Generalized Linear Models: With Applications In Engineering And The Sciences 2nd Edition Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson - Solutions

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Apply the BMA technique to the shingle sales data in Table 4.1. In particular, identify the best model according to the highest posterior model probability. Comments on the effect of each predictor on the response.
Verify equation (9.43).
For the quasar data in Table 5.1, apply a logarithm transformation on the response Y so that Y ← log(Y ). Consider the semi-conjugate Bayesian linear model that includes all five predictors. (a) With specification β0 = 0, V ≈ O, a = 0.0001, and b = 0.0001, obtain the Bayesian estimator of β
Concerning the semi-conjugate Bayesian linear model as given by (9.34), derive the conditional posterior distributions for β and ν.
Show that the Bayesian estimator of β in (9.22) can be viewed as the generalized least squares estimator in the augmented linear model (9.23).
Concerning the Bayesian linear model (9.11), construct a 95% Bayesian confidence interval for the mean response, E(yp|xp, β, ν) = xt pβ, of all individuals that have predictor values equal to x = xp.
Show that the posterior joint density in (9.13) can be rewritten as the form given by (9.14).
Table 8.9 presents minus two times of the maximized log-likelihood score for several candidate models for the prostate cancer data. i. Compute the AIC and BIC for each of the above models. Based on these two criteria, which one is the best? ii. Concerning Model I, use the likelihood ratio test to
We consider a data set collected from a study on prostate cancer, available from http://www.biostat.au.dk/teaching/postreg/AllData.htm. One of study objectives is to see if information collected at a baseline exam of a patient with prostate cancer, could predict whether the tumor has penetrated
In studying the association between smoking (X) and the incidence of some disease (Y ), let π1 denote the probability of getting this disease for smokers and π2 denote the probability of getting this disease for non-smokers. Note that both π1 and π2 are conditional probabilities. The collected
In this manner, the last category, level K, is left as the baseline. Denote πi = Pr{Yi = 1|Zi) and consider model log πi 1 − πi = β0 + β1Zi1 + ··· + βK−1Zi(K−1). Let ORk:k denote the odds ratio of having Y = 1 that compares Level k and Level k(a) Express ORk:K in terms of βj ,
Consider data involves a binary response Y and a categorical predictor X that has K levels. We define (K − 1) dummy variables {Z1, Z2,...,ZK−1} using the reference cell coding scheme such that Zik = " 1 if the ith subject falls into the kth category, 0 otherwise, for k = 1, 2,...,K −
Log-linear model is often applied in the analysis of contingency tables, especially when more than one of the classification factors can be regarded as response variables. Consider a 2 × 2 × K contingency table induced by classification factors (X, Y, V ). Namely, both X and Y have two levels
Let y ∼ binomial(n, π). We consider the maximum likelihood based inference on the parameter π. (a) Write down the log-likelihood function L(π). (b) Find the score function u(π) = y π − n − y 1 − π and the Fisher’s information I(π) = n π · (1 − π) . (c) Find the MLE of π,
Given that Y follows a distribution in the exponential family of form in (8.2), show that E(Y ) = b 0 (θ) and Var(Y ) = b 00(θ)a(φ) by using the following two general likelihood results E µ ∂L ∂θ ¶ = 0 and − E µ ∂ 2L ∂θ2 ¶ = E µ ∂L ∂θ ¶2 ,(a) Class Level Information Class
As a criterion for comparing models, The Cp measure in (5.18) can alternatively be given by Cp ≈ MSE + 2 · (p + 1) n σˆ 2 . Suppose that MSE from the current model is used to estimate the true noise variance, i.e., ˆσ 2 = MSE. Show that Cp is similar to the GCV criterion given in (5.10),
All-subset-regression approach is an exhausted search and it is feasible only in situation where the number of the regressors is not too large. Table 5.8 is the female teacher effectiveness data that has 7 independent variables. In the data set, the response variable y is the quantitative
Consider the quasar data used in Example 5.1. Define Y 0 = log(Y ). Perform variable selection via both the all possible regressions method and the three stepwise procedures. Examine the results and make comments.
