Questions and Answers of Bayesian Biostatistics

Exercise 11.17 Diabetes study: Apply the R program glmBfp to find nonlinear associations with the response y.
Exercise 11.16 Rheumatoid arthritis study: Apply Exercise 11.14 to the REACH data.
Exercise 11.15 Rheumatoid arthritis study: Apply the following Bayesian shrinkage regression approaches: ridge, LASSO, adaptive LASSO and Elastic Net to the REACH data. Use the latent representation
Exercise 11.14 Diabetes study: Apply the WinBUGS program ‘chapter 11 blasso adaptive diabetes.odc’ to perform Bayesian adaptive LASSO estimation on the diabetes data.
Exercise 11.13 Rheumatoid arthritis study: Adapt the WinBUGS program ‘chapter 11 GVS REACH.odc’ to check whether sjc (swollen joint count) should be transformed with log, sqrt or inverse by
Exercise 11.12 Rheumatoid arthritis study: Adapt the WinBUGS program ‘chapter 11 GVS REACH.odc’ to check the importance of the product terms of all regressors with dc (duration of complaints in
Exercise 11.11 Diabetes study: Apply Exercise 11.9 to the diabetes data.
Exercise 11.10 Rheumatoid arthritis study: Adapt the WinBUGS program ‘chapter 11 GVS REACH.odc’ with the approach of Kuo and Mallick (1998).
Exercise 11.9 Rheumatoid arthritis study: Apply the WinBUGS program ‘chapter 11 GVS REACH.odc’ to select variables in a probit regression model using the GVS approach. Compare the selected
Exercise 11.8 Diabetes study: Apply Exercise 11.7 to the diabetes data and compare the selected models from those selected with other BVS approaches.
Exercise 11.7 Rheumatoid arthritis study: Use R2WinBUGS based on ‘chapter 11 NMIG REACH.odc’ to select variables in a logistic and a probit regression model using the approach of Ishwaran and Rao
Exercise 11.6 Diabetes study: Adapt WinBUGS program ‘chapter 11 SSVS REACH.R’ to select variables in a linear regression model and apply it to the diabetes data. Perform a sensitivity analysis by
Exercise 11.5 Diabetes study: Use R2WinBUGS to select the most important variables using a RJMCMC approach and compare it to the solution when applying the R program monomvn. Extend the set of
Exercise 11.4 Rheumatoid arthritis study: Use WinBUGS program ‘chapter 11 RJMCMC REACH.odc’ to check the dependence of the solution on the precision of the regression coefficients, i.e. MAP and
Exercise 11.3 Rheumatoid arthritis study: Find the MP models using the R functions bic.glm and glib. P1: JYS/XYZ P2: ABC JWST177-c11 JWST177-Lesaffre May 26, 2012 11:21 Printer Name: Yet to Come
Exercise 11.2 Diabetes study: Find the MP models using the R functions bic.glm and MC3.REG.
Exercise 11.1 Dietary study: Predict body-mass index (kg/m2) from age, gender, length (cm), weight (kg), and the food intake variables measuring the daily consumption of: alcohol (g/day), calcium
Exercise 10.28 Example X.27: Check the linearity of length, weight and age in predicting the probability of alcohol (consumption) using the B-splines as well as the cubic splines approach.
Exercise 10.27 Example X.25: Check the linearity of the regression model of TBBMC as a function of age with or without assuming linearity in BMI. Apply both the B-splines as well as the cubic splines
Exercise 10.26 Example X.22: Check the choice of the logistic-link function in the toenail data set using WinBUGS.
Exercise 10.25 Example X.18: Verify the normality assumption for the measurement error and the random effects in each of the two groups using a PPC.
Exercise 10.24 Example X.20: In ‘chapter 10 lip cancer PGM PPC theta.odc’ the gamma assumption of the θs is checked using a variety of measures. Use R2WinBUGS.
Exercise 10.23 Exercise 10.12: In ‘chapter 10 caries with residuals.odc’ also a Poisson model is fitted to the dmft indices. Apply a PPC with the χ2-discrepancy measure to check the Poisson
Exercise 10.22 Example X.19: Test the normality assumption with the Shapiro–Francia test (Shapiro and Francia 1972), which consists of computing the Pearson correlation coefficients between the
Exercise 10.21 Example X.8: Perform a sensitivity analysis on the seven models. More specifically, evaluate the effect of varying the distributions of the random effects and measurement error on the
Exercise 10.20 Show that the caterpillar option in WinBUGS can be used to construct a normal probability plot and apply it to the osteoporosis data set.
