Predicting Individual Responses Using Multinomial Logit Analysis Modeling an individual’s response to marketing effort The BookBinders Book Club case 59 59

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 = 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.

Technical Specification of the 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 ).

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

Example Computations (a) (b) (c) (d) (e) Share Share Brand 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

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

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 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.

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 5 5 5

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). 6 6 6

Characteristics of Biological Neural Networks Massively parallel Distributed representation and computation Learning ability Generalization ability Adaptivity Inherent contextual information Fault tolerance Low energy consumption 7 7 7

An Example Artificial Neural Network Inputs In humans: sensory data. In 4Thought: advertising, selling effort, price, etc. Neurons Outputs In humans: muscular reflexes. In 4Thought: sales model. “Synapses” 9 9 9

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). 10 10 10

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.” 11 11 11

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) 21 13 13 13

Role of Hidden Unit in a Two-Dimensional Input Space Description of decision regions Exclusive or Problem Classes with meshed regions General region shapes 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 12 12 12

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. 22 14 14 14

diL = g¢ (ZiL)[tiL – YiL] 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. 15 15 15

Error Propagation Error is propagated back as follows: dil = g¢ (Zil)[ å 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. 16 16 16

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. 17 17 17

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 18 18 18

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.

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 D Average Customer Purchase Purchase Score Customer 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%

Summary of Coefficients

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