Bayesian Methods For Data Analysis 3rd Edition Bradley P. Carlin, Thomas A. Louis - Solutions

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5.6 Answer the following questions using the data in Table 5.14, obtained from the 1993 administration of the GSS:a. What is the marginal association between gender and attitude toward life? Interpret the value of the appropriate marginal odds ratio.b. What is the marginal association between
5.5 Table 5.13 contains data obtained from the 2006 administration of the GSS. Analyze these data and summarize the results. Include a discussion of any associations that exist.Specifically, discuss whether there appears to be homogeneous association or a threeway association, and whether there is
5.4 Answer the following questions using the data in Table 5.12, obtained from the 2006 administration of the GSS:a. What is the marginal association between gender and belief in the afterlife? Interpret the value of the appropriate marginal odds ratio.b. What are the conditional associations (odds
5.3 Analyze the data in Table 5.11 and summarize the associations among the three variables.Specifically, discuss whether there appears to be homogeneous association or a threeway association, and whether there is conditional independence or dependence present.Justify your response with the
5.2 Using Table 5.11, examine the conditional associations between type of employment(someone else or self) and work time (part or full), conditional on gender.a. Report the conditional odds ratios, by gender.b. Provide a substantive interpretation of each conditional odds ratio reported in part
5.1 Table 5.11 contains data obtained from the 2006 administration of the General Social Survey (GSS).a. Provide the marginal table for the association between type of employment (someone else or self) and type of work (part- or full-time).b. Provide the marginal odds ratio from the table in part
4.12 Find a contingency table from a recent news story or study to use for this problem. Cite the source of the data and use it to:a. Provide a research question that can be answered with the data.b. Run the analysis (by computer) to obtain the appropriate results.c. Using the results from part
4.11 If the two officers in Problem 4.10 agreed perfectly, what might the observed contingency table have been? Provide an example of such a table and compute the value of Cohen’s kappa that would be obtained for this table.
4.10 A study was conducted to determine the level of agreement between two personnel officers in their ratings of prospective job candidates. The officers rated each of 20 job candidates as unacceptable (A); acceptable (B); or recommended (C). The results are shown in Table 4.14.a. Compute (by
4.9 A medical study examined the probability of making a full recovery for patients who receive a particular treatment and found that the probability was 0.866 for those over 70 years of age and 0.915 for those under 70 years of age.a. A press release about the study stated that the odds of making
4.10 Select two categories for each variable in Table 4.13 and compute the odds ratio for the resulting sub-table. Interpret this value.
4.9 Use Table 4.13 to compare the odds of finding life exciting rather than routine for those who are married and those who were never married. Report the appropriate odds ratio and interpret its value.
4.8 In a random sample of 996 Americans, individuals were classified according to their marital status and whether they considered life dull, routine, or exciting.The results (based on a subset of the 1993 General Social Survey) are shown in Table 4.13.a. Is there a statistically significant
4.7 Obtain the residuals and contributions to chi-squared for each of the cells in Problem 4.6.a. Which cells show the largest and smallest deviations from the results expected under independence?b. Would any of these deviations be considered large? Explain.
4.6 Use computer software and the data in Table 4.12 to test whether there is a statistically significant association between taking Vitamin C and the incidence of colds.a. Report the results of the Pearson chi-squared statistic and its p-value.b. Report the results of the likelihood ratio test
4.5 Think of two categorical variables that theoretically should be independent and explain why you believe they should be independent.
4.4 Use computer software and the data in Problems 4.1 (Table 4.11) toa. Obtain the odds ratio (as in Problem 4.1).b. Obtain the 95% confidence interval for the odds ratio (as in Problem 4.2).c. Obtain the Pearson chi-squared statistic for testing the association and its p-value.Interpret the
4.3 Is there a statistically significant association between proficiency and NES status based on the data in Problem 4.1? State the null hypothesis and report the conclusions.
4.2 Find the 95% confidence interval for the odds ratio in Problem 4.1 and interpret the result.
4.1 Suppose that fourth-grade students from a national random sample were classified as either proficient or not proficient in mathematics as well as whether each was a native English speaker (NES). Possible results (loosely based on data from the National Center for Education Statistics) are shown
3.14 With dichotomous variables, tests can be conducted using either single proportion methods or the goodness-of-fit methods (with c = 2). Repeat Problem 3.1 using the Pearson chi-squared goodness-of-fit test and state the conclusions.
