Regression Modeling Strategies With Applications To Linear Models Logistic And Ordinal Regression And Survival Analysis 2nd Edition Frank E. Harrell Jr. - Solutions

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5.15 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, C, phthalic anhydride, D, water collection time, E, and solvent distillate collection time, F. The data
5.14 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 distribution. This company has found that the maximum density is a good indicator of overall
5.13 Derive the score equations for generalized linear models when α(φ) is not constant. Find the solution to the resulting score equations.
5.12 Show that Equations (5.7) and (5.9) are in fact equivalent.
5.11 Show that if the model contains an intercept term, then the deviance for both the Poisson and gamma cases reduces to Equation (5.14).
5.10 In linear regression much is made of the so-called Hat matrix and resulting set of Hat diagonals that are used in standard regression diagnostics. If such diagnostics were to be necessary in generalized linear models, what matrix would be used to replace the Hat matrix? Be sure to define all
5.9 Consider an experiment with replications, namely, r¡ observations at each of m design points. In addition, suppose that the response is Poisson and overdispersion is expected. Can you suggest an estimator of the overdispersion parameter σ 2 which is free of the deviance D(ß)? Explain.
5.8 It is well known from principles of statistical inference that Use these expressions to show that for the generalized linear model ΥΒΐ{γ) = ο(θ)α(φ) where b"(ß) =
5.7 Fit a generalized linear model in which the gamma distribution is assumed with an identity link. Write out the score function in terms of linear predictor x'ß. Give an expression for the asymptotic variancecovariance matrix of ß.
5.6 Consider the gamma distribution in the context of the GLM. Put the density function into the form of the exponential family and show that = Ι/μ and Var (y¡) = /¿2 /r.
5.5 Consider an exponential distribution with log link. (a) Write out the score function in terms of the linear predictor χ'β. (b) Show that the information matrix is X'X. Comment.
5.4 Consider a situation in which a normal distribution is assumed with a log link. (a) Describe the score function for maximum likelihood estimation. (b) Give the asymptotic covariance matrix for b and explain all terms.
5.3 Show formally for a normal distribution and identity link that 0(» = Σi=\
5.2 For both logistic and Poisson regression, show that elements in X'y are sufficient statistics for β.
5.1 Consider the binomial distribution with n = 1. (a) Put the probability function in the form of the exponential family and show that b(e) = log(\+ee ) (b) Use b{6) in part (a) to show that = π Var(j) = π(1 — π)
4.21 The following table presents data on the reproduction of Ceriodaphnia organisms in a controlled environment in which a varying concentration of a component of jet engine fuel is introduced. We expect that as the concentration increases the mean number of counts of the organisms should
4.20 The following data come from a dose-response study that investigated a new pharmaceutical product. Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Dose (mg) 180 182 184 186 185 188 190 192 195 194 196 199 200 204 205 Patient Response 0 0 0 0 0 0 1 0 1 0 1 1 0 0 1 Observation 16 17 18 19 20 21
4.19 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 data are shown below.Number of Failures 5 3 0 1 4 0 0 1 Months 18 15 11 14 23 10 5 8 Valve 9 10
4.18 A study was performed to investigate new automobile purchases. A sample of 20 families was selected. Each family was surveyed to determine the age of their oldest vehicle and their total family income. A follow-up survey was conducted 6 months later to determine if they had actually purchased
4.17 The market research department of a soft drink manufacturer is investigating the effectiveness of a price discount coupon on the purchase of a two-liter beverage product. A sample of 5500 customers was given coupons for varying price discounts between 5 and 25 cents. The response variable was
4.16 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 loads. The numbers of fasteners failing at each load were recorded. The complete test data are
4.15 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 indicates yes and y = 0 indicates no). The data are shown below.Household 1 2 3 4 5 6 7 8 9 10
4.14 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). Test 1 2 3 4 5 6 7 8 9 10 11 12 13 Target Speed x (knots) 400 220 490 210 500 270 200 470 480 310 240 490
4.13 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 follow. Perform an appropriate analysis of these data. Dose Control 80 160 235 310 27 33 29
4.12 Slaton, Piegorsch, and Durham (2000) discuss an experiment that examined the in utero damage in laboratory rodents after exposure to boric acid. This particular experiment used four levels of boric acid. The experimenters recorded the number of rodents in the litter and the number of dead
4.11 Zhang and Zelterman (1999) discuss an experiment where female mice were fed extremely low doses of a known carcinogen, 2-acetylaminofluorene (2-AAF). The following table summarizes the results of the incidences of bladder and liver cancers. Perform an appropriate analysis for each cancer.
