Multivariate Data Analysis 7th Edition Jr, Joseph F Hair;Black, William C;Babin, Barry J;Anderson, Rolph E - Solutions

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16. Consider again the stack loss data originally presented in Example 2.16.Suppose we wish to compare the assumptions of normal, t4, and DE errors for these data.(a) Repeat the outlier analysis in Example 4.7 for these data. Do the λi posteriors for the nonnormal models effectively identify the
15. In the previous problem, implement both models in WinBUGS, and use DIC instead of Bayes factors to choose between them. Is your preference between the two models materially altered by this change? What are the advantages and disadvantages of using DIC instead of a Bayes factor here?
14. Consider the dataset of Williams (1959), displayed in Table 4.7. For n = 42 specimens of radiata pine, the maximum compressive strength parallel to the grain yi was measured, along with the specimen’s density, xi, and its density adjusted for resin content, zi (resin contributes much to
13. Consider again the data in Table 3.3. Define model 1 to be the three variable model given in Example 3.7, and model 2 to be the reduced model having m1 = 1 (i.e., the standard logistic regression model). Compute the Bayes factor in favor of model 1 using
12. In the previous problem, suppose we wish to evaluate the model using the model check pD given in equation (2.33) with an independent validation data sample z. Give a computational formula we could use to obtain a simulation-consistent estimate of pD.
11. Suppose we have a convergent MCMC algorithm for drawing samples from p(θ|y) ∝ f(y|θ)π(θ). We wish to locate potential outliers by computing the conditional predictive ordinate f(yi|y(i)) given in equation(2.29) for each i = 1,...,n. Give a computational formula we could use to obtain a
10. Refer again to the cross-protocol data and model in Example 2.12.(a) Create WinBUGS code that will estimate cross validation (“leave one out”) residuals and CPO values via importance sampling, as in equation (3.4). Compare your findings to those obtained via the “exact”and
9. Suppose we are estimating a set of residuals ri using a Monte Carlo approach, as described in Section 4.3. Due to a small sample size n, we are concerned that the approximation E(yi|y(i)) = E(yi|θ)p(θ|y(i))dθ ≈E(yi|θ)p(θ|y)dθwill not be accurate. A faculty member suggests the
8. For the interval null prior partitioning setting of Subsection 4.2.2, derive an expression for the set of priors Hc that correspond to rejecting H0, similar to expression (4.12) for the point null case.
7. For the point null prior partitioning setting described in Subsection 4.2.2, show that requiring G ∈ Hc as given in (4.12) is equivalent to requiring BF ≤ p 1−p 1−ππ, where BF is the Bayes factor in favor of the null hypothesis.
6. Consider the data displayed in Table 4.6, originally collected by Treloar(1974) and reproduced in Bates and Watts (1988). These data record the “velocity” yi of an enzymatic reaction (in counts/min/min) as a function of substrate concentration xi (in ppm), where the enzyme has been treated
5. Consider www.biostat.umn.edu/~brad/data/copresence_data.txt, a data set for which a few records are shown in Table 4.5. Here, Y is a binary variable indicating co-presence of two species in a particular forest at n = 603 sampled locations. The lone predictor variable, X, is the log of the
4. Spiegelhalter et al. (1995b) analyze the flour beetle mortality data in Table 3.3 using WinBUGS. These authors use only the usual, two-parameter parametrization for pi ≡ P(death|wi), but compare the logit, probit, and complementary log-log link functions using the centered covariate zi = wi
3. Consider again the binary dugong modeling of Example 4.4. Suppose we wished to obtain side-by-side boxplots of the posteriors of the effect of log-age, β1, across all three link functions (logit, probit, and cloglog).Modify the WinBUGS code given in that example so that the comparison tool on
2. Refer to the data in Table 2.3, also available on the web in WinBUGS format at http://www.biostat.umn.edu/~brad/data.html. These pharmacokinetic (PK) data of Wakefield et al. (1994) were initially presented in Example 2.13, and analyzed there using an approximate linear hierarchical model.
