Regression Analysis and a Data Analysis Overview - Model Parameters
Regression Analysis and a Data Analysis Overview - Model Parameters
Estimating the Parameters
The parameters, a and 6, are the characteristics of the relationship between X and Y that are of prime interest. One of the goals of the regression analysis is to determine what they are. The procedure is to obtain a random sample of stores and to use the information from this random sample to estimate a and B. For example, assume that a random sample of 20 stores was selected. For each store in the sample, the number of people entering the store on a given Saturday was determined. Further, for each store the expenditure on advertising for the previous day was recorded. The results are plotted in Figure 20-2.
The next step is to obtain a line that has the best "fit" to these points. Of course, a line could be drawn freehand; in practice, however, a computer program is used. The computer program generates a line that has the property that the squared vertical deviations from the line are minimized. Examples of vertical deviations from the line are shown in Figure 20-2 as brackets. Such a line is termed a least-squares line and is denoted by the following expression:
Y = a + bX = 275 + 0.74X
The least-squares criterion is used because it is convenient (for the computer) and because it is logical.
The value a (275) is an estimate of the parameter a and the value b (0.74) is an estimate of the parameter B. These estimates, a and b, are termed regression coefficients and are based on the random sample.
The term Y(read as "Yhat" since it appears as a Y with a hat) indicates the points on the line and is an estimate of store traffic based on the regression model. For example, when X is 600, then
Y = 275 + 0.74X = 719
Thus, if advertising expenditures of 600 were planned, our store traffic estimate would be 719, the point on the line corresponding to an X value of 600. In contrast, Y is the actual sales level. For example, for one store the
advertising expenditure (X) was 500, the store traffic [Y) was 870, and the store traffic estimate [Y] was 645.
If another random sample of 20 stores were obtained, it undoubtedly would contain different stores. As a result, the plot of X and V would differ from Figure 20-2 and the regression coefficients, a and b, would be different. Every random sample would have associated with it a different a and b. A measure of this variation of the parameter estimates, a and b, is given by their standard errors. Thus, just as the sample mean, X, has a standard error, which is estimated by s^, so do a and b have a standard error associated with each of them.3 The computer program will output these terms.
Interpretation of the Parameter Estimates
The parameter estimates have a very precise meaning. The parameter 3 indicates that if the variable X is changed by one unit, the variable Y will change by 3 units. Thus, if $1 is added to the advertising budget, regardless of the level at which the budget is set, an extra 3 customers will be expected to visit the store. Similarly, the parameter a reflects the number of customers expected, on average, if no advertising is run the previous day.
When understanding is the motivation behind the data analysis, the prime interest is in b, the estimate of the 3 parameter. The size of b will be a reflection of the influence of advertising on store traffic. The parameter a is frequently of little interest.
Testing the Hypothesis that p" Equals Zero
If 3 were zero, then there would be no effect of advertising on store traffic and the model should not be used for any purpose with a nonzero b. Before using the model, it is useful to consider the hypothesis that 6 is zero. If the evidence that 8 is nonzero (namely the nonzero estimate b) is unimpressive, we may want to discard the model.
As indicated previously, the estimate b has a variation associated with it (measured by sb) because it is based on a sample of stores. Thus, it could happen that b is nonzero even if the parameter S is actually zero. In fact, it would be highly likely that, even if there were no relationship between advertising (X) and store traffic (Y), any given random sample would produce a nonzero value for b. One way to evaluate the magnitude of b, taking into account its variation, is to use a statistical hypothesis test. The test, like any hypothesis test, is based on a probability question. What is the probability, the p-value, that the absolute value of b would be as large as it is if, in fact, the parameter S were equal to zero?4 If the p-value is high (i. e., 0.25) then the evidence that B is nonzero is unimpressive and the model may well be discarded. Conversely, a hypothesis that the 3 is actually zero should be viewed with suspicion if the p-value were low.
To determine that probability (the p-value) we calculate t, the number of standard errors, sb, represented by b:
t = b + sb
We assume that t is distributed normally.5 Various probabilities are given in Table 20-1. This table shows that if t is equal to 1.0 the probability is 0.317,
4A two-sided test is involved.
5As noted in Chapter 12, it is actually distributed as a t distribution, a distribution similar in shape to a normal distribution that approaches the normal as the sample size gets larger.
a relatively high p-value; whereas if t is above 1.65 then the p-value is below 0.10.
In our example the t-value is
t = b -H sb = 3.2
The probability (the p-value) that t would be that large if 3 were zero is less than 0.01 (in fact it is shown in Table 20-1 to be below .003). Thus, we say that the independent variable is significant at the 0.01 level (which means that the p-value is less than 0.01) and that the evidence suggests that B is not zero. Since the model "passed" the hypothesis test it is appropriate tc make judgments based on it.