The variance of the regression prediction and variance of the least squares estimators tend to be larger when the regression model is overfitted. Show the following results for an over-fitted regression model: (a) Let x1, x2, ··· , xk be the regressors and b0, b1, ··· , bk be the estimates
Use the following simulation experiment (Cook and Weisberg, 1999, pp. 280-282) to inspect how stepwise selection overstates significance. Generate a data set of n = 100 cases with a response and 50 predictors, all from the standard normal distribution independently. Thus none of the predictors is
Complete the remaining steps in the proof of Theorem 5.2.
Verify the following properties of partitioned matrices. Given a symmetric p.d. matrix A is partitioned in the form A = A11 A12 A21 A22 , where A11 and A22 are square matrices, both A11 and A22 are p.d. and the inverse of A is given by A−1 = A11 + A−1 11 A12B−1A21A−1 11 −A−1 11
(a). Compute s 2 , Cp, PRESS residuals, and PRESS statistic for the model y = β0 + β1x1 + β2x2 + β3x3 + β4x4 + ². (b). Compute s 2 , Cp, PRESS residuals, and PRESS statistic for the model y = β0 + β1x1 + β2x3 + β3x4 + ². (c). Compare the regression models from parts (a) and (b) and
6. The following data in Table 4.11 was collected from a study of the effect of stream characteristics on fish biomass. The regressor variables are x1: Average depth of 50 cells x2: Area of stream cover (i.e., undercut banks, logs, boulders, etc.) x3: Percent of canopy cover (average of 12) x4:
(b). Find the distribution of (R − student) 2 j under H0 : σ 2 ∆ 6=
Observation x1 x2 x3 y 1 12.980 0.317 9.998 57.702 2 14.295 2.028 6.776 59.296 3 15.531 5.305 2.947 56.166 4 15.133 4.738 4.201 55.767 5 15.342 7.038 2.053 51.722 6 17.149 5.982 -0.055 60.466 7 15.462 2.737 4.657 50.715 8 12.801 10.663 3.408 37.441 9 13.172 2.039 8.738 55.270 10 16.125 2.271 2.101
The variance inflation model can be written yi = x 0 iβ + ²i fori = 1, 2, · · · , n with model error being normally distributed and E(²j ) = σ 2 + σ 2 ∆ andE(²i) = σ 2 for all i 6= j and E(²i) =
In the partial residual plot discussed in this chapter, show that the least squares slope of the elements ey|Xj regressed against exj |X−j is bj , the slope of xj in the multiple regression of y on X.
Make your conclusions (hint: Use indicator variables).
Consider the data set in Table 4.10. It is known a prior that in observations 10, 16, and 17 there were some difficulties in measuring the response y. Apply the mean shift model and test simultaneously H0 : ∆10 = 0, ∆16 = 0, ∆17 =
Show the following relationship between the leave-one-out residual and the ordinary residual: P RESS = Xn i=1 (ei,−i) 2 = Xn i=1 ³ yi − yˆi 1 − x 0 i (X 0X)−1xi ´2 = Xn i=1 ³ ei 1 − hii ´2 , where hii is the ith diagonal elements of the HAT matrix.
Let the ith residual of the regression model be ei = yi − yˆi . Prove that Var(ei) = s 2 (1 − hii), where s 2 is the mean square error of the regression model and hii is the ith diagonal element of the HAT matrix..7361 0.11564 0.94022 1.0921 0.13904 10 0.3972 0.6555 0.39405 -1.92210 -3.2833
Do you have any reason to change the model given in part (a). (c). Show a partitioning of total degrees of freedom into those attributed to regression, pure error, and lack of fit. (d). Using the model you adopted in part (b), make a test for lack of fit and draw conclusion. (e). Plot residuals of
In an experiment in the civil engineering department of Virginia Polytechnic Institute and State University in 1988, a growth of certain type of algae in water was observed as a function of time and dosage of copper added into the water. The collected data are shown in Table 3.28. (a). Consider
Consider the general linear regression model y = Xβ + ε and the least squares estimate b = (X 0 X) −1X 0 y. Show that b = β + Rε, where R = (X 0 X) −1X 0 .
