Questions and Answers of Regression Analysis

Sketch the expectation function for the logistic growth model (12.34) for \(\theta_{1}=1, \theta_{3}=1\), and values of \(\theta_{2}=1,4,8\), respectively. Overlay these plots on the same \(x-y\)
Consider the Gompertz model in Eq. (12.35). Graph the expectation function for \(\theta_{1}=1, \theta_{3}=1\), and \(\theta_{2}=\frac{1}{8}, 1,8,64\) over the range \(0 \leq x \leq 10\).Equation
For the models shown below, determine whether it is a linear model, an intrinsically linear model, or a nonlinear model. If the model is intrinsically linear, show how it can be linearized by a
Reconsider the regression models in Problem 12.6, parts a-e. Suppose the error terms in these models were multiplicative, not additive. Rework the problem under this new assumption regarding the
Consider the following observations:a. Fit the nonlinear regression model\[y=\theta_{1} e^{\theta_{2} x}+\varepsilon\]to these data. Discuss how you obtained the starting values.b. Test for
Reconsider the data in the previous problem. The response measurements in the two columns were collected on two different days. Fit a new model\[y=\theta_{3} x_{2}+\theta_{1} e^{\theta_{2}
Consider the model\[y=\theta_{1}-\theta_{2} e^{-\theta_{3} x}+\varepsilon\]This is called the Mitcherlich equation, and it is often used in chemical engineering. For example, \(y\) may be yield and
The data below represent the fraction of active chlorine in a chemical product as a function of time after manufacturing.a. Construct a scatterplot of the data.b. Fit the Mitcherlich law (see Problem
Consider the data below.These data were collected in an experiment where \(x_{1}=\) reaction time in minutes and \(x_{2}=\) temperature in degrees Celsius. The response variable \(y\) is
The following table gives the vapor pressure of water for various temperatures, previously reported in Exercise 5.2.Exercise 5.2The following table gives the vapor pressure of water for various
The following data were collected on specific gravity and spectrophotometer analysis for 26 mixtures of NG (nitroglycerine), TA (triacetin), and 2 NDPA (2-nitrodiphenylamine).There is a need to
A major problem associated with many mining projects is subsidence, or sinking of the ground above the excavation. The mining engineer needs to control the amount and distribution of this subsidence.
In the field of ecology, the relationship between the concentration of available dissolved organic substrate and the rate of uptake (velocity) of that substrate by heterotrophic microbial communities
In a study to develop the growth behavior for protozoa colonization in a particular lake, an experiment was conducted in which 15 sponges were placed in a lake and 3 sponges at a time were gathered.
The following data were collected on specific gravity and spectrophotometer analysis for 26 mixtures of NG (nitroglycerine), TA (triacetin and 2 NDPA (2-nitrodiphenylamine).There is a need to
Table B. 15 presents data on air pollution and mortality. Use the all-possibleregressions selection on the air pollution data to find appropriate models for these data. Perform a thorough analysis of
Use the all-possible-regressions selection on the patient satisfaction data in Table B.17. Perform a thorough analysis of the best candidate models. Compare your results with stepwise regression.
Use the all-possible-regressions selection on the fuel consumption data in Table B.18. Perform a thorough analysis of the best candidate models. Compare your results with stepwise regression.
Use the all-possible-regressions selection on the wine quality of young red wines data in Table B.19. Perform a thorough analysis of the best candidate models. Compare your results with stepwise
Use the all-possible-regressions selection on the methanol oxidation data in Table B.20. Perform a thorough analysis of the best candidate models. Compare your results with stepwise regression.
