We’re looking for a professional consultant who can help us forecast sales volume based on the manipulation of marketing variables such as advertising and the number of outside sales representatives.  We also need to assess the contribution of these variables in terms of their effect on sales.  How can ARI help us?  Please give an example.  

A well known sporting goods manufacturer contacted ARI because they wanted to develop a model which would help them predict sales volume based on the manipulation of advertising expenditures and the number of employees working in a sales territory.  An approach based on regression analysis was utilized because the client was interested in the relationship between sales and marketing variables which were frequently manipulated by the company. 

Regression analysis is based on a procedure for deriving a mathematical relationship in the form of an equation between a single dependent variable and one or more independent variables.  The equation used to predict sales based on advertising and the number of employees working in a sales territory is shown below. 

y = a + b₁x₁ + b₂x₂

 Where: 

y = sales, the dependent variable
a = the intercept
x₁ = advertising, an independent variable
x₂ = number of employees, an independent variable
b₁ = the slope for independent variable x₁  
b₂ = the slope for independent variable x₂

Advertising and the number of employees working in a sales territory are the independent variables and they are represented by x₁ and x₂ in this equation.  The intercept, a, is the point on the y-axis that the regression line hits when x = 0.  The slope, b, is the change in sales for any 1-unit change in x₁ and x₂.  In this context the terms independent and dependent are arbitrary designations because this is not a casual relationship.  In other words, there is no cause-and-effect relationship between the dependent and independent variables.

Unlike other consulting projects, a survey was not needed because relevant marketing data was archived by the company.  The matrix shown below was compiled from sales records within the  company’s management information system.

TERRITORY	SALES ($ Thousands) y	ADVERTISING ($ Hundreds) x1	NUMBER OF SALES REPS x2
1	81	27	2
2	94	92	2
3	110	103	4
4	125	115	5
5	90	94	1
6	84	79	1
7	112	105	4
8	99	94	2
9	93	85	2
10	78	54	3
11	114	106	4
12	132	120	6
13	129	118	5
14	79	75	1
15	97	77	3

In most cases the slope and intercept are unknown, therefore they must be estimated with separate equations.  These formulas are complex, however the slope and intercept can be automatically calculated with statistical software using data from the matrix shown above.  The values for these parameters are shown below in the coefficients column.

	Coefficients	Std. Error	Beta	t	Sig.
Constant	49.36	4.46		11.07	0.000
Advertising	0.35	0.06	0.48	5.76	0.000
Sales Reps	6.84	0.95	0.60	7.23	0.000

ARI understands that many marketing managers are not statistically orientated, therefore an interpretation of this needs to be offered in plain English.  The computer has estimated the intercept and slope for advertising and number of sales representatives.  These are the independent variables.  In this example, .35 denotes the change in sales per unit change in advertising when number of sales representatives is held constant.

Similarly 6.84 denotes the change in sales per unit change in number of sales representatives when advertising is held constant.  The estimated intercept is 49.36.  This is the predicted value of sales when the independent variables equal zero.  The t values, shown in the third column, are used to test the statistical significance of the relationship between sales and the independent variables; advertising and number of sales representatives.  The t value is calculated by dividing the slope by the standard error.  i.e. 6.84/.95 = 7.23.  This number is greater than the critical value from the t distribution which is used to reject or not reject the null hypothesis.  The null hypothesis implies that there is no linear relationship between sales and the independent variables.

Since 7.23 is greater than 2.16 (2.16 is the critical value from the t distribution) the null hypothesis for advertising was rejected.  In other words, there is a significant linear relationship between number of sales representatives and sales.  The t value for advertising also exceeds 2.16 which means that it is also a statistically significant variable.  The Sig. abbreviation in the far right column is also a measure of statistical significance.  Values less than .05 are statistically significant.  In this example the Sig. values for both variables are 0.00, hence they are  statistically significant.

Advertising and number of sales representatives were shown to be statistically significant however the client also wanted to determine the relative importance of these variables on sales.  Focus your attention on the beta values in the fourth column of the table.  Since most independent variables are measured with different units, specialized computer software was used to  automatically standardize these values which enabled the client to make direct comparisons between advertising and size of sales force.  The larger the absolute value of the beta number, the more relative importance it assumes in predicting the dependent variable.  Thus number of sales representatives has a greater effect on sales than advertising.

The equation for predicting sales based on advertising expenditures and number of sales representatives is shown below. 

 y = 49.36 + .35x₁ + 6.84x₂ 

How can you use this equation to make a marketing decision?  Imagine you’re the marketing director for the sporting goods manufacturer and you’re faced with declining sales in a western territory.  Should you hire additional sales people or increase your advertising budget?  How might sales be affected in territory one if you hired two additional sales reps and tripled your advertising budget?  The numbers in parenthesis represent proposed changes.  

 y = 49.36 + .35(81) + 6.84(4)

 y = 49.36 + 28.35 + 27.36

 y = 105.07 

The calculated prediction is about $105,000 assuming you manipulate the variables as indicated.  What if you wanted to make some changes in the opposite direction?  In other words, what might you expect if you reduced the advertising budget in territory twelve from $12,000 to $7,500 and transferred one sales rep from the same territory to a different region?  The same equation can be used to make the sales prediction and it is shown below.

 y = 49.36 + .35(7.5) + 6.84(5)

 y = 86.19

Regression analysis is a powerful tool within the realm of predictive research because it can be utilized for many marketing applications.  In addition to sales, regression analysis can be used to predict satisfaction, likelihood of brand purchase and consumer loyalty.  In spite of this,  predictions derived from regression analysis do not imply causality because this statistic is an extension of a correlation.  Finally regression analysis should not be used to make predictions outside the boundaries of the data used to develop the equation.   In other words, advertising values between $2,700 and $12,000 offer the greatest validity.

 What you have just read is a brief example of applied marketing research.
The data shown in the matrix is fictitious.  ARI will never reveal the identity of a client or the actual results of a project under any circumstances without a client’s consent.  This example was developed to help potential clients fathom an application of regression analysis which might be useful to their profession.