3.13 Repeat Problem 3.11 using the likelihood ratio test statistic.
3.12 Repeat Problem 3.10 (parts a andb) using the likelihood ratio test statistic.
3.11 In a study on whether several issues are equally important to voters, a random sample of 200 registered voters were asked to select which of four issues they consider most important. The results obtained are shown in Table 3.5.a. State the null and alternative hypotheses.b. Do the data
3.10 After introducing a new teaching curriculum, a principal would like to determine whether the grade distribution in her school is significantly different than it has been in previous years. Suppose that Table 3.4 represents the distribution over previous years.Further, suppose that in a random
3.9 Use the information in Problem 3.8 to construct and interpret:a. The results of a 95% confidence interval for the proportion of votes for Candidate X.b. The results of a 99% confidence interval for the proportion of votes for Candidate X.c. Based on the results of the confidence intervals, is
3.8 In an election consisting of two candidates, a researcher would like to determine whether it is fair to say that one of the candidates will receive the majority of votes(where majority is defined as more than 50% of the population). In a recent poll for the election, 120 registered voters from
3.7 Repeat Problem 3.1 using the likelihood ratio test. State the null and alternative hypotheses (note that this method requires a two-tailed test). Report the p-value and state the conclusions.
3.6 Use the results of Problems 3.2 and 3.4 to explain why the exact test (based on the binomial distribution) is considered to be more conservative than the score test (based on the normal approximation).
3.5 Use the results of Problems 3.1 and 3.3 to explain why the exact test (based on the binomial distribution) is considered to be more conservative than the score test (based on the normal approximation).
3.4 Repeat Problem 3.2 using the score test.
3.3 Repeat Problem 3.1 using the score test.
3.2 Suppose that in the general population the proportion of obese men who have heart disease is 10%. In a random sample of 50 obese men who were placed on a low-fat diet, two men were found to have heart disease after 6 months.a. What is the maximum likelihood estimate of heart disease rate in the
3.1 In a certain school the proportion of students who are proficient in mathematics is 40%.A new teaching approach is introduced in the school, and after a year it is found that in a random sample of 20 students, 13 are proficient.a. What is the maximum likelihood estimate of the proficiency rate
2.12 Provide a substantive illustration of a situation that would require the use of each of the five probability distributions described in this chapter.
2.11 Telephone calls are received by a college switchboard at the rate of four calls every 3 minutes. What is the probability of obtaining five calls in a 3-minute interval?
2.10 On average, 10 people enter a particular bookstore every 5 minutes.a. What is the probability that only four people enter the bookstore in a 5-minute interval?b. What is the probability that eight people enter the bookstore in a 5-minute interval?
2.9 An owner of a boutique store knows that 45% of the customers who enter her store will make purchases that total less than $200, 15% of the customers will make purchases that total more than $200, and 40% of the customers will simply be browsing. If five customers enter her store on a particular
2.8 The probability that an entering college freshman will obtain his or her degree in four years is 0.4. What is the probability that at least one out of five admitted freshmen will graduate in four years?
2.7 For a multiple-choice test item with four response options, the probability of obtaining the correct answer by simply guessing is 0.25. If a student simply guessed on all 20 items in a multiple-choice test:a. What is the probability that the student would obtain the correct answers to 15 of the
2.6 Researchers at the Food Institute have determined that 67% of women tend to crave sweets over other alternatives. If 10 women are randomly sampled from across the country, what is the probability that only 3 of the women sampled will report craving sweets over other alternatives?
2.5 Suppose that the principal of Learn More School from Problem 2.2 is only able to choose one second-grade student to represent the school in a poetry contest. If she randomly selects a student, what is the probability that the student will be proficient in reading?
2.4 The CEO of a toy company would like to hire a vice president of sales and marketing.Only 2 of the 5 qualified applicants are female, and the CEO would really like to hire a female VP if at all possible to increase the diversity of his administrative cabinet. If he randomly chooses an applicant
2.3 Suppose there are 48 Republican senators and 52 Democrat senators in the United States Senate and the president of the United States must appoint a special committee of 6 senators to study the issues related to poverty in the United States. If the special committee is appointed by randomly
2.2 At Learn More School, 15 of the 20 students in second grade are proficient in reading.a. If the principal of the school were to randomly select two second-grade students to represent the school in a poetry reading contest, what is the probability that both of the students chosen will be
2.1 A divorce lawyer must choose 5 out of 25 people to sit on the jury that is to help decide how much alimony should be paid to his client, the ex-husband of a wealthy business woman. As luck would have it, 12 of the possible candidates are very bitter about having to pay alimony to their
1.8 Provide a substantive research question that would need to be addressed using procedures for categorical data analysis. Be sure to specify how the dependent and independent variables would be measured, and identify the scales of measurement used for these variables.