Generalized Regression Techniques and Variable Selection In Chapter 9, we introduced generalized regression techniques as an approach to handling the multicollinearity problem. The LASSO can
Table B. 22 contains data on 1916 team performance for Major League Baseball. Use all possible regressions to build a model for this data. Perform a residual analysis on the final model and comment
Use stepwise regression to build a model for the 1916 MLB team performance data in Table B.22. Perform a residual analysis on the final model. Compare this model to the all possible regressions model
Table B. 23 contains data from the NBA Combine. Use all possible regressions to build a model for these data. Perform a residual analysis on the final model and comment on model adequacy. Time Run
Use stepwise regression to build a model for the NBA Combine data in Table B.23. Perform a residual analysis on the final model. Compare this model to the all possible regressions model from Problem
Table B. 24 contains data on home rental prices and home sales. Use all possible regressions to build a model for these data. Perform a residual analysis on the final model and comment on model
Use stepwise regression to build a model for the home rental prices and home sales data in Table B.24. Perform a residual analysis on the final model. Compare this model to the all possible
Table B. 25 contains the golf data on strokes gained. Use all possible regressions to build a model for these data. Perform a residual analysis on the final model and comment on model adequacy. SG:
Consider the regression model developed for the National Football League data in Problem 3.1.Data From Problem 3.1Consider the National Football League data in Table B.1.a. Calculate the PRESS
Split the National Football League data used in Problem 3.1 into estimation and prediction data sets. Evaluate the statistical properties of these two data sets. Develop a model from the estimation
Calculate the PRESS statistic for the model developed from the estimation data in Problem 11.2. How well is the model likely to predict? Compare this indication of predictive performance with the
Consider the delivery time data discussed in Example 11.3. Find the PRESS statistic for the model developed from the estimation data. How well is the model likely to perform as a predictor? Compare
Consider the delivery time data discussed in Example 11.3.Data From Example 11.3a. Develop a regression model using the prediction data set.b. How do the estimates of the parameters in this model
In Problem 3.5 a regression model was developed for the gasoline mileage data using the regressor engine displacement \(x_{1}\) and number of carburetor barrels \(x_{6}\). Calculate the PRESS
In Problem 3.6 a regression model was developed for the gasoline mileage data using the regressor vehicle length \(x_{8}\) and vehicle weight \(x_{10}\). Calculate the PRESS statistic for this model.
Consider the gasoline mileage data in Table B.3. Delete eight observations (chosen at random) from the data and develop an appropriate regression model. Use this model to predict the eight withheld
Consider the gasoline mileage data in Table B.3. Split the data into estimation and prediction sets.a. Evaluate the statistical properties of these data sets.b. Fit a model involving \(x_{1}\) and
Refer to Problem 11.2. What are the standard errors of the regression coefficients for the model developed from the estimation data? How do they compare with the standard errors for the model in
Refer to Problem 11.2. Develop a model for the National Football League data using the prediction data set.Data From Problem 11.2Split the National Football League data used in Problem 3.1 into
What difficulties do you think would be encountered in developing a computer program to implement the DUPLEX algorithm? For example, how efficient is the procedure likely to be for large sample
If \(\mathbf{Z}\) is the \(n \times k\) matrix of standardized regressors and \(\mathbf{T}\) is the \(k \times k\) upper triangular matrix in Eq. (11.3), show that the transformed regressors
Show that the least-squares estimate of \(\boldsymbol{\beta}\) (say \(\hat{\boldsymbol{\beta}}_{(i)}\) ) with the \(i\) th observation deleted can be written in terms of the estimate based on all
Consider the heat treating data in Table B.12. Split the data into prediction and estimation data sets.a. Fit a model to the estimation data set using all possible regressions. Select the minimum
Consider the jet turbine engine thrust data in Table B.13. Split the data into prediction and estimation data sets.a. Fit a model to the estimation data using all possible regressions. Select the
Consider the electronic inverter data in Table B.14. Delete the second observation in the data set. Split the remaining observations into prediction and estimation data sets.a. Find the minimum
Table B. 11 presents 38 observations on wine quality.a. Select four observations at random from this data set, then delete these observations and fit a model involving only the regressor flavor and
Consider all 40 observations on the delivery time data. Delete \(10 \%\) (4) of the observations at random. Fit a model to the remaining 36 observations, predict the four deleted values, and
Consider the Michaelis-Menten model introduced in Eq. (12.23). Graph the expectation function for this model for \(\theta_{1}=200\) and \(\theta_{2}=0.04,0.06,0.08\), 0.10 . Overlay these curves on
Consider the Michaelis-Menten model introduced in Eq. (12.23). Graph the expectation function for \(\theta_{1}=100,150,200,250\) for \(\theta_{2}=0.06\). Overlay these curves on the same set of
Graph the expectation function for the logistic growth model (12.34) for \(\theta_{1}=10, \theta_{2}=2\), and values of \(\theta_{3}=0.25,1,2,3\), respectively. Overlay these plots on the same set of
Consider the National Football League data in Table B.1.a. Use the forward selection algorithm to select a subset regression model.b. Use the backward elimination algorithm to select a subset
Consider the National Football League data in Table B.1. Restricting your attention to regressors \(x_{1}\) (rushing yards), \(x_{2}\) (passing yards), \(x_{4}\) (field goal percentage), \(x_{7}\)
In stepwise regression, we specify that \(F_{\mathrm{IN}} \geq F_{\mathrm{OUT}}\left(\right.\) or \(t_{\mathrm{IN}} \geq t_{\mathrm{OUT}}\) ). Justify this choice of cutoff values.