1. Steensma et al. (2005) presented the data in Table 4.4, from a randomized controlled trial comparing two dosing schedules for the drug erythropoiten. Serum hemoglobin (HGB, in g/dL) was recorded for N = 365 cancer patients with anemia over T = 22 weeks. The full data file is available at
3. Which model (or models) should I ultimately choose for the final presentation of my results?
2. How can I tell if my model is providing adequate fit to the data?
1. How can I tell if any of the assumptions I have made (e.g., the specific choice of prior distribution) is having an undue impact on my results?
21. To further study the relationship between identifiability and MCMC convergence, consider again the two-parameter likelihood model Y ∼ N(θ1 + θ2 , 1) , with prior distributions θ1 ∼ N(a1, b2 1) and θ2 ∼ N(a2, b2 2), θ1 and θ2 independent.(a) Clearly θ1 and θ2 are individually
20. Consider the balanced, additive, one-way ANOVA model, Yij = μ + αi + ij , i = 1,...,I, j = 1, . . . , J, (3.34)where ij iid∼ N(0, σ2 e ), μ ∈ , αi ∈ , and σ2 e > 0. We adopt a prior structure that is a product of independent conjugate priors, wherein μhas a flat prior, αi iid∼
19. Consider the following two complete conditional distributions, originally analyzed by Casella and George (1992):f(x|y) ∝ ye−yx, 0
18. Repeat the preceding investigation for a multivariate Nk(0, Ik) target density, p(x) ∝ exp(−1 2xx), where x = (x1,...,xk). Now we wish to compare these three samplers:
17. Consider three approaches for sampling from a N(0, 1) target density, p(x) ∝ exp(−1 2x2):a standard Metropolis algorithm using a Gaussian proposal density, x∗|x(t−1) ∼ N(x(t−1), σ2), the simple slice sampler (3.20), and a Langevin-Hastings algorithm (3.18).(a) Evaluate and compare
16. A random variable Z defined on (0, ∞) is said to have a D-distribution with parameters δ, β > 0 and k ∈ {0, 1, 2,...} if its density function is defined (up to a constant of proportionality) by pZ(z) ∝ zδ−1e−βz(1 − e−z)k . (3.33)This density emerges in many Bayesian
15. Show that Adler’s overrelaxation method (3.21) leaves the desired distribution invariant. That is, show that if θ(g−1)i is conditionally distributed as N(μi, σ2 i ), then so is θ(g)i .
14. Consider the generalized Hastings algorithm that uses the following candidate density:q(v, u) = % p(vi|uj=i) for vj=i = uj=i 0 otherwise .That is, the algorithm chooses (randomly or deterministically) an index i ∈ {1,...,K}, and then uses the full conditional distribution along the ith
13. Returning again to the flour beetle mortality data and model of Example 3.7, note that the decision to use Σ=2 . Σ in equation (3.17)was rather arbitrary. That is, univariate Metropolis “folklore” suggests a proposal density having variance roughly twice that of the true target should
12. Write an R program to reanalyze the flour beetle mortality data in Table 3.3, replacing the multivariate Metropolis algorithm used in Example 3.7 with(a) a Hastings algorithm employing independence chains drawn from a N(θ˜, Σ) candidate density, where ˜ θ˜ = (1.8, −4.0, −1.0)
11. In the previous problem, replace the third-stage prior given above with b1 ∼ G(c1, d1), b2 ∼ G(c2, d2), b1 and b2 independent, thus destroying the conjugacy for these two complete conditionals. Resort to rejection or Metropolis-Hastings subsampling for these two components instead, choosing
10. In the previous problem, assume k is unknown, and adopt the following prior:k ∼ Discrete Uniform(1,...,n), independent of θ and λ .Add k into the sampling chain, and obtain a marginal posterior density estimate for it. What is the effect on the posterior for R?
9. Consider the following hierarchical changepoint model for the number of occurrences Yi of some event during time interval i:Yi ∼% Poisson(θ), i = 1,...,k Poisson(λ), i = k + 1,...,nθ ∼ G(a1, b1), λ ∼ G(a2, b2), θ, and λ independent;b1 ∼ IG(c1, d1), b2 ∼ IG(c2, d2), b1 and b2
8. Repeat the analysis of the previous problem in WinBUGS. Do your answers agree? List two advantages and two disadvantages of MCMC methods relative to noniterative MC methods.