4.10 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 Poisson regression. Temperature 7 5 7 7 6 6 5 6 5 6 5 7 6 6 6 Oil 4 3 3 2 4 3 3 2 4 2 2 3 3 3
4.9 Chowdhury, Gijo, and Raghavan (2000) conducted an experiment to decrease the number of defects on a printed circuit assembly-encoder. The factors were (A) bath temperature, (B) wave height, (Q overhead preheater, (D) preheater 2, (E) preheater 1, (F) air knife, and (G) the vibration of the
4.8 Maruthi and Joseph (1999-2000) conducted an experiment to improve the yield of high, dense, inner-layer circuits in a printed circuit board operation. The factors were (A) surface preparation, (B) preheat, (C) lamination speed, (D) lamination pressure, (E) lamination temperature, (F) exposure
4.7 A major aircraft manufacturer studied the number of failures of an alloy fastener after each fastener was subjected to a pressure load. The data follow. (a) Use the logit link and logistic regression to analyze these data. (b) Repeat this analysis using the probit link. (c) Discuss the two
4.6 Nelson (1982, pp. 407-409) discusses an experiment to determine the relationship of time in use to the number of fissures that develop in furbine wheels. The data follow. (a) Use the logit link and logistic regression to analyze these data. (b) Repeat this analysis using the probit link.(c)
4.5 Anand (1997) discusses an experiment to improve the yield of silica gel in a chemical plant. The factors were (A) sodium silicate density, (B) pH, (C) setting time, and (D) drying temperature. The response was the percent of batches judged to be second grade material out of 1000 batches. The
4.4 Let us suppose that a 2 x 2 factorial with variable coded to ± 1 is used to fit a logistic regression model in a drug study in which 20 subjects were allocated to each of the 4 treatment combinations. If the response is successful, then y = 1, and if the response is unsuccessful, then y = 0.
4.3 In Chapter 3 we learned that the Gauss-Newton procedure for computation of coefficients in a nonlinear model with nonhomogeneous variance involves computing b = bo + (D,V _1D)-1D ,V-1 (y - μ) where V = diag{(j?} and D is the matrix of derivatives of the type
4.2 Show for both Poisson regression and logistic regression that if an intercept is contained in the linear predictor then i = l
4.1 Show that the binomial deviance function D(ß) is given by 2 Ζ*®+%*~»ΗΉ
3.19 The following data were collected on specific gravity and spectrophotometer analysis for 26 mixtures of NG (nitroglycerine), TA (triacetin), and 2 NDPA (2-nitrodiphenylamine).Mixture 16 17 18 19 20 21 22 23 24 25 26 JCI (% NG) 69.98 69.98 69.99 69.99 67.51 67.50 67.48 67.49 64.98 64.98 64.99
3.18 An investigation was made to study age and growth characteristics of selected freshwater mussel species in southwest Virginia. For a particular type of mussel, age and length were measured for 20 females with the following results: Observation 1 2 3 4 5 6 7 Length (in.) 29.2 28.6 29.4 33.0
3.17 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. Then the number of protozoa were counted at 1, 3, 6, 15, and 21 days. In this case the
3.16 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 has been described by the Michaelis-Menten model. The velocity (y) and concentration (x) data
3.15 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. This will ensure that structures on the surface survive the excavation. There are several
3.14 Reconsider the surgical services data introduced in Exercise 2.15. Fit the nonlinear model to these data. Investigate fully the adequacy of the fit of this model to the data. How does the nonlinear model compare to the models obtained in Exercise 2.15?