Consider the solar thermal energy test data in Table B.2.a. Use forward selection to specify a subset regression model.b. Use backward elimination to specify a subset regression model.c. Use stepwise
Consider the gasoline mileage performance data in Table B.3.a. Use the all-possible-regressions approach to find an appropriate regression model.b. Use stepwise regression to specify a subset
Consider the property valuation data found in Table B.4.a. Use the all-possible-regressions method to find the "best" set of regressors.b. Use stepwise regression to select a subset regression model.
Use stepwise regression with \(F_{\mathrm{IN}}=F_{\text {OUT }}=4.0\) to find the "best" set of regressor variables for the Belle Ayr liquefaction data in Table B.5. Repeat the analysis with
Use the all-possible-regressions method to select a subset regression model for the Belle Ayr liquefaction data in Table B.5. Evaluate the subset models using the \(C_{p}\) criterion. Justify your
Analyze the tube-flow reactor data in Table B. 6 using all possible regressions. Evaluate the subset models using the \(R_{p}^{2}, C_{p}\), and \(M S_{\text {Res }}\) criteria. Justify your choice of
Analyze the air pollution and mortality data in Table B. 15 using all possible regressions. Evaluate the subset models using the \(R_{p}^{2}, C_{p}\), and \(M S_{\text {Res }}\) criteria. Justify
Consider the all-possible-regressions analysis of Hald's cement data in Example 10.1. If the objective is to develop a model to predict new observations, which equation would you recommend and
Consider the all-possible-regressions analysis of the National Football League data in Problem 10.2. Identify the subset regression models that are \(R^{2}\) adequate (0.05).Data From Problem
Suppose that the full model is \(y_{i}=\beta_{0}+\beta_{1} x_{i 1}+\beta_{2} x_{i 2}+\varepsilon_{i}, i=1,2, \ldots, n\), where \(x_{i 1}\) and \(x_{i 2}\) have been coded so that
Table B. 11 presents data on the quality of Pinot Noir wine.a. Build an appropriate regression model for quality \(y\) using the all-possibleregressions approach. Use \(C_{p}\) as the model selection
Use the wine quality data in Table B. 11 to construct a regression model for quality using the stepwise regression approach. Compare this model to the one you found in Problem 10.4, part a.Data From
Rework Problem 10.14, part a, but exclude the region information.a. Comment on the difference in the models you have found. Is there indication that the region information substantially improves the
Table B. 12 presents data on a heat treating process used to carburize gears. The thickness of the carburized layer is a critical factor in overall reliability of this component. The response
Table B. 13 presents data on the thrust of a jet turbine engine and six candidate regressors. Use all possible regressions and the \(C_{p}\) criterion to find an appropriate regression model for
Table B. 14 presents data on the transient points of an electronic inverter. Use all possible regressions and the \(C_{p}\) criterion to find an appropriate regression model for these data.
Consider the soft drink delivery time data in Example 3.1.Example 3.1a. Find the simple correlation between cases \(\left(x_{1}\right)\) an distance \(\left(x_{2}\right)\).b. Find the variance
Consider the Hald cement data in Table B.21.a. From the matrix of correlations between the regressors, would you suspect that multicollinearity is present?b. Calculate the variance inflation
Using the Hald cement data (Example 10.1), find the eigenvector associated with the smallest eigenvalue of \(\mathbf{X}^{\prime} \mathbf{X}\). Interpret the elements of this vector. What can you say
Find the condition indices and the variance decomposition proportions for the Hald cement data (Table B.21), assuming centered regressors. What can you say about multicollinearity in these data?
