Question

A store manager believes items with discounts (10% off, 20% off, etc.) sell better than items offered at full price. She also believes items with well-known brand names sell better than items not associated with a well-known brand. Furthermore, she believes that when the prices of brand-name items are discounted sales are much higher than normal.

You think this sounds like she is describing an interaction between brand-named and discounts and you want to use regression to test the effects of discounts, brands, and their interaction.

Assume that the manager provides the following IVs:

• sales                     number of sales (the DV
• item_cost            cost of the item (a continuous IV)
• brand (0,1)          brand=1 for items associated with a well-known brand and 0 otherwise
• discount (0,1)      discount = 1 for items with discounted prices and 0 otherwise

Question: Write out the regression equation(s) you would use to test the hypothesis that there is an interaction between brands and discounts. Please explain.

Answer 1

Let the sales be represented as Y since it is the dependent variable here. Let the item_cost be represented as X1, the brand be represented as X2, and the discount be represented as X3.

Now, the goal is to check the relation between the independent and the dependent variables.

For this, we can employ a regression analysis. The question pertains to find the equation.

A regression equation for n independent variables with no interaction can be represented as:

where are the coefficients of the regression equation and X0 = 1.

However, if there is an interaction between let us say Xk and Xk-1 variables, there will be an additional term in the equation to capture this interaction and the equation becomes:

In the given equation there are three independent variables. The sales as predicted by the store manager are dependent on discount X3, on whether the item belongs to a brand X2 and on the interaction between these two variables i.e. X2X3. There is no prediction based on the item_cost. Hence, our regression equation will capture only two IVs and their interaction.

Therefore, the equation for the model becomes:

This can then be used to fit the existing data and the parameters of the model can be checked for significance.

For X2 = brand, the prediction is that the items sell more if it belongs to a brand. Hence, the hypothesis would be that

For X3 = discount, the prediction is that the items sell more if they are discounted. Hence, the hypothesis would be that

Also, the prediction involves that the sales are even higher when the discount and brand both equal 1. Hence, the hypothesis would be that