Questions and Answers of Linear State Space Systems

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,
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
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
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
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
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
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
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
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
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.
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
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
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
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 =
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
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
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
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
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
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
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,
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
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
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.
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
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)
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
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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.
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
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
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
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
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
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.
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
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.
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
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
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,
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
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
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
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)
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
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)
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,
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
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
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),
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)
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
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
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.
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
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
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),
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
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
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
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
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
Define in words the relationship between the hazard function and the survival function.

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