For the data set given in Table 3.27 Table 3.27 Data Set for Testing Linear Hypothesis y x1 x2 3.9 1.5 2.2 7.5 2.7 4.5 4.4 1.8 2.8 8.7 3.9 4.4 9.6 5.5 4.3 19.5 10.7 8.4 29.3 14.6 14.6 12.2 4.9 8.5 (a). Find the linear regression model.(b). Use the general linear hypothesis test to test H0 : β1 =
A scientist collects experimental data on the radius of a propellant grain (y) as a function of powder temperature, x1, extrusion rate, x2, and die temperature, x3. The data is presented in Table 3.26. (a). Consider the linear regression model yi = β ? 0 + β1(x1i − x¯1) + β2(x2i − x¯2) +
Consider the general linear regression model y = Xβ + ε and the least squares estimate b = (X 0 X) −1X 0 y. Show that b = β + Rε, where R = (X 0 X) −1X 0 .
(a). Find 80 percent, 90 percent, 95 percent, and 99 percent confidence interval for y0, the mean of one future observation at x1 = 9.5 and x2 = 2.5. (b). Find a 90 percent confidence interval for ¯y0, the mean of six observations at x1 = 9.5 and x2 = 2.5.
Assume that the data given in Table 3.25 satisfy the model yi = β0 + β1x1i + β2x2i + εi , where εi ’s are iid N(0, σ2 ). Table 3.25 Data Set for Calculation of Confidence Interval on Regression Prediction y 12.0 11.7 9.3 11.9 11.8 9.5 9.3 7.2 8.1 8.3 7.0 6.5 5.9 x1 3 4 5 6 7 8 9 10 11 12 13
(Hint: Use the fact that the HAT matrix is idempotent.)
Let hii be the ith diagonal elements of the HAT matrix. Prove that (a). For a multiple regression model with a constant term hii ≥ 1/n. (b). Show that hii ≤
Show that the HAT matrix in linear regression model has the property tr(H) = p where p is the total numbers of the model parameters.
The least squares estimators of the regression model Y = Xβ + ε are linear function of the y-observations. When (X 0 X) −1 exists the least squares estimators of β is b = (X 0 X) −1Xy. Let A be a constant matrix. Using Var(Ay) = AVar(y)A 0 and Var(y) = σ 2 I to show that Var(b) = σ 2 (X 0
Let X be a matrix of n × m and X = (X1, X2), where X1 is n × k matrix and X2 is n × (m − k) matrix. Show that (a). The matrices X(X 0 X) −1X 0 and X1(X 0 1X1) −1X 0 1 are idempotent. (b). The matrix X(X 0 X) −1X 0 − X2(X 0 2X2) −1X 0 2 is idempotent. (c). Find the rank of the matrix
Using the matrix form of the simple linear regression to show the unbiasness of theb. Also, calculate the covariance of b using the matrix format of the simple linear regression.
The study “Development of LIFETEST, a Dynamic Technique to Assess Individual Capability to Lift Material” was conducted in Virginia Polytechnic Institute and State University in 1982 to determine if certain static arm strength measures have influence on the “dynamic lift” characteristics
Observations on the yield of a chemical reaction taken at various temperatures were recorded in Table 2.11: (a). Fit a simple linear regression and estimate β0 and β1 using the least squares method.(b). Compute 95% confidence intervals on E(ylz) at 4 levels of temper- atures in the data. Plot
Show that if a ≥ 0 and b2 − ac ≥ 0, then 1 − α confidence interval on x0 is −b − √b2 − ac a ≤ x0 ≤ −b + √b2 − ac a .
