A major source of revenue for any professional sports team is through ticket sales, especially sales to season ticket subscribers. A study performed a regression analysis to determine what factors caused ticket prices to vary among teams in the same league within a given year. The regression equation was
The research gathered data covering a span of seven years (1996-2002). The financial data were gathered through Team Marketing Reports and the rest of the data were collected using publicly available sources such as sports reports. The results of the regression analyses can be seen in the accompanying table. The results suggest that several factors influenced ticket prices. and the largest factor was that the team was playing in a new stadium
As in the preceding example, some independent variables considered in a study often turn out not to be significant. When there are a large number of independent variables and the researcher suspects that not all of them are significant, stepwise regression should be used.
The purpose of stepwise regression is to select, from a large number of predictor variables, a small subset of variables that account for most of the variation in the dependent or criterion variable. In this procedure, the predictor variables enter or are removed from the regression equation one at a time.22 There are several approaches to stepwise regression
1. Forward inclusion. Initially, there are no predictor variables in the regression equation. Predictor variables are entered one at a time, only if they meet certain criteria specified in terms of the F ratio. The order in which the variables are included is based on the contribution to the explained variance.
2. Backward elimination. Initially, all the predictor variables are included in the regression equation. Predictors are then removed one at a time based on the F ratio.
3. Stepwise solution. Forward inclusion is combined with the removal of predictors that no longer meet the specified criterion at each’ step.
Stepwise procedures do not result in regression equations that are optimal, in the sense 0:producing the largest R2, for a given number of predictors. Because of the correlations between predictors, an important variable may never be included, or less important variables may enter the equation.
To identify an optimal regression equation, one would have to compute combinatorial solutions in which all possible combinations are examined. Nevertheless, stepwise regression can be useful when the sample size is large in relation to the number of predictors, as shown in the following example.
Stepping Out … to the Mall
Even in the twenty-first century, browsing is a fundamental part of sbopping-whether it is online or in the mall. Customers like to consider their purchase decisions before actually carrying them out. Many consider store-based retailers to have an advantage over Web-based retailers when it comes to browsing because store-based retailers are larger in size and product offerings. Although the Web appeals to younger shoppers, the mall will remain ahead of the game, especially with so many entertainment factors now being built inside malls. A profile of browsers in regional shopping malls was constructed using three sets of independent variables: demographics, shopping behavior, and psychological and attitudinal variables. The dependent variable consisted of a browsing index. In a stepwise regression including all three sets of variables, demographics were found to be the most powerful predictors of browsing behavior. The final regression equation, which contained 20 of the possible 36 variables, included all of the demographics. The accompanying table presents the regression coefficients, standard errors of the coefficients, and their significance levels
also tend to be somewhat downscale, compared to other mall patrons, exhibiting lower levels of education and income, after accounting for the effects of sex and employment status. Although browsers tend to be somewhat younger than non browsers, they are not necessarily single; those who reported larger family sizes tended to be associated with smaller values of the browsing index.
The downscale profile of browsers relative to other mall patrons indicates that specialty stores in malls should emphasize moderately priced products. This may explain the historically low rate of failure in malls among such stores and the tendency of high-priced specialty shops to be located in only the prestigious malls or upscale non enclosed shopping centers
Stepwise regression and multiple regression are complicated by the presence of multidisciplinary. Virtually all multiple regression analyses done in marketing research involve predictors or independent variables that are related. However, multidisciplinary arises when inter correlations among the predictors are very high. Multidisciplinary can result in several problems, including
1. The partial regression coefficients may not be estimated precisely. The standard errors are likely to be high.
2. The magnitudes as well as the signs of the partial regression coefficients may change from sample to sample.
3. It becomes difficult to assess the relative importance of the independent variables in explaining the variation in the dependent variable.
4. Predictor variables may be incorrectly included or removed in step wise regression.
What constitutes serious multidisciplinary is not always clear, although several rules of thumb and procedures have been suggested in the literature. Procedures of varying complexity have also been suggested to cope with multidisciplinary.I” A simple procedure consists of using only one of the variables in a highly correlated set of variables. Alternatively, the set of independent variables can transformed into a new set of predictor that are mutually independent by using techniques such as principal components analysis More specialized techniques, such as ridge regression. and latent root regression, can also be used.2
Relative Importance of Predictors
When multicollinearity is present, special care is required in assessing the relative importance of independent variables. In applied marketing research, it is valuable to deter:nine the relative importance of the predictors. In other words, how important are the independent variables in accounting for the variation in the criterion or dependent variable?26 Unfortunately, because the predictors are correlated, there is no unambiguous measure of relative importance of the predictors in regression analysisP However, several approaches are commonly used to assess the relative importance of predictor variables.
Given that the predictors are correlated, at least to some extent, in virtually all regression situations, none of these measures is satisfactory. It is also possible that the different measures may indicate a different order of importance of the predictors.28 Yet, if all the measures are examined collectively, useful insights may be obtained into the relative importance of the predictors