Non Linear Modelling An example

Background Ace Snackfoods, Inc. has developed a new snack product called Krunchy Bits. Before deciding whether or not to “go national” with the new product, the marketing manager for Krunchy Bits has decided to commission a year-long test market using IRI’s BehaviorScan service, with a view to getting a clearer picture of the product’s potential. The product has now been under test for 24 weeks. On hand is a dataset documenting the number of households that have made a trial purchase by the end of each week. (The total size of the panel is 1499 households.) The marketing manager for Krunchy Bits would like a forecast of the product’s year-end performance in the test market. First, she wants a forecast of the percentage of households that will have made a trial purchase by week 52.

Data

Approaches to Forecasting Trial French curve “Curve fitting”—specify a flexible functional form fit it to the data, and project into the future. Inspect the data (see Non Linear Modelling.xls)

Proposed Model for this example Y = p 0 (1 – e –bx ) Decreasing returns and saturation. Here: p 0 = saturation proportion b = decreasing returns parameter Widely used in marketing.

Data

Modelled data

How well does the model do?

How well does the model do – forecasting?

Doing the same thing in R NLeg.df=read.csv(file.choose(),header=T) attach(NLeg.df) fit.nls<- nls( propHH ~ p0*(1-exp(- beta*Week )), data = NLeg.df, start = list( p0=.05,beta=.1), trace = TRUE )

Doing the same thing in R > fit.nls<- nls( propHH ~ p0*(1-exp(-beta*Week )), + data = NLeg.df, + start = list( p0=.05,beta=.1), + trace = TRUE ) : : : : : : > > summary(fit.nls) Formula: propHH ~ p0 * (1 - exp(-beta * Week)) Parameters: Estimate Std. Error t value Pr(>|t|) p e-15 *** beta e-10 *** --- Signif. codes: 0 `***' `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1 Residual standard error: on 22 degrees of freedom Correlation of Parameter Estimates: p0 beta

Different types of Models & Their Interpretations

A Simple Model Y (Sales Level) } b(slope of the salesline) } 1 X (Advertising) a (sales level when advertising = 0)

Phenomena P1: Through Origin P4: Saturation P3:Decreasing Returns (concave) P2: Linear Y X Y X Y X Q — Y X

Phenomena P5:Increasing Returns (convex) P8: Super-saturationP7: Threshold P6: S-shape Y X Y X Y X Y X

Aggregate Response Models: Linear Model Y = a + bX Linear/through origin Saturation and threshold (in ranges)

Aggregate Response Models: Fractional Root Model Y = a + bX c c can be interpreted as elasticity when a = 0. Linear, increasing or decreasing returns (depends on c).

Aggregate Response Models: Exponential Model Y = ae bx ; x > 0 Increasing or decreasing returns (depends on b).

Aggregate Response Models: Adbudg Function Y = b + (a–b) S-shaped and concave; saturation effect. Widely used. Amenable to judgmental calibration. X c d + X c

Aggregate Response Models: Multiple Instruments Additive model for handling multiple marketing instruments Y = af (X 1 ) + bg (X 2 ) Easy to estimate using linear regression.

Aggregate Response Models: Multiple Instruments cont’d Multiplicative model for handling multiple marketing instruments Y = aX b X c b and c are elasticities. Widely used in marketing. Can be estimated by linear regression. 1 2