(e). Show that U and Z are independent. (f). Show that W = Z2/A2σˆ2 has the F distribution with degrees of freedom 1 and N. (g). Let S2 1 = (x1j − x¯1)2 and S2 2 = (x2j − x¯2)2, show that the solution of the following quadratic equation about x0, q(x0) = ax2 0 + 2bx0 + c = 0, (βˆ1 −
Consider two simple linear models Y1j = α1 + β1x1j + ε1j , j = 1, 2, · · · , n1 and Y2j = α2 + β2x2j + ε2j , j = 1, 2, · · · , n2 Assume that β1 6= β2 the above two simple linear models intersect. Let x0 be the point on the x-axis at which the two linear models intersect. Also assume
Consider a situation in which the regression data set is divided into two parts as shown in Table 2.10. The regression model is given by yi =    β (1) 0 + β1xi + εi , i = 1, 2, · · · , n1; β (2) 0 + β1xi + εi , i = n1 + 1, · · · , n1 + n2. In other words, there are two
Consider the fixed zero intercept regression model yi = β1xi + εi , (i = 1, 2, · · · , n) The appropriate estimator of σ 2 is given by s 2 = Xn i=1 (yi − yˆi) 2 n − 1 Show that s 2 is an unbiased estimator of σ 2 . Table 2.10 Data for Two Parallel Regression Lines x y x1 y1 . . . . . .
Derive and discuss the (1 − α)100% confidence interval on the slope β1 for the simple linear model with zero intercept.
Consider the zero intercept model given by yi = β1xi + εi , (i = 1, 2, · · · , n) where the εi ’s are independent normal variables with constant variance σ 2 . Show that the 100(1 − α)% confidence interval on E(y|x0) is given by b1x0 + tα/2, n−1s s x 2 P 0 n i=1 x 2 i where s =
Consider a set of data (xi , yi), i = 1, 2, · · · , n, and the following two regression models: yi = β0 + β1xi + ε, (i = 1, 2, · · · , n), Model A yi = γ0 + γ1xi + γ2x 2 i + ε, (i = 1, 2, · · · , n), Model B Suppose both models are fitted to the same data. Show that SSRes, A ≥
8.19 Consider the situation in Exercise 8.15. Use the linear predictor and log link to generate 500 samples of three replicates each. (a) For each of the 500 samples fit the GLM assuming a Poisson response and log link. (b) Calculate the predicted response at X\ = 1 and x2 = -1 for each of the 500
8.18 Reconsider the situation in Exercise 8.17. Rework part (a) using the log link, and then rework parts (b) and (c) using the new data.
8.17 Consider a 23 factorial experiment. Suppose that the response follows an exponential distribution and that the linear predictor is #(μ) = 8.0 +2.0χ1+3.0χ2+*3— 1.5xiX2> where the design variables are coded as +1. (a) Use the inverse link and generate two replicates of this experiment using
8.16 Consider the situation in Exercise 8.15. Suppose now that the linear predictor is #(μ) = 3.5+2.0*1+1.5Χ2+0.5Λ;ΙΛ:2. Rework all the parts of Exercise 8.15 with this new mol.
8.15 Consider a 22 factorial experiment. Suppose that the response follows a Poisson distribution and that the linear predictor is #(μ) = 3.5 +2.0x\+\.5x2, where the design variables are coded as + 1. (a) Use a log link and generate three replicates of this experiment using a Poisson random number
8.14 Consider Exercise 8.11, where we use an identity link with a Poisson response. Assume the model E(y) = 1.0 + 3xi + 2x2-x3 Compute the design efficiencies and the correlation matrix of the estimated parameters.
8.13 Consider Exercise 8.11. Suppose now that the linear predictor also includes the X\x2 and the X\x3 interactions with βλ2 = 0.5 and j?l3 = -2. Calculate all the design efficiencies for the more complicated model.
8.12 Consider a 24 factorial design with ± 1 coding for the design variables. Assume that the response follows a Poisson distribution, and suppose we wish to use the log link. Assume that the true values for the coefficients in the linear predictor are ß0 = 1.5, ß\ = -1.0, ß2 = -3 , ß3 = 1.5,
8.11 Consider a 23 factorial design with ±1 coding for the design variables. Assume that the response follows a gamma distribution, and suppose we wish to use the canonical link. Assume that the true values for the coefficients in the linear predictor are ß0 = 1, ß\ = 3, ß2 = —4, andjS3 =
8.10 Reconsider the situation in Exercise 8.9. Suppose that a uniform prior distribution is selected. Find a Bayesian D-optimal design with 16 runs for this experiment. Compare this to the design found in Exercise 8.9.