1.7 Determine whether procedures for analyzing categorical data are needed to address each of the following research questions. Indicate what analytic procedure (e.g., ANOVA, regression) you would use for those cases that do not require categorical methods, and provide a rationale for each of your
1.6 Determine whether procedures for analyzing categorical data are needed to address each of the following research questions. Provide a rationale for each of your answers by identifying the dependent and independent variables as well as the scale that might have been used to measure each
1.5 Determine whether procedures for analyzing categorical data are needed to address each of the following research questions. Provide a rationale for each of your answers by identifying the dependent and independent variables as well as the scale that might have been used to measure each
1.4 For each of the following research scenarios, identify the dependent and independent variables (or indicate if not applicable) as well as the scale of measurement used for each variable. Explain your answers by describing the scale that might have been used to measure each variable.a. A
1.3 For each of the following research scenarios, identify the dependent and independent variables (or indicate if not applicable) as well as the scale of measurement used for each variable. Explain your answers by describing the scale that might have been used to measure each variable.a. A
1.2 Indicate the scale of measurement used for each of the following variables and explain your answer by describing the probable scale:a. Self-efficacy, as measured by a 10-item scale.b. Race, as measured by a demographic question with six categories.c. Income, as measured by yearly gross income.
1.1 Indicate the scale of measurement used for each of the following variables and explain your answer by describing the probable scale:a. Sense of belongingness, as measured by a 20-item scale.b. Satisfaction with life, as measured by a 1-item scale.c. Level of education, as measured by a
10. Compute the HPD confidence interval in Example B.9 for a general K, and compare its length to the unadjusted and Bonferroni adjusted intervals.
9. In an errors-in-variables simple regression model, the least squares estimate of the regression slope (β) is biased toward 0, an example of attenuation. Specifically, if the true regression (through the origin) is Y = xβ + , but Y is regressed on X, with X = x + δ, then the least squares
8. Consider a linear regression having true model of the form Yi = α +βxi + i, where the i are i.i.d. with mean 0 and variance σ2. Suppose you have a sample of size n, use least-squares estimates (LSEs), and want to minimize the prediction error:PE = E[Ynew − Yˆ (Xnew)2] .Here, Yˆ (x) is
7. Let X be a random variable with mean μ and variance σ2. You want to estimate μ under SEL, and propose an estimate of the form (1 − b)X.(a) Find b∗, the b that minimizes the MSE.(Hint: Use the “variance + (bias)2” representation of MSE.)(b) Discuss the dependence of b∗ on μ and σ2
6. Find the frequentist risk under squared error loss (i.e., the MSE) for the estimator da,c(x) = a+cX¯. What true values of θ favor choosing a small value of c (high degree of shrinkage)? Does your answer change if n is large?
5. For the exponential distribution based on a sample of size n, find the minimax rule for squared error loss.
4. For the binomial distribution with n trials, find the minimax rule for squared error loss and normalized squared error loss, L(θ,a) = (θ−a)2θ(1−θ) .(Hint: The rules are Bayes rules or limits of Bayes rules.)For each of these estimates, compute the frequentist risk (the expected loss with
3. Consider the following loss function:l(θ,a) = % p|θ − a|, θ>a(1 − p)|θ − a|, θ ≤ a
2. Suppose that the posterior distribution of θ, p(θ|x), is discrete with support points {θ1, θ2,...}. Show that the Bayes rule under 0–1 loss(B.2) is the posterior mode.
1. Show that the Bayes estimate of a parameter θ under weighted squared error loss, l(θ,a) = w(θ)(θ − a)2, is given by dπ(x) = E[θw(θ)|x]E[w(θ)|x] .