3.13 Repeat Exercise 3.12 using the model y = ßi(x\)ßl(x2)h (x\X2)ßA+& Discuss the difficulties that you encounter.
3.12 Reconsider the data from Exercise 2.18. Suppose that we now wish to consider fitting a nonlinear model to these data, say, y = (a) Fit the nonlinear model to the data. (b) Test for significance of regression. (c) Does it appear that both of the design variables have important effects? (d)
3.11 The data in the following table represent the fraction of active chlorine in a chemical product at a measured time following manufacturing. Available Chlorine 0.49, 0.48, 0.46, 0.45, 0.44, 0.46, 0.42, 0.41, 0.42, 0.41, 0.41, 0.40, 0.41, 0.40 0.41, 0.40, 0.39 0.39 y 0.49 0.47, 0.46, 0.43, 0.43,
3.10 The model y = β — ß2e~^x + ε is called the Mitcherlich equation, and it is often used in chemical engineering to model the relationship between yield and reaction time. (a) Is this a nonlinear regression model? (b) Discuss how you would obtain starting values for the parameter estimates.
3.9 Reconsider the data from the previous exercise. Suppose that the response data were collected on two different days. Fit a new model to the data, say, y = ßle ** + ßsz + e where z is an indicator variable with z = 0 if the observation was made on the first day and z = 1 if the observation was
3.8 Consider the data shown below. Suppose that we are considering fitting the nonlinear model to these data. x y 0.5 0.68 1.58 12 0.45 2.66 2.50 2.04 4 6.19 7.85 8 56.1 54.2 9 89.8 90.2 10 147.7 146.3 (a) Discuss how you would find starting values for the parameter estimates. (b) Fit the model to
3.7 Reconsider the regression models in Exercise 3.6 parts (a) to (e). Suppose that the error terms in these models were multiplicative. Rework the problem under this new assumption regarding the error structure.
3.6 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 suitable transformation. (a) y = ßxeh+^x + ß. (b) γ = β0 + β χχ+β2 2 3+ε. (c) y = β +
3.5 Consider the Gompertz model in Equation (3.43). Graph the expected value of the response for ß\ = 1, ß3 = 1, and ß2 = 1/8, 1, 8, and 64, respectively, over the range 0 < x < 10. Overlay these plots on the same axes. (a) What is the effect of the parameter ß2 on the expected response? (b)
3.4 Consider the logistic model in Equation (3.42). Graph the expected value of the response for ß\ = 1, ß3 = 1, and ß2 = 1, 4, and 8, respectively. Overlay these plots on the same axes. What is the effect of the parameter 2 on the expected response?
3.3 Consider the logistic model in Equation (3.42). Graph the expected value of the response for ßx = 10, ß2 = 2, and ß3 = 0.5, 1, 2, and 3, respectively. Overlay these plots on the same axes. What is the effect of the parameter j?3 on the expected response?
3.2 Consider the Michaelis-Menten model in Equation (3.12). Graph the expected value of the response for ßx = 100, 150, 200, and 250 for 2 = 0.10. Overlay these plots on the same axes. What is the effect of the parameter ßx on the expected response?
3.1 Consider the Michaelis-Men ten model in Equation (3.12). Graph the expected value of the response for ßx = 100 and β2 = 0.04, 0.06, 0.08, and 0.10. Overlay these plots on the same axes. What is the effect of the parameter β2 on the expected response?
2.32 Suppose that you want to fit a second-order model in three factors using n = 12 runs. Find a I-optimal design for this situation.
2.30 Suppose that you want to fit a second-order model in three factors using n = 15 runs. Find an I-optimal design for this situation. 2.31 Suppose that you want to fit a second-order model in three factors using n = 12 runs. Find a D-optimal design for this situation.
2.29 Suppose that you want to fit a second-order model in three factors using n = 15 runs. Find a D-optimal design for this situation.
2.28 Suppose that you want to fit a main-effects first-order model in three factors over the ± 1 range using n = 8 runs. Find a D-optimal design for this situation.
2.27 Suppose that you want to fit a first-order model in two factors over the ± 1 range using n = 8 runs. The design chosen is a full 22 design augmented with four center runs (x\ = x2 = 0) Is this a D-optimal design?
2.26 Suppose that you want to fit a first-order model in three factors with all of the two-factor interactions in over the +1 range using n = 12 runs. Find an I-optimal design for this situation.
2.25 Suppose that you want to fit a first-order model with all of the two-factor interactions in three factors over the +1 range using n = 12 runs. Find a D-optimal design for this situation.
2.24 Consider the model = Χ + εwhere Ε(ε) = 0, and Var[e] = V. Show that ^ = (x'y-l xy l x'\-l y is BLUE.
2.23 Show that the variance of the /th residual et in a multiple regression model is
2.22 Show that we can express the residuals from a multiple regression model as e = (I-H)y , where H = XCX'X)"1 ^
2.21 A regression model is used to relate a response y to k = 4 regressors. What is the smallest value of R2 that will result in a significant regression if a = 0.05? Use the results of the previous exercise. Are you surprised by how small the value of R2 is?
