Example using R: Heart Valves Study

Transcription

1 Example using R: Heart Valves Study Goal: Show that the thrombogenicity rate (TR) is less than two times the objective performance criterion R and WinBUGS Examples p. 1/27

2 Example using R: Heart Valves Study Goal: Show that the thrombogenicity rate (TR) is less than two times the objective performance criterion Data: From both the current study and a previous study on a similar product (St. Jude mechanical valve). R and WinBUGS Examples p. 1/27

3 Example using R: Heart Valves Study Goal: Show that the thrombogenicity rate (TR) is less than two times the objective performance criterion Data: From both the current study and a previous study on a similar product (St. Jude mechanical valve). Model: Let T be the total number of patient-years of followup, and θ be the TR per year. We assume the number of thrombogenicity events Y P oisson(θt): f(y θ) = e θt (θt) y y!. R and WinBUGS Examples p. 1/27

4 Example using R: Heart Valves Study Goal: Show that the thrombogenicity rate (TR) is less than two times the objective performance criterion Data: From both the current study and a previous study on a similar product (St. Jude mechanical valve). Model: Let T be the total number of patient-years of followup, and θ be the TR per year. We assume the number of thrombogenicity events Y P oisson(θt): f(y θ) = e θt (θt) y y!. Prior: Assume a Gamma(α,β) prior for θ: p(θ) = θα 1 e θ/β Γ(α)β α,θ > 0. R and WinBUGS Examples p. 1/27

5 Heart Valves Study The gamma prior is conjugate with the likelihood, so the posterior emerges in closed form: p(θ y) θ y+α 1 e θ(t+1/β) The study objective is met if where OPC = θ 0 = Gamma(y + α, (T + 1/β) 1 ). P(θ < 2 OPC y) 0.95, R and WinBUGS Examples p. 2/27

6 Heart Valves Study The gamma prior is conjugate with the likelihood, so the posterior emerges in closed form: p(θ y) θ y+α 1 e θ(t+1/β) The study objective is met if where OPC = θ 0 = Gamma(y + α, (T + 1/β) 1 ). P(θ < 2 OPC y) 0.95, Prior selection: Our gamma prior has mean M = αβ and variance V = αβ 2. This means that if we specify M and V, we can solve for α and β as α = M 2 /V and β = V/M. R and WinBUGS Examples p. 2/27

7 Heart Valves Study A few possibilities for prior parameters: R and WinBUGS Examples p. 3/27

8 Heart Valves Study A few possibilities for prior parameters: Suppose we set M = θ 0 = and V = 2θ 0 (so that 0 is two standard deviations below the mean). Then α = 0.25 and β = 0.152, a rather vague prior. R and WinBUGS Examples p. 3/27

9 Heart Valves Study A few possibilities for prior parameters: Suppose we set M = θ 0 = and V = 2θ 0 (so that 0 is two standard deviations below the mean). Then α = 0.25 and β = 0.152, a rather vague prior. Suppose we set M = 98/5891 =.0166, the overall value from the St. Jude studies, and V = M (so 0 is one sd below the mean). Then α = 1 and β = , a moderate (exponential) prior. R and WinBUGS Examples p. 3/27

10 Heart Valves Study A few possibilities for prior parameters: Suppose we set M = θ 0 = and V = 2θ 0 (so that 0 is two standard deviations below the mean). Then α = 0.25 and β = 0.152, a rather vague prior. Suppose we set M = 98/5891 =.0166, the overall value from the St. Jude studies, and V = M (so 0 is one sd below the mean). Then α = 1 and β = , a moderate (exponential) prior. Suppose we set M = 98/5891 =.0166 again, but set V = M/2. This is a rather informative prior. R and WinBUGS Examples p. 3/27

11 Heart Valves Study A few possibilities for prior parameters: Suppose we set M = θ 0 = and V = 2θ 0 (so that 0 is two standard deviations below the mean). Then α = 0.25 and β = 0.152, a rather vague prior. Suppose we set M = 98/5891 =.0166, the overall value from the St. Jude studies, and V = M (so 0 is one sd below the mean). Then α = 1 and β = , a moderate (exponential) prior. Suppose we set M = 98/5891 =.0166 again, but set V = M/2. This is a rather informative prior. We also consider event counts that are lower (1), about the same (3), and much higher (20) than for St. Jude. R and WinBUGS Examples p. 3/27