7. Actually perform the point and interval estimation of the age corresponding to minimum property tax in the previous problem using a
6. In the previous problem, suppose that instead of the intercept β0, the parameter of interest is the age corresponding to minimum tax under this model. Describe how you would modify the above analysis in this case.
5. Use quadratic regression to analyze the data in Table 3.4, which relate property taxes and age for a random sample of 19 homes. The response variable Y is property taxes assessed on each home (in dollars), while the explanatory variable X is age of the home (in years). Assume a normal likelihood
4. Suppose we have a sample {θ1,...,θN } from a posterior p1(θ|y), and we want to see the effect of changing the prior from π1(θ) to π2(θ).Describe how the weighted bootstrap procedure could be helpful in this regard. Are there any restrictions on the shape of the new prior π2 for this
3. Show that the probability of accepting a given θj candidate in the rejection sampling is c/M, where c = L(θ)π(θ)dθ, the normalizing constant for the posterior distribution h(θ).
2. Suppose that f(y|θ) = n i=1 f(yi|θ). Describe how equation (3.1) could be used to investigate the effect of single case deletion (i.e., removing xi from the dataset, for i = 1,...,n) on the posterior expectation of g(θ).
1. Use Theorem 3.1 to obtain a first-order approximation to the Bayes factor BF, given in equation (2.19). Is there any computational advantage in the case where the two models are nested, i.e., θ1 = (θ2, γ)?
23. Refer to Example 2.18.(a) Repeat the analysis of the stack loss data assuming a slightly generalized model where we letπj iid∼ Bernoulli(θ) , where θ ∼ Unif(0, 1). That is, allow prior variable inclusion probabilities other than 1/2. Do you find any substantial changes to the posteriors
22. Refer to Example 2.16. Repeat the exact “leave one out” analysis of the stack loss data using the BRugs language. Here we would avoid the“trickery” of the fully WinBUGS solution given in the example, and instead simply format the data using R to leave out the ith observation. We then
21. Refer to Example 2.15. Repeat the analysis of the MAC treatment data assuming a model having(a) three sets of random effects ri, Wi, and β1i, adding clinic-specific treatment effect parameters.(b) two sets of random effects Wi and β1i, but setting r1 = 1 for all i.(c) one set of random
20. The following are increased hours of sleep for 10 patients treated with soporific B compared with soporific A (Cushny and Peebles, presented by Fisher in Statistical Methods for Research Workers):1.2, 2.4, 1.3, 1.3, 0.0, 1.0, 1.8, 0.8, 4.6, 1.4 Perform a Bayesian analysis of this data and draw
19. The data in Table 2.11 come from a 1980 court case (Reynolds v. Sheet Metal Workers) involving alleged applicant discrimination. Perform a Bayesian analysis of these data. Is there evidence of discrimination?Selected Rejected Total Black 14 30 44 White 41 39 80 Total 55 69 124 Table 2.11
18. For the Dirichlet process prior described in Section 2.6, use the properties of the Dirichlet distribution to show that for any measurable set B,(a) E[G(B)] = G0(B)(b) V ar[G(B)] = G0(B)(1 − G0(B))/(ν + 1)What might be an appropriate choice of vague prior distribution on ν?
17. Consider the two-stage binomial problem wherein θ ∼ G, and Xi|θ iid∼Bernoulli(θ), i = 1,...,n.(a) Find the joint marginal distribution of Xn = (X1,...,Xn).(b) Are the components independent? In any case, describe features of the marginal distribution.(c) Let Sn = X1 + ··· + Xn. Under
15. Suppose that Y |θ ∼ G(1, θ) (i.e., the exponential distribution with meanθ), and that θ ∼ IG(α, β).(a) Find the posterior distribution of θ.(b) Find the posterior mean and variance of θ.(c) Find the posterior mode of θ.(d) Write down two integral equations that could be solved to
14. Refer to Example 2.10. Compare pD and DIC for the two prior distributions considered for the error terms, namely, the Gamma(.1, .1) prior on the error precision τ, and the Unif(.01, 100) prior on the error standard deviation σ. Which prior emerges as “best”? In your opinion, is it
13. The classic Behrens-Fisher problem is that of estimating normal means from independent populations with unknown (and thus possibly unequal) variances. Here we consider this problem in the context of an?
