Chapter 20 Curved Patterns Copyright © 2011 Pearson Education, Inc 20.1 Detecting Nonlinear Patterns What improvement in mileage should a manufacturer expect from reducing the weight of a car? Use regression analysis to find an equation that summarizes the association between gas mileage and weight This pattern of association is not linear, but curved of 38 Copyright © 2011 Pearson Education, Inc 20.1 Detecting Nonlinear Patterns Recognizing Nonlinearity Will changes in the explanatory variable result in equal sized changes in the estimated response, regardless of the value of x? Does trimming 200 pounds from a large SUV have the same effect on mileage as trimming 200 pounds from a small compact? of 38 Copyright © 2011 Pearson Education, Inc 20.1 Detecting Nonlinear Patterns Scatterplot with Fitted Line of 38 Copyright © 2011 Pearson Education, Inc 20.1 Detecting Nonlinear Patterns Mileage (MPG) vs Weight (1000’s pounds) The least squares fitted line is Estimated MPG City = 35.6 – 4.52 Weight (000 lb) The line has an r2 = 0.57 and se = 2.9 MPG The equation estimates that mileage would increase 0.904 MPG, on average, by reducing car weight by 200 pounds of 38 Copyright © 2011 Pearson Education, Inc 20.1 Detecting Nonlinear Patterns Residual Plot Easier to spot curved pattern in the residuals of 38 Copyright © 2011 Pearson Education, Inc 20.2 Transformations Transformation: re-expression of a variable by applying a function to each observation Transformations allow the use of regression analysis to describe a curved pattern Two nonlinear transformations useful in business applications: reciprocal and logarithms of 38 Copyright © 2011 Pearson Education, Inc 20.2 Transformations Choosing An Appropriate Transformation The process of choosing the right transformation is usually iterative Among the possible choices, select the one that captures the curvature of the data and produces an interpretable equation of 38 Copyright © 2011 Pearson Education, Inc 20.2 Transformations Choosing An Appropriate Transformation Match the pattern in a scatterplot of y on x to one of the shapes to find an appropriate transformation 10 of 38 Copyright © 2011 Pearson Education, Inc 20.4 Logarithm Transformation Comparing Equations The curve produced by taking log transformations of the data provides a better fit to the data 24 of 38 Copyright © 2011 Pearson Education, Inc 20.4 Logarithm Transformation Comparing Equations The log-log equation shows that customers are more price sensitive at low prices For example, an average price change from $0.80 to $0.90 leads to a drop of more than 27,000 cans in estimated sales volume In contrast, a change from $1.10 to $1.20 leads to a smaller drop in sales of about 9,500 cans 25 of 38 Copyright © 2011 Pearson Education, Inc 20.4 Logarithm Transformation Elasticity Elasticity: measure that relates % change in x to % change in y; slope in a log-log regression equation Can use elasticity to find the optimal price of pet food for the supermarket chain 26 of 38 Copyright © 2011 Pearson Education, Inc 20.4 Logarithm Transformation Elasticity Optimal Price = Cost Elasticity Elasticity 1 = = 2.44 0.60 2.44 $1.017 Estimated profit is maximized at this price 27 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Motivation How much should a convenience store charge for a half-gallon of orange juice? The orange juice costs the chain $1 to stock and sell 28 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Method In order to determine the optimal price, need to estimate elasticity from a regression of log sales on log price The chain collected data on sales of orange juice from stores at 50 different locations The stores sold orange juice at different prices 29 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Method 30 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Method 31 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Mechanics The least squares regression of log sales on log price is Estimated log Sales = 4.81 - 1.75 log Price 32 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Mechanics – Check Conditions All conditions satisfied 33 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Mechanics – Optimal Price c /(1 ) $1( 1.75) /(1 1.75) $2.33 At $2.33, each store can expect to sell 28 cartons of orange juice for an estimated profit of $37.24 34 of 38 Copyright © 2011 Pearson Education, Inc 4M Example 20.1: OPTIMAL PRICING Message The chain would make higher profits by decreasing the price of a half-gallon of orange juice from its current price of $3.00 to $2.33 35 of 38 Copyright © 2011 Pearson Education, Inc Best Practices Anticipate whether the association between y and x is linear Check that a line summarizes the relationship between the explanatory variable and the response both visually and substantively Stick to models you can understand and interpret 36 of 38 Copyright © 2011 Pearson Education, Inc Best Practices (Continued) Interpret the slope carefully Graph your model in the original units 37 of 38 Copyright © 2011 Pearson Education, Inc Pitfalls Don’t think that regression only fits lines Don’t forget to look for curves, even in models with high values of r2 Don’t forget lurking variables Don’t compare r2 between models with different responses 38 of 38 Copyright © 2011 Pearson Education, Inc ... nonlinear transformations useful in business applications: reciprocal and logarithms of 38 Copyright © 201 1 Pearson Education, Inc 20. 2 Transformations Choosing An Appropriate Transformation ... charge for a national brand of pet food? 18 of 38 Copyright © 201 1 Pearson Education, Inc 20. 4 Logarithm Transformation Timeplots of Sales and Price of Pet Food 19 of 38 Copyright © 201 1 Pearson... Copyright © 201 1 Pearson Education, Inc 20. 2 Transformations Transformation: re-expression of a variable by applying a function to each observation Transformations allow the use of regression analysis