12 Heart Valves Study A few possibilities for prior parameters: Suppose we set M = θ 0 = and V = 2θ 0 (so that 0 is two standard deviations below the mean). Then α = 0.25 and β = 0.152, a rather vague prior. Suppose we set M = 98/5891 =.0166, the overall value from the St. Jude studies, and V = M (so 0 is one sd below the mean). Then α = 1 and β = , a moderate (exponential) prior. Suppose we set M = 98/5891 =.0166 again, but set V = M/2. This is a rather informative prior. We also consider event counts that are lower (1), about the same (3), and much higher (20) than for St. Jude. The study objective is not met with the bad data unless the posterior is rescued by the informative prior (lower right corner, next page). R and WinBUGS Examples p. 3/27

13 Heart Valves Study Priors and posteriors, Heart Valves ADVANTAGE study, Poisson-gamma model for various prior (M, sd) and data (y) values; T = 200, 2 OPC = vague prior posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = M, sd = ; Y = 1 M, sd = ; Y = 3 M, sd = ; Y = 20 moderate prior posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = M, sd = ; Y = 1 M, sd = ; Y = 3 M, sd = ; Y = 20 informative prior posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = M, sd = ; Y = M, sd = ; Y = M, sd = ; Y = 20 R and WinBUGS Examples p. 4/27

14 Heart Valves Study Priors and posteriors, Heart Valves ADVANTAGE study, Poisson-gamma model for various prior (M, sd) and data (y) values; T = 200, 2 OPC = vague prior posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = M, sd = ; Y = 1 M, sd = ; Y = 3 M, sd = ; Y = 20 moderate prior posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = M, sd = ; Y = 1 M, sd = ; Y = 3 M, sd = ; Y = 20 informative prior posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = posterior prior P(theta < 2 OPC y) = M, sd = ; Y = 1 M, sd = ; Y = 3 M, sd = ; Y = 20 S code to create this plot is available in brad/hv.s try it yourself in S-plus or R ( R and WinBUGS Examples p. 4/27

15 Alternate hierarchical models One might be uncomfortable with our implicit assumption that the TR is the same in both studies. To handle this, extend to a hierarchical model: Y i Poisson(θ i T i ), i = 1, 2, where i = 1 for St. Jude, and i = 2 for the new study. R and WinBUGS Examples p. 5/27

16 Alternate hierarchical models One might be uncomfortable with our implicit assumption that the TR is the same in both studies. To handle this, extend to a hierarchical model: Y i Poisson(θ i T i ), i = 1, 2, where i = 1 for St. Jude, and i = 2 for the new study. Borrow strength between studies by assuming θ i iid Gamma(α,β), i.e., the two TR s are exchangeable, but not identical. R and WinBUGS Examples p. 5/27

17 Alternate hierarchical models One might be uncomfortable with our implicit assumption that the TR is the same in both studies. To handle this, extend to a hierarchical model: Y i Poisson(θ i T i ), i = 1, 2, where i = 1 for St. Jude, and i = 2 for the new study. Borrow strength between studies by assuming θ i iid Gamma(α,β), i.e., the two TR s are exchangeable, but not identical. We now place a third stage prior on α and β, say α Exp(a) and β IG(c,d). R and WinBUGS Examples p. 5/27

18 Alternate hierarchical models One might be uncomfortable with our implicit assumption that the TR is the same in both studies. To handle this, extend to a hierarchical model: Y i Poisson(θ i T i ), i = 1, 2, where i = 1 for St. Jude, and i = 2 for the new study. Borrow strength between studies by assuming θ i iid Gamma(α,β), i.e., the two TR s are exchangeable, but not identical. We now place a third stage prior on α and β, say α Exp(a) and β IG(c,d). Fit in WinBUGS using the pump example as a guide! R and WinBUGS Examples p. 5/27

19 Example 2.5 revisited again! Pump Example ind Y i θ i Poisson(θ i t i ), ind θ i G(α,β), α Exp(µ), β IG(c,d), i = 1,...,k, where µ,c,d, and the t i are known, and Exp denotes the exponential distribution. R and WinBUGS Examples p. 6/27

