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Predicting Individual Responses Using Multinomial Logit Analysis Modeling an individual’s response to marketing effort The BookBinders Book Club case Marketing Engineering, Spring 1999 1 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. Marketing Engineering, Spring 1999 2 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. Marketing Engineering, Spring 1999 3 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 ). Marketing Engineering, Spring 1999 4 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 Marketing Engineering, Spring 1999 5 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 Marketing Engineering, Spring 1999 6 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 ) Marketing Engineering, Spring 1999 7 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 Marketing Engineering, Spring 1999 8 MNL Model of Response to Direct Mail Probability of responding to = direct mail solicitation function of (past response behavior, marketing effort, characteristics of customers) Marketing Engineering, Spring 1999 9 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 Marketing Engineering, Spring 1999 10 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. Marketing Engineering, Spring 1999 11 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. Marketing Engineering, Spring 1999 12 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 Marketing Engineering, Spring 1999 13 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). Marketing Engineering, Spring 1999 14 Characteristics of Biological Neural Networks Massively parallel Distributed representation and computation Learning ability Generalization ability Adaptivity Inherent contextual information Fault tolerance Low energy consumption Marketing Engineering, Spring 1999 15 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” Marketing Engineering, Spring 1999 16 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). Marketing Engineering, Spring 1999 17 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.” Marketing Engineering, Spring 1999 18 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) Marketing Engineering, Spring 1999 19 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) Marketing Engineering, Spring 1999 20 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. Marketing Engineering, Spring 1999 21 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. Marketing Engineering, Spring 1999 22 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. Marketing Engineering, Spring 1999 23 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. Marketing Engineering, Spring 1999 24 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 Marketing Engineering, Spring 1999 25 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 kbijk w RFM and Regression models can be implemented in Excel. Also, all three scoring procedures for “probability of purchase” can be implemented in Excel. Marketing Engineering, Spring 1999 26 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. Marketing Engineering, Spring 1999 27 Scoring Customers for their Potential Profitability A Customer Purchase Probability B Average Purchase Volume C Margin D Customer Score =ABC 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 Marketing Engineering, Spring 1999 28 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 Marketing Engineering, Spring 1999 29 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 + + - Marketing Engineering, Spring 1999 30 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%. Marketing Engineering, Spring 1999 31