Regression Analysis and a Data Analysis Overview - Regression Model
Regression Model
The construction of a regression model usually starts with the specification of the dependent variable and the independent variable or variables. Suppose that our organization, Midwest Stereo, has 200 retail stores that sell hi-fi and related equipment. Our goal is to determine the impact of advertising on store traffic, that is, the number of people who come into the store as a result of the advertising. More specifically, we are concerned with the number of people entering the store on a Saturday as a result of advertising placed the day before. The following regression model might then be hypothesized:
Y = a + fiX + e
where
Y = the number of people entering the store on Saturday X = the amount of money the store spent on advertising on Friday e = an error term a, (3 = model parameters
There are several aspects of the model worth emphasizing. First, the hypothesized relationship is linear; it represents a straight line, as shown in Figure 20-1. Such an assumption is not as restrictive as it might first appear. Even if the actual relationship is curved, as illustrated by the dotted arc in Figure 20-1, the relationship still may be close to linear in the range of advertising expenditures of interest. Thus, a linear relationship still may be very adequate.1
1 Further, a simple transformation of the independent variable can change some types of nonlinear relationships into linear ones. For example, instead of advertising, we might replace the advertising term with its square root or with the logarithm of advertising. The result would be a model such as
Y = a + p logX + e or Y = a + (3 VX + e
2There are several assumptions surrounding the error term. First, we assume that it is, on average, zero. The line is hypothesized to be positioned so that errors are as likely to occur above the line as below. Second, we assume that the error is not larger for large values of X than for small values of X. Third, for some of the judgments, subsequent analyses will assume tha: the error has a normal distribution. These three assumptions actually are of minor practical
The error term is central to the model. In reality store traffic is affected by variables other than advertising expenditures. It also is affected by store size and location, the weather, the nature of what is advertised, whether the advertising is in newspapers or on radio, and other factors. Thus, even if advertising expenditures are known, and our hypothesized linear relationship between advertising expenditures and store traffic is correct, it will be impossible to predict store traffic exactly. There still will be a margin of error. The error term explicitly reflects the error.
The construction of a regression model usually starts with the specification of the dependent variable and the independent variable or variables. Suppose that our organization, Midwest Stereo, has 200 retail stores that sell hi-fi and related equipment. Our goal is to determine the impact of advertising on store traffic, that is, the number of people who come into the store as a result of the advertising. More specifically, we are concerned with the number of people entering the store on a Saturday as a result of advertising placed the day before. The following regression model might then be hypothesized:
Y = a + fiX + e
where
Y = the number of people entering the store on Saturday X = the amount of money the store spent on advertising on Friday e = an error term a, (3 = model parameters
There are several aspects of the model worth emphasizing. First, the hypothesized relationship is linear; it represents a straight line, as shown in Figure 20-1. Such an assumption is not as restrictive as it might first appear. Even if the actual relationship is curved, as illustrated by the dotted arc in Figure 20-1, the relationship still may be close to linear in the range of advertising expenditures of interest. Thus, a linear relationship still may be very adequate.1
1 Further, a simple transformation of the independent variable can change some types of nonlinear relationships into linear ones. For example, instead of advertising, we might replace the advertising term with its square root or with the logarithm of advertising. The result would be a model such as
Y = a + p logX + e or Y = a + (3 VX + e
2There are several assumptions surrounding the error term. First, we assume that it is, on average, zero. The line is hypothesized to be positioned so that errors are as likely to occur above the line as below. Second, we assume that the error is not larger for large values of X than for small values of X. Third, for some of the judgments, subsequent analyses will assume tha: the error has a normal distribution. These three assumptions actually are of minor practical
The error term is central to the model. In reality store traffic is affected by variables other than advertising expenditures. It also is affected by store size and location, the weather, the nature of what is advertised, whether the advertising is in newspapers or on radio, and other factors. Thus, even if advertising expenditures are known, and our hypothesized linear relationship between advertising expenditures and store traffic is correct, it will be impossible to predict store traffic exactly. There still will be a margin of error. The error term explicitly reflects the error.
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