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 = p0 (1 – e–bx)

Decreasing returns and saturation.

Here: p0 = saturation proportion

b = decreasing returns parameter

Widely used in marketing.

Data

Cumulative Trial vs Week

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Week

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Week # HHs Propn. of Households1 8 0.0052 14 0.0093 16 0.0114 32 0.0215 40 0.0276 47 0.0317 50 0.0338 52 0.0359 57 0.038

10 60 0.04011 65 0.04312 67 0.04513 68 0.04514 72 0.04815 75 0.05016 81 0.05417 90 0.06018 94 0.06319 96 0.06420 96 0.06421 96 0.06422 97 0.06523 97 0.06524 101 0.067

Modelled data

Week # HHs Propn. of Households Modelled Proportion diff p0 beta1 8 0.005 0.005 9.08E-10 0.0862 0.0642852 14 0.009 0.010 1.12E-06 LS 0.0001283 16 0.011 0.015 1.98E-054 32 0.021 0.020 3.25E-065 40 0.027 0.024 8.94E-066 47 0.031 0.028 1.42E-057 50 0.033 0.031 4.49E-068 52 0.035 0.035 9.82E-109 57 0.038 0.038 2.49E-08

10 60 0.040 0.041 7.23E-0711 65 0.043 0.044 1.13E-0712 67 0.045 0.046 2.72E-0613 68 0.045 0.049 1.2E-0514 72 0.048 0.051 9.74E-0615 75 0.050 0.053 1.09E-0516 81 0.054 0.055 1.81E-0617 90 0.060 0.057 7.5E-0618 94 0.063 0.059 1.3E-0519 96 0.064 0.061 1.06E-0520 96 0.064 0.062 2.8E-0621 96 0.064 0.064 3.58E-0822 97 0.065 0.065 2.86E-0723 97 0.065 0.067 3.38E-0624 101 0.067 0.068 1.56E-07

How well does the model do?

Cumulative Trial vs Week

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Propn. of Households

Modelled Proportion

"R^2" 0.985

How well does the model do – forecasting?

Cumulative Trial vs Week

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Propn. of Households

Modelled Proportion

forecast region-->

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 )0.00450657 : 0.05 0.10 0.003392708 : 0.08067993 0.04530598 0.0002417058 : 0.07951665 0.06836061 0.0001274101 : 0.08637640 0.06347303 0.0001261767 : 0.08661768 0.06369747 0.0001261767 : 0.08661641 0.06369891 > > summary(fit.nls)Formula: propHH ~ p0 * (1 - exp(-beta * Week))Parameters: Estimate Std. Error t value Pr(>|t|) p0 0.086616 0.004462 19.41 2.49e-15 ***beta 0.063699 0.005721 11.13 1.65e-10 ***---Signif. codes: 0 `***' 0.001 `**' 0.01 `*' 0.05 `.' 0.1 ` ' 1

Residual standard error: 0.002395 on 22 degrees of freedom

Correlation of Parameter Estimates: p0beta -0.9798

Different types of Models

&

Their Interpretations

A Simple Model

Y (Sales Level)} b (slope of the salesline)

}1

X (Advertising)

a(sales level whenadvertising = 0)

Phenomena

P1: Through Origin

P4: SaturationP3: 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 + bXc

c can be interpreted as elasticity when a = 0.

Linear, increasing or decreasing returns (depends on c).

Aggregate Response Models:Exponential Model

Y = aebx; 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.

Xc

d + Xc

Aggregate Response Models:Multiple Instruments

• Additive model for handling multiple marketing instruments

Y = af (X1) + bg (X2)

Easy to estimate using linear regression.

Aggregate Response Models:Multiple Instruments cont’d

• Multiplicative model for handling multiple marketing instruments

Y = aXb Xc

b and c are elasticities.

Widely used in marketing.

Can be estimated by linear regression.

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