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regression analysis
Questions and Answers of
Regression Analysis
Bailer and Piegorsch [2000] report on an experiment that examines the effect of a herbicide, nitrofen, on the umber of offspring produced by a particular freshwater invertebrate zooplankton. The data
Chapman [1997-98] conducted an experiment using accelerated life testing to determine the estimated shelf life of a photographic developer. The data follow. Lifetimes often follow an exponential
Gupta and Das [2000] performed an experiment to improve the resistivity of a urea formaldehyde resin. The factors were amount of sodium hydroxide, \(A\), reflux time, \(B\), solvent distillate,
Bast et al. [1983] measured the levels of the antibody CA 125 in blood serums of patients with specific cancers. Antigen levels of 35 and of 65 units per \(\mathrm{mL}\) were considered significant.
Schubert et al. [1992], conducted an experiment using a catapult to determine the effects of hook \(\left(x_{1}\right)\), arm length \(\left(x_{2}\right)\), start angle \(\left(x_{3}\right)\), and
Table B. 17 contains data on the global mean surface air temperature anomaly and the global \(\mathrm{CO}_{2}\) concentration. Fit a regression model to these data, using the global
Table B. 18 contains hourly yield measurements from a chemical process and the process operating temperature. Fit a regression model to these data with the Cochrane-Orcutt method, using the
The data in the table below give the percentage share of market of a particular brand of canned peaches \(\left(y_{t}\right)\) for the past 15 months and the relative selling price
The data in the following table gives the monthly sales for a cosmetics manufacturer \(\left(y_{t}\right)\) and the corresponding monthly sales for the entire industry \(\left(x_{t}\right)\). The
Consider the simple linear regression model \(y_{t}=\beta_{0}+\beta_{1} x+\varepsilon_{t}\), where the error are generated by the second-order autoregressive process\[\varepsilon_{t}=ho_{1}
Consider the weighted least squares normal equations for the case of simple linear regression where time is the predictor variable. Suppose that the variances of the errors are proportional to the
Consider a simple linear regression model where time is the predictor variable. Assume that the errors are uncorrelated and have constant variance \(\sigma^{2}\). Show that the variances of the model
Consider the global surface air temperature anomaly data and the \(\mathrm{CO}_{2}\) concentration data in Table B.17. Fit a time series regression model to these data, using global surface air
Consider the global mean surface temperature and \(\mathrm{CO}_{2}\) concentration from Table B.17. Fit the lagged variables regression models in Eqs. (14.26) and (14.27) to these data. Compare these
Consider the chemical process data in Table B.18. Fit the lagged variables regression models in Eqs. (14.26) and (14.27) to the yield response using temperature as the predictor. Compare these models
Consider the global mean surface temperature and \(\mathrm{CO}_{2}\) concentration from Table B.17. Suppose that you want to use the Cochrane-Orcutt model from Exercise 14.1 to predict the global
Explain why an estimator with a breakdown point of \(50 \%\) may not give satisfactory results in fitting a regression model.
Consider the continuous probability distribution \(f(x)\). Suppose that \(\theta\) is an unknown location parameter and that the density may be written as \(f(x-\theta)\) for \(-\infty
Tukey's Biwelght. A popular \(\psi\) function for robust regression is Tukey's biweight, where\[\psi(z)= \begin{cases}z\left[1-(z / a)^{2}\right]^{2}, & |z| \leq a \\ 0, & |z|>a\end{cases}\]with
The U.S. Air Force uses regression models for cost estimating, an application that almost always involves outliers. Simpson and Montgomery [1998a] present 19 observations on first-unit satellite cost
Table B. 14 presents data on the transient points of an electronic inverter. Fit a model to those data using an \(M\)-estimator. Is there an indication that observations might have been incorrectly
Consider the regression model in Problem 2.10 relating systolic blood pressure to weight. Suppose that we wish to predict an individual's weight given an observed value of systolic blood pressure.
Consider the regression model in Problem 2.4 relating gasoline mileage to engine displacement.Data From Problem 2.4Table B. 3 presents data on the gasoline mileage performance of 32 different
Consider a regression model relating total heat flux to radial deflection for the solar energy data in Table B.2.a. Suppose that the observed total heat flux is \(250 \mathrm{~kW}\). Find a point
Consider the soft drink delivery time data in Example 3.1. Find an approximate \(95 \%\) bootstrap confidence interval on the regression coefficient for distance using \(m=1000\) bootstrap samples.
Consider the soft drink delivery time data in Example 3.1. Find the bootstrap estimate of the standard deviation of \(\hat{\beta}_{1}\) using the following numbers of bootstrap samples: \(m=100,
Describe how you would find a bootstrap estimate of the standard deviation of the estimate of the mean response at a particular point, say \(\mathbf{x}_{0}\).
Describe how you would find an approximate bootstrap confidence interval on the mean response at a particular point, say \(\mathbf{x}_{0}\).
