marketing engineering, spring 1999 1 predicting individual responses using multinomial logit...

Marketing Engineering, Spring 1999 1

Predicting Individual ResponsesUsing Multinomial Logit Analysis

Modeling an individual’s response to marketing effort

The BookBinders Book Club case


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.


Technical Specification of the Multinomial Logit Model

Individual i’s probability of choosing brand 1(Pi1) is given by:

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.

Pe

ei

A

A

j

i

ij1

1


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 ).

U Aij ij ij


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

EstimatedImportanceWeight (wk) 2.0 1.7 1.3 2.2


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

eAij


An Important Logit Model Implication

Marginal Impact of a Marketing Action ( )

Probability of Choosing Alternative 1 ( )

0.0 0.5 1.0

Low

High

dP

dbw P Pil

ijkk il il ( )1

Pi1

dP

dbil

ijk


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


MNL Model of Response to Direct Mail

Probability of function of (past response behavior,

responding to = marketing effort,

direct mail characteristics of

solicitation customers)


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


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 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.


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.


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


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).


Characteristics of Biological Neural Networks

Massively parallel

Distributed representation and computation

Learning ability

Generalization ability

Adaptivity

Inherent contextual information

Fault tolerance

Low energy consumption


An Example Artificial Neural Network

Inputs

In humans:sensory data.

In 4Thought:advertising, selling effort, price, etc.

Outputs

In humans:muscular reflexes.

In 4Thought:sales model.

Neurons

“Synapses”


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).


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.”


Transfer Function Formulations

Hard limiter (Y = 1 if Z T; else = 0)

Sigmoidal (0 Y 1)

1Y = g(Z) = ––––––––

1 + e–(Z–T)

Tanh (–1 Y 1)

Y = g(Z) = tanh (Z – T)


Role of Hidden Unit in a Two-Dimensional Input Space

Exclusive orProblem

Classes with meshed regions

General region shapes

Description of decision regions

Structure

Single layer

Two layer

Three layer

Half planebounded byhyperplane

Arbitrary (complexity

limited by number of hidden units)

Arbitrary (complexity limited

by number of hidden units)


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.


Error Propagation

The error is calculated at each node for each input set k:

The error at the output node is equal to

iL = g (Zi

L)[tiL – Yi

L]

where:

TiL = Target value on the i-th output node (layer L

of network);

iL = Error to be back propagated from node i in

layer L;

g = gradient of transfer function.


Error Propagation

Error is propagated back as follows:

il = g(Zi

l)[wijl+1 j

l+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.


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.


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


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 ofpurchase” can be implemented in Excel.

Score for respondent i w w bk ijkk

( ) 0

i s obability of purchasee

e

w w b

w w b

k ijk

k ijk'

pr

0

01


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.


Scoring Customers for their Potential Profitability

A B C DAverage Customer

Purchase Purchase ScoreCustomer Probability 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% $60.00 0.47 6.20

7 11% $77.00 0.38 3.22

8 13% $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


Develop Tables such as the Following (Example Shown for Mailing to the Top 60%

Model

Number of hits(favorable responses at

60th percentile ofordered scores)

Expected responserate by mailing the

top 60% of customersin the ordered list

% of favorablerespondentsrecovered at

60th percentileRFM

Regression

MNL

Neural Net


Summary of Coefficients

Coefficient RegressionModel

MNL NeuralNetwork

Gender - - -Amount Purchased NS - -Months since first purchase NS NS NSMonths 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 - NS -


Economics of Mailings

Note: If we mailed to everyone on the list, we can expect a response rate of 8.9%.

FinancialComponent

Regression MNL NeuralNetwork

RFM

Cost of Book +Overhead (a)

$86978.25* 85608.00 85999.00 70861.50

Mailing costs(30,000*0.65) (b)

19500.00 19500.00 19500.00 19500.00

Expected sales (c) 127768.05 125755.20 126330.30 104093.10Net revenue (d) 21289.80 20647.20 20831.30 13731.60ROI = d/(a+b) 19.99% 19.64% 19.75% 15.20% Computed as follows: (50000 0.6) 0.1333 (15 + 15 0.45)

marketing engineering, spring 1999 1 predicting individual responses using multinomial logit...

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