Estimating the Parameters
The parameters, a and 6, are the characteristics of the relationship between X and Y that are of prime interest. One of the goals of the regression analysis is to determine what they are. The procedure is to obtain a random sample of stores and to use the information from this random sample to estimate a and B. For example, assume that a random sample of 20 stores was selected. For each store in the sample, the number of people entering the store on a given Saturday was determined. Further, for each store the expenditure on advertising for the previous day was recorded. The results are plotted in Figure 20-2.
The next step is to obtain a line that has the best "fit" to these points. Of course, a line could be drawn freehand; in practice, however, a computer program is used. The computer program generates a line that has the property that the squared vertical deviations from the line are minimized. Examples of vertical deviations from the line are shown in Figure 20-2 as brackets. Such a line is termed a least-squares line and is denoted by the following expression:
Y = a + bX = 275 + 0.74X
The least-squares criterion is used because it is convenient (for the computer) and because it is logical.
The value a (275) is an estimate of the parameter a and the value b (0.74) is an estimate of the parameter B. These estimates, a and b, are termed regression coefficients and are based on the random sample.
The term Y(read as "Yhat" since it appears as a Y with a hat) indicates the points on the line and is an estimate of store traffic based on the regression model. For example, when X is 600, then
Y = 275 + 0.74X = 719
Thus, if advertising expenditures of 600 were planned, our store traffic estimate would be 719, the point on the line corresponding to an X value of 600. In contrast, Y is the actual sales level. For example, for one store the
advertising expenditure (X) was 500, the store traffic [Y) was 870, and the store traffic estimate [Y] was 645.
If another random sample of 20 stores were obtained, it undoubtedly would contain different stores. As a result, the plot of X and V would differ from Figure 20-2 and the regression coefficients, a and b, would be different. Every random sample would have associated with it a different a and b. A measure of this variation of the parameter estimates, a and b, is given by their standard errors. Thus, just as the sample mean, X, has a standard error, which is estimated by s^, so do a and b have a standard error associated with each of them.3 The computer program will output these terms.
Interpretation of the Parameter Estimates
The parameter estimates have a very precise meaning. The parameter 3 indicates that if the variable X is changed by one unit, the variable Y will change by 3 units. Thus, if $1 is added to the advertising budget, regardless of the level at which the budget is set, an extra 3 customers will be expected to visit the store. Similarly, the parameter a reflects the number of customers expected, on average, if no advertising is run the previous day.
When understanding is the motivation behind the data analysis, the prime interest is in b, the estimate of the 3 parameter. The size of b will be a reflection of the influence of advertising on store traffic. The parameter a is frequently of little interest.
Testing the Hypothesis that p" Equals Zero
If 3 were zero, then there would be no effect of advertising on store traffic and the model should not be used for any purpose with a nonzero b. Before using the model, it is useful to consider the hypothesis that 6 is zero. If the evidence that 8 is nonzero (namely the nonzero estimate b) is unimpressive, we may want to discard the model.
As indicated previously, the estimate b has a variation associated with it (measured by sb) because it is based on a sample of stores. Thus, it could happen that b is nonzero even if the parameter S is actually zero. In fact, it would be highly likely that, even if there were no relationship between advertising (X) and store traffic (Y), any given random sample would produce a nonzero value for b. One way to evaluate the magnitude of b, taking into account its variation, is to use a statistical hypothesis test. The test, like any hypothesis test, is based on a probability question. What is the probability, the p-value, that the absolute value of b would be as large as it is if, in fact, the parameter S were equal to zero?4 If the p-value is high (i. e., 0.25) then the evidence that B is nonzero is unimpressive and the model may well be discarded. Conversely, a hypothesis that the 3 is actually zero should be viewed with suspicion if the p-value were low.
To determine that probability (the p-value) we calculate t, the number of standard errors, sb, represented by b:
t = b + sb
We assume that t is distributed normally.5 Various probabilities are given in Table 20-1. This table shows that if t is equal to 1.0 the probability is 0.317,
4A two-sided test is involved.
5As noted in Chapter 12, it is actually distributed as a t distribution, a distribution similar in shape to a normal distribution that approaches the normal as the sample size gets larger.
a relatively high p-value; whereas if t is above 1.65 then the p-value is below 0.10.
In our example the t-value is
t = b -H sb = 3.2
The probability (the p-value) that t would be that large if 3 were zero is less than 0.01 (in fact it is shown in Table 20-1 to be below .003). Thus, we say that the independent variable is significant at the 0.01 level (which means that the p-value is less than 0.01) and that the evidence suggests that B is not zero. Since the model "passed" the hypothesis test it is appropriate tc make judgments based on it.
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