Exercise 10.19 Example IX.18: Assess the impact of varying the distribution of the measurement error. Consider a t-distribution, a logistic distribution, etc. compare these models to the basic model
Exercise 10.18 Example IX.18: Assess the impact of varying the prior of the measurement error variance (σ2) and the variance of the random intercept (σ2 b ) in φ2i to (1) σ2 ∼ IG(10−3,
Exercise 10.17 Example X.16: Use ‘chapter 10 lip cancer with residuals.R’ and reproduce Figure 10.5. Verify whether the graphs change with a log-t regression model.
Exercise 10.15 Example X.13: Perform the various Bayesian residual analyses by making use of the SAS procedure GENMOD. Exercise 10.16 Example X.15: Perform the various Bayesian residual analyses by
Exercise 10.14 In ‘chapter 10 osteo study-outliers.R’ PPOi and CPOi are computed for the linear regression applied to the osteoporosis data. Apply this program. In addition, compare PPO 179 based
Exercise 10.13 In ‘chapter 10 caries with residuals.R’ the dmft index is fitted with a Poisson regression model using R2WinBUGS. Use the approach of Exercise 10.12 to find outliers, if any.
Exercise 10.12 In ‘chapter 10 caries with residuals.odc’ a logistic regression model is fitted to the CE data of the Signal-Tandmobiel study of Example VI.9 with some additional covariates, i.e.
Exercise 10.11 Example X.13: Verify that the posterior of the raw residuals is indeed a tn−d−1-distribution. Produce also a similar (and additional) graph as Figure 10.3.
Exercise 10.10 Example X.11: Suppose that ψ = 1/θα with α > 0. Show that the prior of ψ is Pareto(α, 1) if the prior for θ is uniform on [0, 1]. Observe the change in pD and DIC when α is
Exercise 10.9 Choosing a prior distribution in Exercise 10.7 for the Poisson model which is in conflict with the data and observe the change in pD and DIC.
Exercise 10.8 Based on the data in Exercise 10.7 and using DIC, select the best fitting model from the following list: (1) Poisson model, (2) negative binomial model, (3) Poisson-gamma, (4)
Exercise 10.6 Assume in Example X.10 alternative distributions for θ and choose the best model in combination with a normal distribution for chol or log(chol) using DIC. Exercise 10.7 Evaluate which
Exercise 10.5 In ‘chapter 10 dmft scores full data set.odc’ a Poisson-gamma and a negative binomial distribution are fit to the dmft index of the 4352 seven-year-old children of the
Exercise 10.4 Example X.8: Write a SAS program based on procedure MCMC to obtain parameter estimates for models M1 to M7. Procedure MIXED allows for a Bayesian mixed analysis. Check when this
Exercise 10.3 Example X.8: Assume in model M1 that (a) variances for random effects grow to infinity or (b) variances for random effects grow to zero. This can be done by fixing the respective
Exercise 10.2 Example X.8: Assume in model M1 three choices for the inverse Wishart prior IW(D, 2) for G the covariance matrix of the random effects, varying by the choice of D: (a) D = diag(0.001,
Exercise 10.1 Example X.8: Perform the longitudinal analyses using ‘chapter 10 PotthoffRoy growthcurves.odc’. Run also the extra analyses in this program. Further, since normality of the random
Exercise 9.16 The R program ‘chapter 9 dietary study-R2WB.R’ uses the R2WinBUGS program for the analysis of dietary study. Run this program and produce your own trace plots, histograms, scatter
Exercise 9.15 Example IX.18: Although in the first 10 000 iterations the trace plots behaved well, when running the chains further it became apparent that the variance parameter σ got stuck at zero
Exercise 9.14 Gelman (2006) suggested to reformulate the Gaussian hierarchical model as yi j ∼ N(µ + ξ ηi, σ2 ) ξ ∼ N(0, σ2 η ).In WinBUGS program ‘chapter 9 dietary study chol
Exercise 9.13 Example IX.12: Check the impact of choosing an Inverse Wishart prior for the covariance matrix of the random intercept and slope on the estimation of the model parameters, by varying
Exercise 9.12 In the WinBUGS program ‘chapter 9 generated.odc’ a Bayesian–Gaussian hierarchical model is given using fictive data from Tiao and Tan (1965). The data were generated such that the
Exercise 9.11 Use the WinBUGS program ‘chapter 9 dietary study chol2.odc’ to examine the dependence of the posterior of σθ on the choice of ε in the inverse gamma ‘noninformative’ prior
Exercise 9.10 Determine the predicted profiles and the distribution of future observations for the BNLMM of Example IX.18. Estimate the one missing response in the data.
