Introduction to Linear Regression Analysis. Douglas C. Montgomery

is

(2.45)

The prediction interval (2.45) is of minimum width at and widens as increases. By comparing (2.45) with (2.43), we observe that the prediction interval at x₀ is always wider than the CI at x₀ because the prediction interval depends on both the error from the fitted model and the error associated with future observations.

      Example 2.7 The Rocket Propellant Data

      We find a 95% prediction interval on a future value of propellant shear strength in a motor made from a batch of sustainer propellant that is 10 weeks old. Using (2.45), we find that the prediction interval is

      which simplifies to

      Therefore, a new motor made from a batch of 10-week-old sustainer propellant could reasonably be expected to have a propellant shear strength between 2048.32 and 2464.32 psi.

Figure 2.5 The 95% confidence and prediction intervals for the propellant data.

Figure 2.5 shows the 95% prediction interval calculated from (2.45) for the rocket propellant regression model. Also shown on this graph is the 95% CI on the mean [that is, E(y|x) from Eq. (2.43). This graph nicely illustrates the point that the prediction interval is wider than the corresponding CI.

      We may generalize (2.45) somewhat to find a 100(1 − α) percent prediction interval on the mean of m future observations on the response at x = x₀. Let be the mean of m future observations at x = x₀. A point estimator of is . The 100(1 − α)% prediction interval on is

      (2.46)

2.6 COEFFICIENT OF DETERMINATION

      The quantity

(2.47)

is called the coefficient of determination. Since SS_T is a measure of the variability in y without considering the effect of the regressor variable x and SS_Res is a measure of the variability in y remaining after x has been considered, R² is often called the proportion of variation explained by the regressor x. Because 0 ≤ SS_Res ≤ SS_T, it follows that 0 ≤ R² ≤ 1. Values of R² that are close to 1 imply that most of the variability in y is explained by the regression model. For the regression model for the rocket propellant data in Example 2.1, we have

      that is, 90.18% of the variability in strength is accounted for by the regression model.

      The statistic R² should be used with caution, since it is always possible to make R² large by adding enough terms to the model. For example, if there are no repeat points (more than one y value at the same x value), a polynomial of degree n − 1 will give a “perfect” fit (R² = 1) to n data points. When there are repeat points, R² can never be exactly equal to 1 because the model cannot explain the variability related to “pure” error.

      Although R² cannot decrease if we add a regressor variable to the model, this does not necessarily mean the new model is superior to the old one. Unless the error sum of squares in the new model is reduced by an amount equal to the original error mean square, the new model will have a larger error mean square than the old one because of the loss of one degree of freedom for error. Thus, the new model will actually be worse than the old one.

      The magnitude of R² also depends on the range of variability in the regressor variable. Generally R² will increase as the spread of the x’s increases and decrease as the spread of the x’s decreases provided the assumed model form is correct. By the delta method (also see Hahn 1973), one can show that the expected value of R² from a straight-line regression is approximately

      Clearly the expected value of R² will increase (decrease) as Sxx (a measure of the spread of the x’s) increases (decreases). Thus, a large value of R² may result simply because x has been varied over an unrealistically large range. On the other hand, R² may be small because the range of x was too small to allow its relationship with y to be detected.

      There are several other misconceptions about R². In general, R² does not measure the magnitude of the slope of the regression line. A large value of R² does not imply a steep slope. Furthermore, R² does not measure the appropriateness of the linear model, for R² will often be large even though y and x are nonlinearly related. For example, R² for the regression equation in Figure 2.3b will be relatively large even though the linear approximation is poor. Remember that although R² is large, this does not necessarily imply that the regression model will be an accurate predictor.

2.7 A SERVICE INDUSTRY APPLICATION OF REGRESSION

      A hospital is implementing a program to improve service quality and productivity. As part of this program the hospital management is attempting to measure and evaluate patient satisfaction. Table B.17 contains some of the data that have been collected on a random sample of 25 recently discharged patients. The response variable is satisfaction, a subjective response measure on an increasing scale. The potential regressor variables are patient age, severity (an index measuring the severity of the patient’s