STAT3613 Tutorial 01

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT2313/3613 Marketing Engineering Tutorial 1 Market Response Model Review Marketing engineering • Use of decision models for making marketing decisions. • Apply computer models to help transform objective and subjective data about the marketing envi- ronment into insights, decisions and implementation of decisions. Data are facts, beliefs or observations used in making decision • Dependent or output variables are those determined by a set of independent variables • Products sales (dependent variable) are driven by the level of advertising spending (independent variable 1) and the quality of the product (independent variable 2) Mathematical models • Specify the relationships embodied in a model in the form of equations Inputs • Independent variables • the marketing actions that the marketer can control Outputs • Dependent variables • Measurable outputs of concern to the firm • Customers awareness levels, product perceptions, sales levels, and profits Response model • The linkage from those inputs to the outputs • Relationship, specification, mathematical form • equations or sets of equations that relate dependent variables to independent variables in a model 1

Least Square Errors • y i is the observed value and ˆ y i is the predicted value under the model • Define sum of squared error ( SSE ) as SSE = n X i =1 ( y i - ˆ y i ) 2 • Define total sum of squares ( SST ) as SST = n X i =1 ( y i - ¯ y i ) 2 , ¯ Y = 1 n n X i =1 y i • SSE is the amount of the variation not explained by the model. • R 2 , R-square, defined as 1 - ratio of variation not explained by the model to the total variation R 2 = 1 - SSE SST • In general, the higher the R 2 , the better is the model. • R 2 = 0 , only as good as the average of y ; R 2 = 1 for the perfect model Type of Models Model Function Range of parameters Remarks Linear Y = a + bX a ≥ 0 linear relationship Power series Y = a + bX + cX 2 + dX 3 + . . . a ≥ 0 takes many shapes Fractional root Y = a + bX c c = 1 2 ⇒ square root model c = - 1 ⇒ reciprocal model Semilog Y = a + b ln X X > 0 threshold model Exponential Y = ae bX a > 0 , X > 0 Modified exponential Y = a (1 - e - bX ) + c a ≥ 0 Logistic Y = a 1+ e - ( b + cX ) + d Gompertz Y = ab c x + d a > 0 , 0 < b < 1 , c > 1 ADBUDG Y = a + b X c X c + d Phenomena a) Linear: Linear, power series and fractional root models b) Concave (decreasing returns): Power series, fractional root, semilog, modified exponential and ADBUDG models c) Saturation: Fractional root, modified exponential, logistic, Gompertz and ADBUDG models d) Convex (increasing returns): Power series, fractional root, semilog and exponential model e) S-shape: Power series, logistic, Gompertz and ADBUDG models f) Threshold: Semilog model g) Super saturation: Power series and ADBUDG models Example A company develops promotional response model tools to help it decide the level and allocation of promotional spending using response modeling and optimization with SAS. The managers constructed a response model, relating promotional spending with sales. They explored the promotional spending response analysis using the following information: 2

Promotional Spending X Sales Y ($’000,000) (’000,000 units) 0.00 6.3 0.44 6.7 0.87 8.0 1.31 9.3 2.50 10.9 5.00 11.8 1. Plot the sales Y against the promotional spending X . Describe the relationship between Y and X . Which response models seem appropriate? 2. Find the starting values for each model. 3. Estimate the parameters of the response models and choose the best model with largest R 2 . Solutions 1. Use SAS "‘proc sgplot"’ to draw a plot of Y against X . Sales increases slowly from X = 0 to X = 0 . 44 and much faster thereafter until X = 1 . 31 . However there is no observation between X = 1 . 31 and X = 2 . 5 and between X = 2 . 5 to X = 5 . The lacking of information of X during these intervals may reduce the model accuracy. Since there is a saturation in the data, models with S-shape seem appropriate. The response models to be considered will be power series, logistic, Gompertz and ADBUDG models. 2. The initial values of parameters for power series and Gompertz models are given by arbitrary values, but not extraordinary ones. For logistic and ADBUDG models, initial values of parameters are estimated by specific formula given in the lecture notes. The data are normalized with promotional spending = 1 . 31 as reference level. Promotional Spending X Sales Y ($’000,000) (’000,000 units) 0.00 0.68 0.34 0.72 0.66 0.86 1.00 1.00 1.91 1.17 3.82 1.27 (a) For ADBUDG model, a = min( Y ) = 0 . 68 , b = max( Y ) - min( Y ) = 1 . 27 - 0 . 68 = 0 . 59 , d = b 1 - a - 1 = 0 . 84 , y 0 ( slope at x = 1) = 1 . 17 - 0 . 86 1 . 91 - 0 . 66 = 0 . 248 , c = y 0 ( d +1) 2 b × d = 1 . 69 . Note that slope of curve at X = 1 is an approximation only. If you use the original data, d = b 1 - a - 1 does not hold any more and slope at other point should be used instead. (b) For logistic model, select (1 , 1) as the inflection point, a = max( Y ) - min( Y ) = 0 . 59 , slope at the point of inflection = 1 . 17 - 0 . 86 1 . 91 - 0 . 66 = 0 . 248 , c = 4 × 0 . 248 /a = 1 . 68 , 3

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b = - 1 × c = - 1 . 68 , d = min( Y ) = 0 . 68 . Note that the inflection point is approximated at X = 1 in this example. It may not be the case in the other situations. The selection of inflection point is important. 3. Use SAS "‘proc model"’ to estimate each model. Numerical estimation is presented below: Power Series Logistic Gompertz ADBUDG a 0.6550 0.7255 0.6075 0.6733 b 0.3266 -1.5426 0.0278 0.6192 c -0.0088 2.0052 0.1668 2.2249 d -0.0091 0.5340 0.6523 0.9310 R 2 0.9868 0.9931 0.9962 0.9994 Thus, the best model is ADBUDG model with the largest R 2 = 0 . 9994 . 4