Exercise 9.9 Show that the random intercept of the logistic random intercept model of Example IX.14 has a bimodal distribution, with one mode around −2.4. Show also that the subjects with an
Exercise 9.8 Compare the analysis of Example IX.14 with an analysis where treatment is included as main effect.
Exercise 9.7 Example IX.17: In Gelman and Meng (1996, p. 193) it is argued that a uniform prior on the degrees of freedom (ν) essentially puts all of its mass on infinity. As an alternative, the
Exercise 9.6 Compare the analysis of Example IX.12 with an analysis where treatment is included as main effect.
Exercise 9.5 Compare the Bayesian analysis of Example IX.8 to a frequentist analysis and verify that the results are quite similar.
Exercise 9.4 Perform a WinBUGS analysis of the dietary cholesterol data under assumptions A1 and A2.
Exercise 9.3 Determine the WinBUGS samplers for the nodes µ, θi, σ2 and σ2 θ in the WinBUGS program ‘chapter 9 dietary study chol.odc’.
Exercise 9.2 Write a R function to Gibbs sample from the Poisson-gamma hierarchical model based on GDR lip cancer data.
Exercise 9.1 Show that the MLE of θ under assumption A2 in Section 9.2.2 is given by i yi/ i ei. Derive also the posterior summary measures for different priors on θ, (1) flat prior, (2)
Exercise 8.13 Take the Dugongs Example of the WinBUGS document ‘Examples Vol II’ and observe what happens when the prior distribution for γ is changed into (a) Beta(1, 3) and (b) Beta(3, 1).
Exercise 8.12 Take the Dugongs Example of the WinBUGS document ‘Examples Vol II’ and observe what happens when centering the covariate age. What is your conclusion?
Exercise 8.11 Analyze the interval-censored data (contained in file ‘chapter 7 interval censoring.R’) with WinBUGS and PROC MCMC.
Exercise 8.10 Analyze the first event in the Kidney Example of the WinBUGS document ‘Examples Vol I’ with the Bayesian option in the SAS procedure LIFEREG and procedure MCMC.
Exercise 8.9 Take the Kidney Example of the WinBUGS document ‘Examples Vol I’. Look only at the first event (i.e. change the model and the data structure). Run the WinBUGS program and explore
Exercise 8.8 Analyze the caries data of Example VII.9 with PROC GENMOD and PROC MCMC. Try out different MH samplers with PROC MCMC. Compare the convergence rates and posterior summary measures
Exercise 8.7 Perform the analysis of Exercise 7.8 in SAS using PROC MCMC.
Exercise 8.6 Perform the analysis of Exercise 7.7 using WinBUGS. Repeat the analyses with PROC MCMC.
Exercise 8.5 Run the Mice Example of the WinBUGS document ‘Examples Vol I’ in SAS using the Bayesian option in PROC LIFEREG and PROC MCMC. Try out different MH samplers with PROC MCMC. Compare
Exercise 8.4 Run the Mice Example of the WinBUGS document ‘Examples Vol I’ with WinBUGS and replace the Weibull distribution with a lognormal distribution. Produce the autocorrelation function in
Exercise 8.3 Perform the analysis of Exercises 7.1 and 7.2 in SAS using PROC GENMOD and PROC MCMC.
Exercise 8.2 Apply some acceleration techniques to improve the convergence rate in Exercise 8.1 of the WinBUGS sampler when the blocking options are switched off.
Exercise 8.1 Perform the analysis of Exercises 7.1 and 7.2 in WinBUGS with blocking options both switched on and off.
Exercise 7.9 Apply the Gibbs sampler of Example VII.9 on the caries data set (caries.txt) analyzed in Example VI.9 and compare your results with the MCMC analyses of Example VI.9. You can also
Exercise 7.8 Joseph et al. (1995) wished to estimate the prevalence of Strongyloides infection using data from a survey of all Cambodian refugees who arrived in Montreal during an 8- month period.
Exercise 7.7 Repeat the analysis of Example VII.6 for the intercrosses AB/ab × AB/ab (coupling). In this case (see Rao (1973)), the probabilities are (a) for AB: (3 − 2π + π2 )/4, (b) for Ab:
Exercise 7.6 Assess the convergence properties of the block Gibbs sampler of Exercise 6.6.