15. Table 7.7 presents observed (Yi) and expected (Ei) cases of lip cancer in 56 counties in Scotland, which have been previously analyzed by Clayton and Kaldor (1987) and Breslow and Clayton (1993). The table also shows a spatial covariate, xi, the percentage of each county’s population engaged
14. The data in Table 7.6 were originally analyzed by Tolbert et al. (2000), and record the horizontal (s1) and vertical (s2) coordinates (in meters×104) of 10 ozone monitoring stations in the greater Atlanta metro area.Figure 7.12 shows the station locations on a greater Atlanta zip code ?
12. Suppose we have a collection of binary responses yi ∈ {0, 1}, i = 1,...,n, and associated k-dimensional predictor variables xi. Define the latent variables y∗i as y∗i = xT i β + i, i = 1,...,n, where the i are independent mean-zero errors having cumulative distribution function F, and
11. Consider again the model of Example 7.2. Suppose that instead of a straight-line growth curve, we wish to investigate the exponential growth
10. Table 7.4 gives a dataset of male mortality experience originally presented and analyzed by Broffitt (1988). The rates are for one-year intervals, ages 35 to 64 inclusive (i.e., bracket i corresponds to age 34 + i at last birthday).(a) Use the results of the previous problem to obtain a
9. Consider the estimation of human mortality rates between ages x and x + k, where x ≥ 30. Data available from a mortality study of a group of independent lives includes di, the number of deaths observed in the unit age interval [x + i − 1, x + i], and ei, the number of person-years the lives
8. (Devroye, 1986, p. 38) Suppose X is a random variable having cdf F, and Y is a truncated version of this random variable with support restricted to the interval [a, b]. Then Y has cdf G(y) =⎧⎨⎩0 , yF (b)−F (a) , a ≤ y ≤ b 1 , y>b.Show that Y can be generated as F −1(F(a) + U[F(b)
7. Prove that if the target parameters (θ1,...,θk) are independent and stochastically ordered, then for all j, P˜j (γ) = Pˆj (and therefore P˜j (γ)does not depend on γ). (Hint: Use the equivalence of P˜ and P∗.)
6. Table 7.1 shows that the AR(1) model “calms” longitudinal variation in the ranks. Discuss why this calming can be advantageous and why it might be disadvantageous.
5. Identify applications in which SEL in estimating ranks is the appropriate metric, and others in which (above γ)/(below γ) loss is appropriate. For the latter, discuss how you would select γ.
4. Let G(0.5)k (t) be the posterior median of Gk(t).(a) Show that for each t, G(0.5)k (t) minimizes expected, integrated absolute loss, E 8 | Gest k (t) − Gk(t) | dt9.(b) Outline an algorithm to compute G(0.5)k (t) using MCMC output.(c) Let k = 100 and assume that, a posteriori (θ1,...,θk) are
3. Compute and compare the mean and variance induced by edfs formed from the coordinate-specific MLEs, the standard Bayes estimates, and G¯k for the normal/normal, Poisson/gamma, and beta/binomial twostage models.
2. Let G∗k be a discrete distribution with mass 1/k at mass points (u1
1. Let Gk(t) be defined as in Subsection 7.1.2. Show that the posterior mode of Gk(t), denoted G˜k(t), has equal mass at exactly k mass points that are derived from an expression which for each t is the likelihood of independent, non-identically distributed Bernoulli variables.
5. Consider again the simple beta-binomial setting of Example 6.5, where we have a binomial response and the preliminary data from Study A given in Table 6.1. We again seek to base posterior inference on the lower 0.025 quantile of the posterior for π, after utilizing the existing data from the n1
4. Consider a clinical trial of a cholesterol-lowering drug, where we wish to use equal sample sizes in the experimental and control arms. Assume that LDL values (Y ) in the population are Y ∼ Normal(μ, σ2), where the population variance is assumed known (σ = 4.5). We consider the trial result
3. Consider again the clinical trial design setting of O’Hagan and Stevens(2001), as described in Subsection 6.2.2.(a) Confirm that expression (6.11) is the result of the analysis objective P(β > 0|y) > ω for this model. (Hint: Use expression (4.2), the Lindley and Smith (1972) result on the
2. Using equations (6.2) and (6.4), verify that ˜n ≥ nθ. Will this relation hold for all G? (Hint: see equation (6.3).)
1. Actually carry out the analysis in Example 6.4, where the prior g(θ) in(6.6) is(a) the Unif(0, 1) prior used in the example, g(θ) = 1.(b) a mixture of two Beta priors, g(θ) = .5 · 3θ2 + .5 · 30θ2(1 − θ2).What is the optimal sample size ˜n in each case? Is either prior amenable to a
17. Show that if the sampling variance is unknown and has an inverse gamma prior distribution, then in the limit as information in this inverse gamma goes to 0 and as τ 2 → ∞, Bayesian intervals are Student’s t-intervals.