8.9 Suppose that you want to design an experiment to fit a Poisson regression model in two predictors. A first-order model with interaction is assumed. Based on prior information about the experimental situation, reasonable ranges for the model parameters are 0.5 < ß0 < 3 1 < ß\ < 3 0.25 < ß2 <
8.8 Reconsider the situation in Exercise 8.7. Suppose that a uniform prior distribution is selected. Find a Bayesian D-optimal design with 16 runs for this experiment. Compare this to the design found in Exercise 8.7.
8.7 Suppose that you want to design an experiment to fit a Poisson regression model in two predictors. A first-order model is assumed. Based on prior information about the experimental situation, reasonable ranges for the model parameters are 0.5 < ß0 < 3 1 < ß\ < 3 0.25 < ß2 < 0.75 and the
8.6 Rework Exercise 8.5 using a uniform prior. What difference does this make in the designs obtained?
8.5 Consider the spermatozoa survival study in Table 8.7. Suppose that we want to construct a Bayesian D-optimal design for this experiment. We have some prior information about the anticipated results, which we have summarized with the following ranges of the parameters: - 1(b) Find a Bayesian
8.4 Reconsider the situation in Exercise 8.3. Suppose that a uniform prior distribution is selected. Find a Bayesian D-optimal design with 16 runs for this experiment. Compare this to the design found in Exercise 8.3.
8.3 Suppose that you want to design an experiment to fit a logistic regression model in two predictors. A first-order model with interaction is assumed. Based on prior information about the experimental situation, reasonable ranges for the model parameters are 0.5 < j80 < 8 1 < ß\ < 5 1 < ß2 < 3
8.2 Reconsider the situation in Exercise 8.1. Suppose that a uniform prior distribution is selected. Find a Bayesian D-optimal design with 10 runs for this experiment. Compare this to the design found in Exercise 8.1.
8.1 Suppose that you want to design an experiment to fit a logistic regression model in two predictors. A first-order model is assumed. Based on prior information about the experimental situation, reasonable ranges for the model parameters are 0.5
7.9 McKnight and van den Eeden (1993) analyzed a multi period, twotreatment crossover experiment to establish whether the artificial sweetener aspartame caused headaches. The trial involved randomly assigning 27 patients to different sequences of placebo and aspartame. The variables in
7.8 Somner (1982) reported on an Indonesian Children's Health Study designed to determine the effects of vitamin A deficiency in preschool children. The investigators were particularly interested in whether children with vitamin A deficiency were at increased risk of developing respiratory
7.7 In Example 6.5, a milling process for corn was discussed in terms of yield of grits. (a) Assume Batch is a random effect and fit an appropriate linear mixed model. (b) Produce an appropriate set of residual diagnostic plots and comment on whether or not you feel there are any unusual
7.6 In Exercise 6.8, an arthritis clinical trial was described. (a) Assume the patients represent a random sample from a population of possible patients and refit the data using an appropriate logistic mixed effects model (be sure to argue whether a model with random intercepts, random slopes, or
7.5 In Exercise 6.2 a process for steel normalization was discussed. (a) Propose an appropriate mixed model for the data described. (b) Provide an estimate of the marginal mean strength of normalized steel and compare this estimate to the GEE estimate. (c) What advantages can you see to analyzing
7.4 Weiss (2005, p. 353) describes a patient controlled analgesia study presented by Henderson and Shimakura (2003). The number of selfadministered doses in a 4-hour period was recorded for each patient for each of 12 consecutive periods. Two groups of patients were considered: a 1-milligram (mg)
7.3 Schall (1991) considers data from an experiment to measure the mortality of cancer cells under radiation. Four hundred cells were placed on a dish, and three dishes were irradiated at a time, or occasion. The zero-dose data provided here were obtained from OzDASL and are found in
7.2 Robinson, Wulff, Montgomery, and Khuri (2006) consider a wafer etching process in semi conductor manufacturing. During the etching process, some of the variables are not perfectly controllable and the net effect is that wafers produced on any given day (i.e., within the same batch) may be
7.1 Jensen, Birch, and Woodall (2008) considered profile monitoring of a calibration data set in which the data consists of 22 calibration samples. One of the purposes of the experiment was to determine the relationship between an absorbance measure (absorbance) of a chemical solution to the volume