Exercise 7.5 Apply thinning (=10) to the sampling algorithms of Exercise 6.1 and assess their convergence properties. Assess also their performance when centering BMI.
Exercise 7.4 Import the data of ‘osteoporosismultiple.txt’ into R and perform a Bayesian probit regression analysis using the DA approach of Example VII.9 predicting overweight (BMI > 25) from
Exercise 7.3 Import the data of the Mice Example of the WinBUGS document Examples Vol I into R. Write an R program that implements the Gibbs sampler for the model specified in the Mice Example. Then
Exercise 7.2 Export the Markov chains obtained in Exercise 7.1 to CODA or BOA and explore the stationarity of the chains. Let the Gibbs sampler run long enough such that, upon convergence, the MC
Exercise 7.1 Take extreme starting values for the parameters of the Bayesian regression analyses of Exercise 6.1, e.g. ‘beta0=100,beta1=100,tau=1/0.05’ and observe the initial monotone behavior
Exercise 6.12 Consult Waagepetersen and Sorensen (2001) which gives a more elaborate explanation of the Reversible Jump MCMC approach. In that paper, Example 4.2 describes the use of RJMCMC to choose
Exercise 6.11 Vary in the RJMCMC program of Example VI.10 the settings of the prior distributions and evaluate the sensitivity of the results. Evaluate also the dependence of the results to the
Exercise 6.10 Perform a Bayesian regression of TBBMC on age, length, weight, and BMI (data in ‘osteoporosismultiple.txt’) with the classical NI priors using the following: Basic Gibbs sampler
Exercise 6.9 Repeat Exercise 6.8 but now employ a Random Walk Metropolis algorithm.
Exercise 6.8 Perform a Bayesian logistic regression on the caries data of Example VI.9 (data in ‘caries.txt’) and with the same priors. Make use of the R function ars to apply the basic Gibbs
Exercise 6.7 Apply the Slice sampler to a beta distribution. Compare its performance to the classical R built-in sampler rbeta. Apply the Slice sampler also to a mixture of two beta distributions
Exercise 6.6 Apply the block Gibbs sampler introduced in Section 6.2.4 to the osteoporosis data (see also Exercise 6.1), with blocks (β0, β1 ) and σ2. Compare its performance to the basic Gibbs
Exercise 6.5 Program the random-scan Gibbs sampler introduced in Section 6.2.4. Apply the procedure on the osteoporosis data (see also Exercise 6.1) and compare the performance of the basic Gibbs
Exercise 6.4 Write an R program for the reversible Gibbs sampler introduced in Section 6.2.4. Apply the procedure on the osteoporosis data (see also Exercise 6.1) and compare the performance of the
Exercise 6.3 Sample from the auto-exponential model of Besag (1974) which is defined for positive (y1, y2, y3 ) with density f(y1, y2, y3 ) ∝ exp [−(y1 + y2 + y3 + ψ12y1y2 + ψ13y1y3 + ψ23y2y3
Exercise 6.1 Derive the posterior distribution (with the same NI priors as in the chapter) of (β0, β1, σ2 ) in the osteoporosis study (data are in ‘osteop.txt’) by: (a) Gibbs sampler and (b)
Exercise 5.16 Derive the conjugate prior for the distribution (4.33) using the rule explained in Section 5.3.1.
Exercise 5.15 Holzer et al. (2006) analyzed a retrospective cohort study for the efficacy and safety of endovascular cooling in unselected survivors of cardiac arrest compared to controls. The
Exercise 5.13 Show that Jeffreys prior distribution for σ2 of a normal distribution with given mean, results in a proper posterior distribution when there are at least two observations.Exercise 5.14
Exercise 5.12 Show that in the binomial case the arcsin( √·)-transformation yields an approximate data-translated likelihood. Show also that this prior is locally uniform in the original scale
Exercise 5.11 Show that for the multinomial model, Jeffreys rule suggests to take for a noninformative prior p(θ) ∝ (θ1 × ... × θp)−1/2.
Exercise 5.9 Show graphically that the data-translated likelihood principle is satisfied on the original scale of µ for a normal likelihood with σ given. Exercise 5.10 Show that Jeffreys rule for
Exercise 5.8 Prove that all one-dimensional marginal distributions of a Dirichlet distribution are beta distributions.
Exercise 5.7 Show that the natural conjugate for the multinomial distribution is the Dirichlet distribution and derive Jeffreys prior and the resulting posterior distribution.

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