Standardization is the process by which the raw data are transformed into new variables that have a mean of 0 and a variance of I. When the data are standardized. the intercept assumes a value of. The term beta coefficient of beta weight is used to denote the standardized regression coefficient. In this case. the slope obtained by the regression of Yon X. Byr is the same as the slope obtained by the regression of X on Y. Bxy Moreover. each of these regression coefficients is equal to the simple correlation between X and Y.

There is a simple relationship between the standardized and non standardized regression coefficients:

For the regression results given in Table 17.2, the value of the beta coefficient is estimated as 0.9361. Note that this is also the value of r calculated earlier in this Once the parameters have been estimated, they can be tested for .signiflcance

**Determine the Strength and Significance of Association**

A related inference involves determining the strength and significance of the association between Yand X. The strength of association is measured by the coefficient of determination, il. In bivariate regression, il is the square of the simple correlation coefficient obtained by correlating the two variables. The ‘coefficient il varies between 0 and I. It signifies the proportion of the total variation in Ythat is accounted for by the variation in X. The decomposition of the total variation in Yis similar to that for analysis of variance .As shown in Figure 17.6, the total variation, SSy’ may be decomposed into the variation accounted for by the regression line, SSrrg’ and the error or residual variation, SS<rroror SSrrJ’ as follows

where

The strength of association may then be calculated as follows:

To illustrate the calculations of let us consider again the regression of attitude toward the city on the duration of residence. It may be recalled from earlier calculations of the simple correlation coefficient that:

The predicted values (f) can be calculated using the regression equation:

Attitude (f) = I JJ793 + 0.5897 (Duration of residence)

For the first observation in Table 17.1, this value is:

(Y) = 1.0793 + 0.5897 X 10 = 6.9763

For each successive observation, the predicted values are, in order, 8.1557,8.1557,3.4381, 8.1557,4.6175,5.7969,2.2587,11.6939,6.3866,11.1042, and 2.2587. Therefore

**Multiple Regression**

**Multiple regression** involves a single dependent variable and two or more independent variables. The questions raised in the context of bivariate regression can also be answered via multiple regression by considering additional independent variables

Can variation in sales be explained in terms of variation in advertising expenditures, prices, and level of distribution?

• Can variation in market shares be accounted for by the size of the sales force, advertising expenditures, and sales promotion budgets?

• Are consumers’ perceptions of quality determined by their perceptions of prices, brand image, and brand attributes?

Additional questions can also be answered by multiple regression

• How much of the variation in sales can be explained by advertising expenditures, prices, and level of distribution?

• What is the contribution of advertising expenditures in explaining the variation in sales when the levels of prices and distribution are controlled?

• What levels of sales may be expected, given the levels of advertising expenditures, prices, and level of distribution?

**Global Brands-Local Ads**

Europeans welcome brands from other countries. but when it comes to advertising, they prefer the homegrown variety. A survey done by and Partners and its affiliates finds that most European consumers’ favorite commercials are for local brands even though they are more than likely to buy foreign brands. Respondents in France. Germany. and the United Kingdom named Coca-Cola as the most often purchased soft drink. However. the French selected the famous award-winning spot for France” Perrier bottled water as their favorite commercial. Similarly, in Germany, the favorite advertising was for a German brand of nonalcoholic beer However, in the United Kingdom. Coca-Cola was the favorite soft drink and also the favorite advertising. In light of such findings, the important question is-does advertising help? Does it help increase the purchase probability of the brand er does it merely maintain a high brand recognition rate? One way of finding out is by running multiple regressions where the dependent variable is the likelihood of brand purchase and the independent variables are brand attribute evaluations and advertising evaluations. Separate models with and without advertising can be run to assess an} significant difference in the contribution. Individual t tests could also be examined to find out the significant contribution of both the brand attributes and advertising. The results will indicate the degree to which advertising plays an imponant part in brand purchase decisions. In conjunction with thCse results, a recent study revealed that attempting to build brand loyalty purchases by means of a sales promotion is not a desirable way to achieve such an objective. According to the study, sales promotions only encourage momentary brand switching and merely enhance short-term performance for companies. Furthermore, over the long run, a sales promotion may imply a low quality or unstable brand image to consumers or it may confuse consumers, which could also lead to a decline in brand loyalty. The results of this study show that sacrificing advertising and relying on sales promotions reduces brand associations, which ultimately leads to a decrease in brand loyalty purchase

The general form of the multiple regression model is as follows

which is estimated by the following equation:

As before, the coefficient a represents the intercept, but the bs are now the partial regression coefficients. The least-squares criterion estimates the parameters in such a way as to minimize the total error, SSm. This process also maximizes the correlation between the actual values of Y and the predicted values, Y. All the assumptions made in bi variate regression also apply in multiple regression. We define some associated statistics and then describe the procedure for multiple regression analysis