Consider the nonlinear regression model fit to the data in Problem 12.11. Find the bootstrap standard errors for the regression coefficients \(\hat{\theta}_{1}, \hat{\theta}_{2}\), and
Consider the NFL team performance data in Table B.1. Construct a regression tree for this data set. Team y X2 X4 Xs X6 X7 Xg Xg Washington 10 2113 1985 38.9 64.7 +4 868 59.7 2205 1917 Minnesota 11
A Designed Experiment for Linear Regression. You wish to fit a simple linear regression model over the region \(-1 \leq x \leq 1\) using \(n=10\) observa-tions. Four experimental designs are under
An analyst is fitting a simple linear regression model with the objective of obtaining a minimum-variance estimate of the intercept \(\beta_{0}\). How should the data collection experiment be
Suppose that you are fitting a simple linear regression model that will be used to predict the mean response at a particular point such as \(x_{0}\). How should the data collection experiment be
Consider the linear regression model \(y=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\varepsilon\), where the regressors have been coded so that\[\sum_{i=1}^{n} x_{i 1}=\sum_{i=1}^{n} x_{i 2}=0 \quad
Suppose that you plan to run a \(2^{3}\) factorial experiment with 8 runs. You discover that your budget has been increased so that you can perform an additional 4 runs. Where would be the best
Continuation of Exercise 15.20. Suppose that a colleague suggests that you put the additional 4 runs at the center of the design region (assume that the three factors are continuous so that this is
Suppose that you want to fit a first-order regression model with an interaction term to two continuous factors. You plan to conduct the experiment over the usual \(-1,+1\) region in both factors, but
Consider the situation described in Exercise 15.22. Rework this problem assuming that the model you want to fit is second-order and you can afford to perform 12 runs.Data From Exercise 15.22Suppose
The table below presents the test-firing results for 25 surface-to-air antiaircraft missiles at targets of varying speed. The result of each test is either a hit \((y=1)\) or a miss \((y=0)\).a. Fit
A study was conducted attempting to relate home ownership to family income. Twenty households were selected and family income was estimated, along with information concerning home ownership ( \(y=1\)
The compressive strength of an alloy fastener used in aircraft construction is being studied. Ten loads were selected over the range \(2500-4300\) psi and a number of fasteners were tested at those
The market research department of a soft drink manufacturer is investigating the effectiveness of a price discount coupon on the purchase of a twoliter beverage product. A sample of 5500 customers
A study was performed to investigate new automobile purchases. A sample f 20 families was selected. Each family was surveyed to determine the age of their oldest vehicle and their total family
A chemical manufacturer has maintained records on the number of failures of a particular type of valve used in its processing unit and the length of time (months) since the valve was installed. The
Myers [1990] presents data on the number of fractures (y) that occur in the upper seams of coal mines in the Appalachian region of western Virginia. Four regressors were reported: \(x_{1}=\) inner
Reconsider the model for the soft drink coupon data from Problem 13.4, part a. Construct plots of the deviance residuals from the model and comment on these plots. Does the model appear satisfactory
Reconsider the model for the aircraft fastener data from Problem 13.3, part a. Construct plots of the deviance residuals from the model and comment on these plots. Does the model appear
The gamma probability density function is\[f(y, r, \lambda)=\frac{\lambda^{r}}{\Gamma(r)} e^{-\lambda y} y^{r-1} \quad \text { for } y, \lambda \geq 0\]Show that the gamma is a member of the
The exponential probability density function is\[f(y, \lambda)=\lambda e^{-\lambda y} \quad \text { for } y, \lambda \geq 0\]Show that the exponential distribution is a member of the exponential
The negative binomial probability mass function is\[\begin{aligned}& \qquad f(y, \pi, \alpha)=\left(\begin{array}{c}y+\alpha-1 \\\alpha-1\end{array}\right) \pi^{\alpha}(1-\pi)^{y} \\& \text { for }
The data in the table below are from an experiment designed to study the advance rate \(y\) of a drill. The four design factors are \(x_{1}=\) load, \(x_{2}=\) flow, \(x_{3}=\) drilling speed, and
Reconsider the drill data from Problem 13.16. Remove any regressors from the original model that you think might be unimportant and rework parts b-e of Problem 13.16. Comment on your findings.Data
The table below shows the predicted values and deviance residuals for the Poisson regression model using \(x_{2}=\) bomb load as the regressor fit to the aircraft damage data in Example 13.8. Plot
Consider a logistic regression model with a linear predictor that includes an interaction term, say \(\mathbf{x}^{\prime} \boldsymbol{\beta}=\beta_{0}+\beta_{1} x_{1}+\beta_{2} x_{2}+\beta_{12} x_{1}
The theory of maximum-likelihood states that the estimated large-sample covariance for maximum-likelihood estimates is the inverse of the information matrix, where the elements of the information
Reconsider the pneumoconiosis data in Table 13.1. Fit models using both the probit and complimentary log-log functions. Compare these models to the one obtained in Example 13.1 using the logit.Table
On 28 January 1986 the space shuttle Challenger was destroyed in an explosion shortly after launch from Cape Kennedy. The cause of the explosion was eventually identified as catastrophic failure of
A student conducted a project looking at the impact of popping temperature, amount of oil, and the popping time on the number of inedible kernels of popcorn. The data follow. Analyze these data using
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
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