16. Study the frequentist and Bayesian coverage probabilities for Bayesian intervals based on the Gaussian/Gaussian model with a N(μ, τ 2) prior and a sampling distribution, conditional on θ, that is N(θ, σ2).(a) Show you can assume μ = 0 and σ2 = 1 without loss of generality .(b) Show that
15. In the interval estimation setting involving a beta prior and a binomial likelihood,(a) Under what condition on the prior will the HPD interval for θ be one-sided when X = n? Find this interval.(b) Outline a specific interval-halving algorithm to find the HPD interval(θL, θU ). Use your
14. In the beta/binomial point estimation setting of Subsection 5.6.2,(a) For μ = .5 and general n and M, find the region where the Bayes rule has smaller risk than the MLE.
13. In the bivariate Gaussian example of Subsection 5.5.2, show that if Σ =I, then B has the form (5.39).
12. Do a simulation comparison of the MLE and three EB approaches: Robbins, the Poisson/gamma, and the Gaussian/Gaussian model for estimating the rate parameter in a Poisson distribution. Note that the Gaussian/Gaussian model is incorrect. Investigate two values of k (10, 20) and several true prior
11. In the Gaussian/Gaussian EM example,
10. Consider again the data in Table 3.1, also reproduced here for convenience (Table 5.7). These are the numbers of pump failures, Yi, observed in ti thousands of hours for k = 10 different systems of a certain nuclear power plant. The observations are listed in increasing order of raw failure
9. Consider the PEB model Yi|θi ind∼ P oisson(θiti), θi iid∼ G(a, b), i = 1,..., k.(a) Find the marginal distribution of Y = (Y1,...,Yk)T .(b) Use the method of moments to obtain closed form expressions for the hyperparameter estimates ˆa and ˆb. (Hint: Define the rates ri ≡ Yi/ti, and
8. Consider again Fisher’s sleep data:1.2, 2.4, 1.3, 1.3, 0.0, 1.0, 1.8, 0.8, 4.6, 1.4 .Suppose these k = 10 observations arose from the Gaussian/Gaussian PEB model, Yi|θi ind∼ N(θi, σ2), i = 1,...,k ,θi iid∼ N(μ, τ 2), i = 1,...,k .Assuming σ2 = 1 for these data, compute(a) θˆJS(b)
7. Prove result (5.26) on the maximum coordinate-specific loss for the James-Stein estimate.
6. Show how to evaluate (5.24) for a general θ. (Hint: A non-central chisquare can be represented as a Poisson mixture of central chi-squares with mixing on the degrees of freedom.)
5. Prove result (5.24) using the completeness of the noncentral chi-square distribution.
4. In the Gaussian/Gaussian model (5.5), if σ2 = 1 and μ = 0, the Yis are marginally independent with distribution Yi|τ iid∼ N(0, 1 + τ 2) ≡ N(0, 1/B) .(a) Find the marginal MLE of B, B MLE, and the resulting PEB point estimates ˆθEB i , i = 1,...,k.(b) Show that while B in (5.25) is not
3. Consider the gamma/inverse gamma model, i.e., Y1,...,Yk ind∼ G(α, θi),α a known tuning constant, and θ1,...,θk iid∼ IG(η, 1).(a) Find the marginal density of yi, m(yi|η).(b) Suppose α = 2. Find the marginal MLE of η, ˆη.
2. Under the compound sampling model (5.27) with fi = f for all i, show that (5.28) holds (i.e., that the Yis are marginally i.i.d.). What is the computational significance of this result for parametric EB data analysis?
1. (Berger, 1985, p. 298) Suppose Yi ind∼ N(θi, 1), i = 1,...,k, and that theθi are i.i.d. from a common prior G. Define the marginal density m(yi)in the usual way as m(yi) = N(yi | θi, 1) dG(θi) .(a) Show that, given G, the posterior mean for θi can be written as E(θi|yi) = yi

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Bayesian Methods For Data Analysis 3rd Edition Bradley P. Carlin, Thomas A. Louis - Solutions

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