. A multiple regression model was used to relate y viscosity of a chemical product to x1

An article entitled A Method for Improving the Accuracy of Polynomial Regression Analysis in the Journal of Quality Technology (1971, pp. 149155) reported the following data on y ultimate shear strength of a rubber compound (psi) and x cure temperature (F).

12-47. Consider the following data, which result from an experiment to determine the effect of x test time in hours at a particular temperature on y change in oil viscosity:

When fitting polynomial regression models, we often subtract from each x value to produce a centered regressor . This reduces the effects of dependencies among the model terms and often leads to more accurate estimates of the regression coefficients. Using the data from Exercise 12-46, fit the model Use the results to estimate the coefficients in the uncentered model .

12-49. Suppose that we use a standardized variable , where sx is the standard deviation of x, in constructing a polynomial regression model. Using the data in Exercise 12-46 and the standardized variable approach, fit the model . (a) What value of y do you predict when ? (b) Estimate the regression coefficients in the unstandardized model . (c) What can you say about the relationship between SSE and R2 for the standardized and unstandardized models? (d) Suppose that is used in the model along with . Fit the model and comment on the relationship between SSE and R2 in the standardized model and the unstandardized model

The following data shown were collected during an experiment to determine the change in thrust efficiency (y, in percent) as the divergence angle of a rocket nozzle (x) changes: x y 1 y  y2sy Y 0  1x  11x 2   x 285F Y * 0  * 1x  * 111x2 2   x 1x  x2sx Y 0  1x  11x2   Y * 0  * 1x  * 11 1x2 2  . x x  x x (a) Fit a second-order model to the data. (b) Test for significance of regression and lack of fit using 0.05. (c) Test the hypothesis that 11 0, using 0.05. (d) Plot the residuals and comment on model adequacy. (e) Fit a cubic model, and test for the significance of the cubic term using 0.05.

12-51. An article in the Journal of Pharmaceuticals Sciences (Vol. 80, 1991, pp. 971977) presents data on the observed mole fraction solubility of a solute at a constant temperature and the dispersion, dipolar, and hydrogen bonding Hansen partial solubility parameters. The data are as shown in the following table, where y is the negative logarithm of the mole fraction solubility, x1 is the dispersion partial solubility, x2 is the dipolar partial solubility, and x3 is the hydrogen bonding partial solubility. (a) Fit the model (b) Test for significance of regression using 0.05. (c) Plot the residuals and comment on model adequacy

Consider the gasoline mileage data in Exercise 12-5. (a) Discuss how you would model the information about the type of transmission in the car. (b) Fit a regression model to the gasoline mileage using engine displacement, horsepower, and the type of transmission in the car as the regressors. (c) Is there evidence that the type of transmission affects gasoline mileage performance?

Consider the tool life data in Example 12-12. Test the hypothesis that two different regression models (with different slopes and intercepts) are required to adequately model the data. Use indicator variables in answering this question

Use the National Football League Team Performance data in Exercise 12-4 to build regression models using the following techniques: (a) All possible regressions. Find the equations that minimize MSE and that minimize Cp. (b) Stepwise regression.

12-55. Use the gasoline mileage data in Exercise 12-5 to build regression models using the following techniques: (a) All possible regressions. Find the minimum Cp and minimum MSE equations. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the various models obtained.

Consider the electric power data in Exercise 12-6. Build regression models for the data using the following techniques: (a) All possible regressions. (b) Stepwise regression. (c) Forward selection. (d) Backward elimination. (e) Comment on the models obtained. Which model would you prefer?

12-57. Consider the wire bond pull strength data in Exercise 12-8. Build regression models for the data using the following methods: (a) Stepwise regression. (b) Forward selection. (c) Backward elimination. (d) Comment on the models obtained. Which model would you prefer?

Consider the NHL data in Exercise 12-11. Build regression models for these data using the following methods: (a) Stepwise regression. (b) Forward selection. (c) Backward elimination. (d) Which model would you prefer?

12-59. Consider the data in Exercise 12-51. Use all the terms in the full quadratic model as the candidate regressors. (a) Use forward selection to identify a model. (b) Use backward elimination to identify a model. (c) Compare the two models obtained in parts (a) and (b). Which model would you prefer and why?

Find the minimum Cp equation and the equation that maximizes the adjusted R2 statistic for the wire bond pull strength data in Exercise 12-8. Does the same equation satisfy both criteria?

12-61. For the NHL data in Exercise 12-11. (a) Find the equation that minimizes Cp. (b) Find the equation that minimizes MSE. (c) Find the equation that maximizes the adjusted R2 . Is this the same equation you found in part (b)?

We have used a sample of 30 observations to fit a regression model. The full model has nine regressors, the variance estimate is and .

A sample of 25 observations is used to fit a regression model in seven variables. The estimate of 2 for this full model is MSE 10. (a) A forward selection algorithm has put three of the original seven regressors in the model. The error sum of squares for the three-variable model is SSE 300. Based on Cp, would you conclude that the three-variable model has any remaining bias? (b) After looking at the forward selection model in part (a), suppose you could add one more regressor to the model. This regressor will reduce the error sum of squares to 275. Will the addition of this variable improve the model? Why?

The data shown in the table on page 464 represent the thrust of a jet-turbine engine ( y) and six candidate regressors: x1 = primary speed of rotation, x2 secondary speed of rotation, x3 fuel flow rate, x4 pressure, x5 exhaust temperature, and x6 ambient temperature at time of test. (a) Fit a multiple linear regression model using x3 fuel flow rate, x4 pressure, and x5 exhaust temperature as the regressors. (b) Test for significance of regression using 0.01. Find the P-value for this test. What are your conclusions? (c) Find the t-test statistic for each regressor. Using 0.01, explain carefully the conclusion you can draw from these statistics. (d) Find R2 and the adjusted statistic for this model. (e) Construct a normal probability plot of the residuals and interpret this graph. (f) Plot the residuals versus Are there any indications of inequality of variance or nonlinearity? (g) Plot the residuals versus x3. Is there any indication of nonlinearity? (h) Predict the thrust for an engine for which x3 1670, x4 170, and x5 1589.

