NS
NS
+
+
-
MNL
NS
+
+
NS
Neural
Network
NS
+
+
-
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Economics of Mailings
Financial
Regression
MNL
Neural
RFM
Component
Network
Cost of Book +
$86978.25*
85608.00
85999.00
70861.50
Overhead (a)
Mailing costs
19500.00
19500.00
19500.00
19500.00
(30,000*0.65) (b)
Expected sales (c)
127768.05
125755.20
126330.30
104093.10
Net revenue (d)
21289.80
20647.20
20831.30
13731.60
ROI = d/(a+b)
19.99%
19.64%
19.75%
15.20%
 Computed as follows: (50000  0.6)  0.1333  (15 + 15  0.45)
Note: If we mailed to everyone on the list, we can expect a response rate of 8.9%.
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