6.8 Lipsitz, Kim, and Zhao (1994) analyzed data from an arthritis clinical trail. Patients were randomly assigned to receive either auranofin or a placebo. They surveyed each patient at baseline, 1 month, 2 months, and 3 months on how they felt. A value of 3 indicated "good," a value of 2 indicated
6.7 Thall and Vail (1990) compared a new treatment medication for epilepsy to a placebo. They recorded each patient's age and 8-week baseline seizure counts. They then monitored the 2-week seizure counts for each patient during four successive but nonoverlapping periods. The data follow. Perform a
6.6 Potthoff and Roy (1964) analyzed a longitudinal study of dental growth in children. Specifically, they measured the distance from the center of the pituitary gland to the maxillary fissure for children at ages 8, 10, 12, and 14. The data follow. Individual Girl 1 2 3 4 5 6 7 8 9 10 11 8 21 21
6.5 Stiger, Barnhart, and Williamson (1999) analyzed a double-blind clinical trail that compared a hypnotic drug to a placebo in subjects with insomnia. The researchers surveyed each patient about the amount of time it took them to go to sleep prior to treatment and again after 2 weeks of
6.4 Reiczigel (1999) analyzed an experiment that studies the effect of cholagogues on changes in gallbladder volume (GBV) of dogs. The experiment used two different cholagogues (A and B) and tap water (W) as a control. Six healthy dogs were assigned to each treatment. The GBVs were measured by
6.3 An experiment is designed to study pigment dispersion in paint. Four different mixes of a particular pigment are studied. The procedure consists of preparing a particular mix and then applying that mix to a panel by three application methods (brushing, spraying, and rolling). The response
6.2 Steel is normalized by heating above the critical temperature, soaking, and then air cooling. This process increases the strength of the steel, refines the grain, and homogenizes the structure. An experiment is performed to determine the effect of temperature and heat treatment time on the
6.1 An engineer studied the effect of pulp preparation and processing temperature on the strength of paper. The nature of the process dictated that pulp preparation was hard-to-change. As a result one large batch of pulp was split into smaller batches, which then receive the temperature treatment.
5.23 Byrne and Taguchi (1987) discuss an experiment to see the effect of interference (x\), connector wall thickness (x2), insertion depth (x3), and amount of adhesive (x4) on the pull-off force for an elastometric connector to a nylon tube in an automotive application. The data follow. *1 -1 -1 -1
5.22 Schubert et al. (1992) conducted an experiment using a catapult to determine the effects of hook (x\)9 arm length (x2), start angle (x3), and stop angle (x4) on the distance that the catapult throws a ball. They threw the ball three times for each setting of the factors. The data follow. X[ X2
5.21 The negative binomial distribution is often used to model the number of trials until the r th success. However, this distribution provides an interesting alternative to the Poisson distribution in the presence of overdispersion. The probability function for the negative binomial distribution
5.20 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 mL were considered significant. The data follow. Fit an appropriate GLM and perform the diagnostic analysis of these data. Group Healthy
5.19 Consider the nitrogen dioxide data in Exercise 2.17. Analyze these data assuming a Poisson distribution. Use both the canonical and the log links. Perform appropriate residual analyses of your final models. Discuss your results.
5.18 Consider the cathodic bonding of elastomeric metal bond data in Exercise 2.16. Fit a GLM to these data. Perform appropriate residual analyses of your final models. Analyze these data with a power link.
5.17 Consider the staffing of naval hospitals data in Exercise 2.15. Analyze these data assuming a Poisson distribution. Use both the canonical and the power links. Perform appropriate residual analyses of your final models. Discuss your results.
5.16 Consider the semiconductor data in Exercise 2.9. Assume a gamma distribution. Analyze these data using both the canonical and log links. Perform appropriate residual analyses of your final models. Discuss your results.

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Generalized Linear Models: With Applications In Engineering And The Sciences 2nd Edition Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson - Solutions

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