2.20 Consider a multiple regression model with k regressors. Show that the test statistic for significance of regression can be written as (1 -#)/(*-* - 1) Suppose that n = 20, k = 4, and R2 = 0.90. If a = 0.05, what conclusion would you draw about the relationship between y and the four regressors?
2.19 Reconsider the data from the factorial experiment in Exercise 2.18. Use the Box-Cox method to determine if a transformation on the response is necessary.
2.18 The data in the following table come from a factorial experiment conducted to study the effect of reaction time and reaction temperature on the concentration of a chemical product. Temperature Time(h) 50°C 75°C 100°C 2 4.70,2.68 5.52,3.75 3.98,4.22 4 6.35,6.10 5.88,7.69 6.28,7.12 6
2.17 Consider the following data set in which nitrogen dioxide concentrations in parts per million are collected for 26 days in September 1984 at a monitoring facility in the San Francisco Bay area. These data were taken from Chatterjee and Hadi (1988).*i Windspeed X2 Maximum x3 Insolation ;e mph)
2.16 In a study in collaboration with the Engineering Sciences and Me- chanics Department at Virginia Tech, personnel at the Statistics Consulting Center were called upon to analyze a data set dealing with cathodic debonding of elastomeric metal bonds. A model relating the amount of debonding as a
2.15 In an effort to develop a preliminary personnel equation for estimation of worker-hours per month expended in surgical services at Naval hospitals, the U.S. Navy collected data on y (worker-hours per month) and x (surgical cases) from 15 hospitals. The data (taken from the Navy's Procedures
2.14 An article in the Journal of Pharmaceuticals Sciences (Vol. 80, 1991, pp. 971-977) presents data on the observed mole fraction solubility of a solute at a constant temperature to the dispersion, dipolar, and hydrogen bonding Hansen partial solubility parameters. The data are as
2.13 Reconsider the heat treating data in Exercises 2.11 and 2.12, where we fit a model to PITCH using regressors xx = SOAKTIME x SOAKPCT and x2 = DIFFTIME x DIFFPCT. (a) Using the model with regressors χ and x2, find a 95% confidence interval on mean PITCH when SOAKTIME = 1.00, SOAKPCT = 1.10,
2.12 Reconsider the heat treating data from Exercise 2.11. (a) Fit a new model to the response PITCH using new regressors x, = SOAKTIME x SOAKPCT and x2 = DIFFTIME x DIFFPCT. (b) Test the model in part (a) for significance of regression using a = 0.05. Also calculate the /-test for each regressor,
2.11 Heat treating is often used to carburize metal parts, such as gears. The thickness of the carburized layer is considered an important feature of the gear, and it contributes to the overall reliability of the part. Because of the critical nature of this feature, two different lab tests are
2.10 Consider the semiconductor hFE data in Exercise 2.9. (a) Plot the residuals from this model versus y. Comment on the information in this plot. (b) What is the value of R2 for this model? (c) Refit the model using In hFE as the response variable. (d) Plot the residuals versus predicted In hFE
2.9 An engineer at a semiconductor company wants to model the relationship between the device gain or hFE(y) and three parameters: emitter-RS (JCI), base-RS (x2), and emitter-to-base-RS (x3). The data are shown below: * i *i *3 y Emitter-RS Base-RS E-B-RS hFE-lM-5V 14.620 226.00 7.000 128.40 15.630
2.8 Consider the wire bond pull strength data in Exercise 2.4. Fit a regressor model using all six regressors. (a) Is there an indication that this model is superior to the one from Exercise 2.4? (b) Find a 95% confidence interval on mean pull strength when x0 = 6.0, x2 = 20, x3 = 30, x4 = 90, x5 =
2.7 Consider the wire bond pull strength data in Exercise 2.4. (a) Find 95% confidence intervals on the regression coefficients. (b) Find a 95% confidence interval on mean pull strength when x2 = 20, x3 = 30, x4 = 90, and x5 = 2.0.
