An example

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.

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.

Week
8
9
10
11
12
13
14
6
7
4
5
1
2
3
20
21
22
23
24
15
16
17
18
19
# HHs
75
81
90
94
96
65
67
68
72
96
96
97
97
101
32
40
47
50
8
14
16
Propn. of Households
0.005
0.009
0.011
0.021
0.027
52
57
60
0.031
0.033
0.035
0.038
0.040
0.064
0.064
0.065
0.065
0.067
0.043
0.045
0.045
0.048
0.050
0.054
0.060
0.063
0.064
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0
Cumulative Trial vs Week
5 10
Week
15 20 25

Week
9
10
11
12
13
14
15
7
8
5
6
3
4
1
2
16
17
18
19
20
21
22
23
24
# HHs
96
96
97
97
101
81
90
94
96
67
68
72
75
40
47
50
52
8
14
16
32
Propn. of Households Modelled Proportion diff
0.005
0.009
0.005
0.010
9.08E-10 p0
1.12E-06 LS
0.011
0.021
0.027
0.031
0.015
0.020
0.024
0.028
1.98E-05
3.25E-06
8.94E-06
1.42E-05
57
60
65
0.033
0.035
0.038
0.040
0.043
0.031
0.035
0.038
0.041
0.044
4.49E-06
9.82E-10
2.49E-08
7.23E-07
1.13E-07
0.045
0.045
0.048
0.050
0.054
0.060
0.063
0.064
0.046
0.049
0.051
0.053
0.055
0.057
0.059
0.061
2.72E-06
1.2E-05
9.74E-06
1.09E-05
1.81E-06
7.5E-06
1.3E-05
1.06E-05
0.064
0.064
0.065
0.065
0.067
0.062
2.8E-06
0.064
3.58E-08
0.065
2.86E-07
0.067
3.38E-06
0.068
1.56E-07 beta
0.0862 0.064285
0.000128

"R^2" 0.985
Cumulative Trial vs Week
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0
Propn. of Households
Modelled Proportion
5 20 25 10
Week
15

How well does the model do – forecasting?
Cumulative Trial vs Week forecast region-->
Propn. of Households
Modelled Proportion
10.00%
9.00%
8.00%
7.00%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
0 10 20
Week
30 40 50

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 )

> 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: p0 beta -0.9798

&
Their Interpretations

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

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

Y
Y
P5: Increasing Returns
(convex)
P6: S-shape
Y
X
P7: Threshold
X
P8: Super-saturation
Y
X 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 )
X c d + X c
S-shaped and concave; saturation effect.
Widely used. Amenable to judgmental calibration.

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
1 2 b and c are elasticities.
Widely used in marketing.
Can be estimated by linear regression.