12. Consider www.biostat.umn.edu/~brad/data/land_data.txt, for which a few records are shown in Table 2.9. Here, Y represents log-transformed?
11. In a pediatric clinical study carried out to see how effective aspirin is in reducing temperature, n = 12 five-year-old children suffering from influenza had their temperature taken immediately before (X) and 1 hour after (Y ) administration of aspirin. The data are in Table 2.8, and
9. Show that the Jeffreys prior based on the binomial likelihoodf(x|θ) = n xθx(1 − θ)n−x is given by the Beta(.5, .5) distribution.10. Suppose that in the situation of Example 2.3 we adopt the noninformative prior p(θ, σ) = 1σ , θ ∈ , σ> 0 (i.e., p(θ, σ2) = 1σ2 , θ ∈ , σ2 >
8. Suppose that f(x|θ, σ) is normal with mean θ and standard deviation σ.(a) Suppose that σ is known, and show that the Jeffreys prior (2.12) forθ is given by p(θ)=1, θ ∈ .(b) Next, suppose that θ is known, and show that the Jeffreys prior forσ is given by p(σ) = 1σ , σ> 0 .(c)
7. Let θ be a univariate parameter of interest, and let γ = g(θ) be a 1-1 transformation. Use (2.12) and (2.13) to show that (2.14) holds, i.e., that the Jeffreys prior is invariant under reparametrization. (Hint: What is the expectation of the so-called score statistic, d log f(x|θ)/dθ?)
6. For the following densities from Appendix A, state whether each represents a location family, scale family, location-scale family, or none of these. For any of the first three situations, suggest a simple noninformative prior for the unknown parameter(s):(a) X ∼ DE(1, σ2)(b) X ∼ Unif(θ
6. For the following densities from Appendix A, state whether each represents a location family, scale family, location-scale family, or none of these. For any of the first three situations, suggest a simple noninformative prior for the unknown parameter(s):(a) X ∼ DE(1, σ2)(b) X ∼ Unif(θ
5. For each of the following densities from Appendix A, provide a conjugate prior distribution for the unknown parameter(s), if one exists:(a) X ∼ Bin(n, θ), n known(b) X ∼ NegBin(r, θ), r known(c) X ∼ Mult(n, θ), n known(d) X ∼ G(α, β), α known(e) X ∼ G(α, β), β known(f) X ∼
4. Let θ be the height of the tallest individual currently residing in the city where you live. Elicit your own prior density for θ by(a) a histogram approach(b) matching an appropriate distributional form from Appendix A.What difficulties did you encounter under each approach?
3. Confirm expression (2.11) for the conditional posterior distribution of σ2 given θ in the conjugate normal model. If your random number generator could produce only χ2 random variates, how could you obtain samples from (2.11)?
2. Confirm expression (2.3) for the conditional posterior distribution of θgiven σ2 in the conjugate normal model, and show that the precision in this posterior is the sum of the precisions in the likelihood and the normal prior, where precision is defined to be the reciprocal of the variance.