12-65. Consider the engine thrust data in Exercise 12-64. Refit the model using as the response variable and ln x3 as the regressor (along with x4 and x5). (a) Test for significance of regression using 0.01. Find the P-value for this test and state your conclusions.

The transient points of an electronic inverter are influenced by many factors. Table 12-20 gives data on the transient point (y, in volts) of PMOS-NMOS inverters and five candidate regressors:, x1 width of the NMOS device, x2 length of the NMOS device, x3 width of the PMOS device, x4 length of the PMOS device, and x5 temperature (C). (a) Fit the multiple linear regression model to these data. Test for significance of regression using 0.01. Find the P-value for this test and use it to draw your conclusions. (b) Test the contribution of each variable to the model using the t-test with 0.05. What are your conclusions? (c) Delete x5 from the model. Test the new model for significance of regression. Also test the relative contribution of each regressor to the new model with the t-test. Using 0.05, what are your conclusions?

12-67. Consider the inverter data in Exercise 12-66. Delete observation 2 from the original data. Define new variables as follows: and (a) Fit a regression model using these transformed regressors (do not use x5). (b) Test the model for significance of regression using 0.05. Use the t-test to investigate the contribution of each variable to the model ( 0.05). What are your conclusions? (c) Plot the residuals versus and versus each of the transformed regressors. Comment on the plots.

Following are data on y green liquor (g/l) and x paper machine speed (feet per minute) from a Kraft paper machine. (The data were read from a graph in an article in the Tappi Journal, March 1986.)

Consider the jet engine thrust data in Exercise 12-64. (a) Use all possible regressions to select the best regression equation, where the model with the minimum value of MSE is to be selected as best. (b) Repeat part (a) using the CP criterion to identify the best equation. (c) Use stepwise regression to select a subset regression model. (d) Compare the models obtained in parts (a), (b), and (c) above.

Consider the electronic inverter data in Exercise 12-66 and 12-67. Define the response and regressors variables as in Exercise 12-67, and delete the second observation in the sample. (a) Use all possible regressions to find the equation that minimizes Cp. (b) Use all possible regressions to find the equation that minimizes MSE. (c) Use stepwise regression to select a subset regression model. (d) Compare the models you have obtained.

12-71. Consider the three-variable regression model for the jet engine thrust data in Exercise 12-65. Calculate the variance inflation factors for this model. Would you conclude that multicollinearity is a problem in this model?

. A multiple regression model was used to relate y viscosity of a chemical product to x1 temperature and x2 reaction time. The data set consisted of n 15 observations. (a) The estimated regression coefficients were and . Calculate an estimate of mean viscosity when x1 100F and x2 2 hours. (b) The sums of squares were SST 1230.50 and SSE 120.30. Test for significance of regression using 0.05. What conclusion can you draw?

12-73. An article in the Journal of the American Ceramics Society (Vol. 75, 1992, pp. 112116) describes a process for immobilizing chemical or nuclear wastes in soil by dissolving the contaminated soil into a glass block. The authors mix CaO and Na2O with soil and model viscosity and electrical conductivity. The electrical conductivity model involves six regressors, and the sample consists of n 14 observations. (a) For the six-regressor model, suppose that SST 0.50 and R2 0.94. Find SSE and SSR, and use this information to test for significance of regression with 0.05. What are your conclusions? (b) Suppose that one of the original regressors is deleted from the model, resulting in R2 0.92. What can you conclude about the contribution of the variable that was removed? Answer this question by calculating an F-statistic. (c) Does deletion of the regressor variable in part (b) result in a smaller value of MSE for the five-variable model, in comparison to the original six-variable model? Comment on the significance of your answer.

Consider a multiple regression model with k regressors. Show that the test statistic for significance of regression can be written as Suppose that n 20, k 4, and R2 0.90. If 0.05, what conclusion would you draw about the relationship between y and the four regressors?

A regression model is used to relate a response y to k 4 regressors with n 20. What is the smallest value of R2 that will result in a significant regression if 0.05? Use the results of the previous exercise. Are you surprised by how small the value of R2 is?

A regression model is used to relate a response y to k 4 regressors with n 20. What is the smallest value of R2 that will result in a significant regression if 0.05? Use the results of the previous exercise. Are you surprised by how small the value of R2 is?

Show that the variance of the ith residual ei in a multiple regression model is and that the covariance between ei and ej is  2 hij, where the hs are the elements of H X(X X) 1 X .

Consider the multiple linear regression model y X  . If denotes the least squares estimator of , show that where .

Constrained Least Squares. Suppose we wish to find the least squares estimator of in the model y X  subject to a set of equality constraints, say,

Piecewise Linear Regression (I). Suppose that y is piecewise linearly related to x. That is, different linear relationships are appropriate over the intervals and . Show how indicator variables can be used to fit such a piecewise linear regression model, assuming that the point is known.

Piecewise Linear Regression (II). Consider the piecewise linear regression model described in Exercise 12-79. Suppose that at the point a discontinuity occurs in the regression function. Show how indicator variables can be used to incorporate the discontinuity into the model.

Piecewise Linear Regression (III). Consider the piecewise linear regression model described in Exercise 12-79. Suppose that the point x* is not known with certainty and must be estimated. Suggest an approach that could be used to fit the piecewise linear regression model.