Repeat Problem 9.4 without centering the regressors and compare the results. Which approach do you think is better?Data From Problem 9.4Find the condition indices and the variance decomposition
Use the regressors \(x_{2}\) (passing yardage), \(x_{7}\) (percentage of rushing plays), and \(x_{8}\) (opponents' yards rushing) for the National Football League data in Table B.1.a. Does the
Consider the gasoline mileage data in Table B.3.a. Does the correlation matrix give any indication of multicollinearity?b. Calculate the variance inflation factors and the condition number of
Using the gasoline mileage data in Table B. 3 find the eigenvectors associated with the smallest eigenvalues of \(\mathbf{X}^{\prime} \mathbf{X}\). Interpret the elements of these vectors. What can
Use the gasoline mileage data in Table B. 3 and compute the condition indices and variance-decomposition proportions, with the regressors centered. What statements can you make about
Analyze the housing price data in Table B. 4 for multicollinearity. Use the variance inflation factors and the condition number of \(\mathbf{X}^{\prime} \mathbf{X}\). y X1 X2 X3 X4 X5 X6 X7 Xg 25.9
Analyze the chemical process data in Table B. 5 for evidence of multicollinearity. Use the variance inflation factors and the condition number of \(\mathbf{X}^{\prime} \mathbf{X}\). Run No. y X x2 X3
Analyze the patient satisfaction data in Table B. 17 for multicollinearity. Satisfaction Age Severity Surgical-Medical Anxiety 68 55 50 0 2.1 77 46 24 1 2.8 96 30 46 1 3.3 80 35 48 1 4.5 43 59 58 0 2
Analyze the fuel consumption data in Table B. 18 for multicollinearity. y X2 X3 X4 xs X6 X7 Xg 343 0 52.8 811.7 2.11 220 261 87 1.8 356 1 52.8 811.7 2.11 220 261 87 1.8 344 0 50.0 821.3 2.11 223 260
Analyze the wine quality of young red wines data in Table B. 19 for multicollinearity. y X2 X3 X4 x6 X7 Xg Xg x10 19.2 0 3.85 66 9.35 5.65 2.40 3.25 0.33 19 0.065 18.3 0 3.73 79 11.15 6.95 3.15 3.80
Analyze the methanol oxidation data in Table B. 20 for multicollinearity. x1 X2 X3 X4 xs 0 454 8.8 3.90 1.30 1.1 474 8.2 3.68 1.16 4.2 524 7.0 2.78 1.25 94.2 503 7.4 2.27 1.57 20.7 493 7.6 2.40 1.55
The table below shows the condition indices and variance decomposition proportions for the acetylene data using centered regressors. Use this information to diagnose multicollinearity in the data and
Apply ridge regression to the Hald cement data in Table B.21.a. Use the ridge trace to select an appropriate value of \(k\). Is the final model a good one?b. How much inflation in the residual sum of
Use ridge regression on the Hald cement data (Table B.21) using the value of \(k\) in Eq. (9.8). Compare this value of \(k\) value selected by the ridge trace in Problem 9.17. Does the final model
Estimate the parameters in a model for the gasoline mileage data in Table B. 3 using ridge regression.a. Use the ridge trace to select an appropriate value of \(k\). Is the resulting model
Estimate the parameters in a model for the gasoline mileage data in Table B. 3 using ridge regression with the value of \(k\) determined by Eq. (9.8). Does this model differ dramatically from the one
Estimate model parameters for the Hald cement data (Table B.21) using principal-component regression.a. What is the loss in \(R^{2}\) for this model compared to least squares?b. How much shrinkage in
Estimate the model parameters for the gasoline mileage data using principalcomponent regression.a. How much has the residual sum of squares increased compared to least squares?b. How much shrinkage
Consider the air pollution and mortality data given in Table B.15.a. Is there a problem with collinearity? Discuss how you arrived at this decision.b. Perform a ridge trace on these data.c. Select a
Consider the air pollution and mortality data given in Table B.15.a. Is there a problem with collinearity? Discuss how you arrived at this decision.b. Perform a ridge trace on these data.c. Select a
The pure shrinkage estimator is defined as \(\hat{\beta}_{s}=c \hat{\beta}\), were \(0 \leq c \leq 1\) is a constant chosen by the analyst. Describe the kind of shrinkage that this estimator
Show that the pure shrinkage estimator (Problem 9.25) is the solution toData From Problem 9.25The pure shrinkage estimator is defined as \(\hat{\beta}_{s}=c \hat{\beta}\), were \(0 \leq c \leq 1\) is
The mean square error criterion for ridge regression is\[E\left(L_{1}^{2}\right)=\sum_{j=1}^{p} \frac{\lambda_{j}}{\left(\lambda_{j}+k\right)^{2}}+\sum_{j=1}^{p} \frac{\alpha_{j}^{2}
Consider the mean square error criterion for generalized ridge regression. Show that the mean square error is minimized by choosing \(k_{j}=\sigma^{2} / \alpha_{j}^{2}, j=1\), \(2, \ldots, p\).
Show that if \(\mathbf{X}^{\prime} \mathbf{X}\) is in correlation form, \(\boldsymbol{\Lambda}\) is the diagonal matrix of eigenvalues of \(\mathbf{X}^{\prime} \mathbf{X}\), and \(\mathbf{T}\) is the
Formally show that\[D_{i}=\frac{r_{i}}{p} \frac{h_{i i}}{1-h_{i i}}\]
Table B. 14 contains data concerning the transient points of an electronic inverter. Fit a regression model to all 25 observations but only use \(x_{1}-x_{4}\) as the regressors. Investigate this

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