1. Let Y1, Y2.... be a sequence of random variables (not necessarily independent) taking on the values 0 and 1. Let Sn = Y1 + ...Yn.(a) Represent P(Yn+1 = 1|Sn = s) in terms of the joint distribution of the Yi.(b) Further refine the representation when all permutations of the Yi such that Sn = s
8. In analyzing data from a Bin(n, θ) likelihood, the MLE is ˆθMLE = Y /n, which has MSE = Ey|θ(ˆθMLE −θ)2 = V ary|θ(ˆθMLE) = θ(1−θ)/n. Find the MSE of the estimator ˆθBayes = (Y + 1)/(n + 2) and discuss in what contexts you would prefer it over ˆθMLE. (ˆθBayes is the
7. In the basic diagnostic test setting, a disease is either present (D = 1)or absent (D = 0), and the test indicates either disease (T = 1) or no disease (T = 0). Represent P(D = d|T = t) in terms of test sensitivity, P(T = 1|D = 1), specificity, P(T = 0|D = 0), and disease prevalence, P(D = 1),
6. In the Normal/Normal example of Subsection 1.5.1, let σ2 = 2, μ = 0, and τ 2 = 2.(a) Suppose we observe y = 4. What are the mean and variance of the resulting posterior distribution? Sketch the prior, likelihood, and posterior on a single set of coordinate axes.(b) Repeat part (a) assuming τ
5. Here is an example in a vein similar to that of Example 1.2, and originally presented by Berger and Berry (1988). Consider a clinical trial established to study the effectiveness of vitamin C in treating the common cold. After grouping subjects into pairs based on baseline variables such as
4. For predictive distributions of survival time associated with two medical treatments, propose treatment selection criteria that are meaningful to you (or if you prefer, to society).
3. Assume you have developed predictive distributions of the length of time it takes to drive to work, one distribution for Route A and one for Route B. What summaries of these distributions would you use to select a route(a) to maximize the probability that the drive takes less than 30 minutes?(b)
2. Repeat the journal publication thought problem from Subsection 1.2.1 for the situation where(a) you have won a lottery on your first try.(b) you have correctly predicted the winner of the first game of the World Series (professional baseball).
1. Let θ be the true proportion of men in your community over the age of 40 with hypertension. Consider the following “thought experiment”:(a) Though you may have little or no expertise in this area, give an initial point estimate of θ.(b) Now suppose a survey to estimate θ is established in
=+What types of uncontrolled variables or covariates might be operating in each of these situations?
=+5. Describe some data analysis situations in which MANOVA and MANCOVA would be appropriate in your areas of interest.
=+ How would you design a study to ensure adequate power?
=+4. How is statistical power affected by statistical and research design decisions?
=+dependent variables; and (c) examination of the discriminant functions. Describe the practical advantages and disadvantages of each of these approaches.
=+is computed after eliminating the effects of the previous
=+contrast procedures; (b) stepdown analysis, which is similar to stepwise regression in that each successive F statistic
=+3. Besides the overall, or global, significance, at least three approaches to follow-up tests include (a) use of Scheffé
=+What would a significant interaction tell you?
=+What are the different sources of variance in your experiment?
=+2. Design a two-way factorial MANOVA experiment.
=+1. What are the differences between MANOVA and discriminant analysis? What situations best suit each multivariate technique?
=+• Is the direct distribution system always preferred over the indirect system across the customer groups of X1?
=+• Do the two distribution systems show differences for customers of 5 years or more?
=+• Is the direct distribution system more effective for newer customers?
=+3. What is the relationship between the distribution system and these relationships with customers in terms of the purchase outcomes?
=+2. Is HBAT establishing better relationships with its customers over time, as reflected in customer satisfaction and other purchase outcomes?
=+1. What differences are present in customer satisfaction and other purchase outcomes between the two channels in the distribution system?
=+ Describe the purpose of multivariate analysis of covariance (MANCOVA).
=+ Interpret interaction results when more than one independent variable is used in MANOVA.
=+ Describe the purpose of post hoc tests in ANOVA and MANOVA.
=+ Discuss the different types of test statistics that are available for significance testing in MANOVA.
=+ State the assumptions for the use of MANOVA.
=+ Discuss the advantages of a multivariate approach to significance testing compared to the more traditional univariate approaches.
=+ Explain the difference between the univariate null hypothesis of ANOVA and the multivariate null hypothesis of MANOVA.
=+5. Why is it important to examine the results of a measurement model before proceeding to test a structural model?
=+4. How is the validity of a SEM model estimated?
=+3. What is the distinguishing characteristic of a nonrecursive SEM model?
=+item be incorporated into a SEM model?
=+2. How can a measured variable represented with a single

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Multivariate Data Analysis 7th Edition Jr, Joseph F Hair;Black, William C;Babin, Barry J;Anderson, Rolph E - Solutions

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