2.6 For the regression model for the wire bond pull strength data in Exercise 2.4: (a) Plot the residuals versus y and versus the regressors used in the model. What information is provided by these plots? (b) Construct a normal probability plot of the residuals. Are there reasons to doubt the
2.5 Consider the wire bond pull strength data in Exercise 2.4. (a) Estimate σ 2 for this model. (b) Find the standard errors for each of the regression coefficients. (c) Calculate the ¿-test statistic for each regression coefficient. Using = 0.05, what conclusions can you draw? Do all variables
2.4 The pull strength of a wire bond is an important characteristic. The table below gives information on pull strength (y), die height (x\), post height (x2), loop height (x3), wire length (x4), bond width on the die (xs), and bond width on the post (x6). (a) Fit a multiple linear regression model
2.3 Suppose that we wish to use the models from Exercises 2.1 and 2.2 to estimate the mean bearing wear when xx = 25 and x2 = 1000. (a) Compute point estimates of the mean wear using both models. (b) Find a 95% confidence interval on the mean response at the point x\ = 25 and x2 = 1000 for both
2.2 Reconsider the bearing data from Exercise 2.1. Expand the multiple regression model to include an interaction term. (a) Test for significance of regression. (b) Compute /-statistics for each of the regression coefficients and provide an interpretation. Specifically, does the model require an
2.1 The following data were collected on the wear of a bearing y, the oil viscosity x\, and load x2. y 193 230 172 91 113 125 *1 1.6 15.5 22.0 43.0 33.0 40.0 x2 851 816 1058 1201 1357 1115 (a) Fit a multiple linear regression model to these data. (b) Test for significance of regression. (c) Compute
Verify, in an unbiased fashion, your “final” model, for either calibration or discrimination. Validate intermediate steps, not just the final parameter estimates.
Derive a final Cox PH model. Stratify on polytomous factors that do not satisfy the PH assumption. Decide whether to categorize and stratify on continuous factors that may strongly violate PH. Remember that in this case you can still model the continuous factor to account for any residual
For factors that remain, assess the PH assumption using at least two methods, after ensuring that continuous predictors are transformed to be as linear as possible. In addition, for polytomous predictors, derive log cumulative hazard estimates adjusted for continuous predictors that do not assume
Assess the nature of the association of several predictors of your choice.For polytomous predictors, perform a log-rank-type score test (or k-sample ANOVA extension if there are more than two levels). For continuous predictors, plot a smooth curve that estimates the relationship between the
For the same data in Problem 1, compute MLEs of parameters of a Weibull distribution. Also compute the MLEs of S(3) and T0.5.
For the failure times (in days)133+ 6+ 7+compute MLEs of the following parameters of an exponential distribution by hand: λ, μ, T0.5, and S(3 days). Compute 0.95 confidence limits for λand S(3), basing the latter on log[Λ(t)].
Do preliminary analyses of survival time using the Mayo Clinic primary biliary cirrhosis dataset described in Section 8.9. Make graphs of Altschuler–Nelson or Kaplan–Meier survival estimates stratified separately by a few categorical predictors and by categorized versions of one or two
The commonly used log-rank test for comparing survival times between groups of patients is a special case of the test of association between the grouping variable and survival time in a Cox proportional hazards regression model. Depending on how one handles tied failure times, the log-rankχ2
Repeat Problem 6 except for tertiles of meanbp.
Plot Kaplan–Meier survival function estimates stratified by dzclass. Estimate the median survival time and the first quartile of time until death for each of the four disease classes.
Consider long-term follow-up of patients in the support dataset. What proportion of the patients have censored survival times? Does this imply that one cannot make accurate estimates of chances of survival? Make a histogram or empirical distribution function estimate of the censored followup times.
A placebo-controlled study is undertaken to ascertain whether a new drug decreases mortality. During the study, some subjects are withdrawn because of moderate to severe side effects. Assessment of side effects and withdrawal of patients is done on a blinded basis. What statistical technique can be
In a study of the life expectancy of light bulbs as a function of the bulb’s wattage, 100 bulbs of various wattage ratings were tested until each had failed. What is wrong with using the product-moment linear correlation test to test whether wattage is associated with life length concerning
Define in words the relationship between the hazard function and the survival function.

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Regression Modeling Strategies With Applications To Linear Models Logistic And Ordinal Regression And Survival Analysis 2nd Edition Frank E. Harrell Jr. - Solutions

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