20 Example 2.5 revisited again! Pump Example ind Y i θ i Poisson(θ i t i ), ind θ i G(α,β), α Exp(µ), β IG(c,d), i = 1,...,k, where µ,c,d, and the t i are known, and Exp denotes the exponential distribution. We apply this model to a dataset giving the numbers of pump failures, Y i, observed in t i thousands of hours for k = 10 different systems of a certain nuclear power plant. R and WinBUGS Examples p. 6/27

21 Example 2.5 revisited again! Pump Example ind Y i θ i Poisson(θ i t i ), ind θ i G(α,β), α Exp(µ), β IG(c,d), i = 1,...,k, where µ,c,d, and the t i are known, and Exp denotes the exponential distribution. We apply this model to a dataset giving the numbers of pump failures, Y i, observed in t i thousands of hours for k = 10 different systems of a certain nuclear power plant. The observations are listed in increasing order of raw failure rate r i = Y i /t i, the classical point estimate of the true failure rate θ i for the i th system. R and WinBUGS Examples p. 6/27

22 Pump Data i Y i t i r i Hyperparameters: We choose the values µ = 1, c = 0.1, and d = 1.0, resulting in reasonably vague hyperpriors for α and β. R and WinBUGS Examples p. 7/27

23 Pump Example Recall that the full conditional distributions for the θ i and β are available in closed form (gamma and inverse gamma, respectively), but that no conjugate prior for α exists. R and WinBUGS Examples p. 8/27

24 Pump Example Recall that the full conditional distributions for the θ i and β are available in closed form (gamma and inverse gamma, respectively), but that no conjugate prior for α exists. However, the full conditional for α, [ k ] p(α β, {θ i },y) g(θ i α,β) h(α) i=1 [ k i=1 ] θi α 1 Γ(α)β α e α/µ can be shown to be log-concave in α. Thus WinBUGS uses adaptive rejection sampling for this parameter. R and WinBUGS Examples p. 8/27

25 WinBUGS code to fit this model model { for (i in 1:k) { theta[i] dgamma(alpha,beta) lambda[i] <- theta[i]*t[i] Y[i] dpois(lambda[i]) } alpha dexp(1.0) beta dgamma(0.1, 1.0) } DATA: list(k = 10, Y = c(5, 1, 5, 14, 3, 19, 1, 1, 4, 22), t = c(94.320, 15.72, 62.88, , 5.24, 31.44, 1.048, 1.048, 2.096, 10.48)) INITS: list(theta=c(1,1,1,1,1,1,1,1,1,1), alpha=1, beta=1) R and WinBUGS Examples p. 9/27

26 Pump Example Results Results from running 1000 burn-in samples, followed by a production run of 10,000 samples (single chain): node mean sd MC error 2.5% median 97.5% alpha beta theta[1] E theta[5] theta[6] theta[10] R and WinBUGS Examples p. 10/27

27 Pump Example Results Results from running 1000 burn-in samples, followed by a production run of 10,000 samples (single chain): node mean sd MC error 2.5% median 97.5% alpha beta theta[1] E theta[5] theta[6] theta[10] Note that while θ 5 and θ 6 have very similar posterior means, the latter posterior is much narrower (smaller sd). R and WinBUGS Examples p. 10/27

28 Pump Example Results Results from running 1000 burn-in samples, followed by a production run of 10,000 samples (single chain): node mean sd MC error 2.5% median 97.5% alpha beta theta[1] E theta[5] theta[6] theta[10] Note that while θ 5 and θ 6 have very similar posterior means, the latter posterior is much narrower (smaller sd). This is because, while the crude failure rates for the two pumps are similar, the latter is based on a far greater number of hours of observation (t 6 = 31.44, while t 5 = 5.24). Hence we know more about pump 6! R and WinBUGS Examples p. 10/27

29 PK Example Wakefield et al. (1994) consider a dataset for which Y ij x ij = plasma concentration of the drug Cadralazine = time elapsed since dose given where i = 1,...,10 indexes the patient, while j = 1,...,n i indexes the observations, 5 n i 8. R and WinBUGS Examples p. 11/27

30 PK Example Wakefield et al. (1994) consider a dataset for which Y ij x ij = plasma concentration of the drug Cadralazine = time elapsed since dose given where i = 1,...,10 indexes the patient, while j = 1,...,n i indexes the observations, 5 n i 8. Attempt to fit the one-compartment nonlinear pharmacokinetic (PK) model, η ij (x ij ) = 30α 1 i exp( β i x ij /α i ). where η ij (x ij ) is the mean plasma concentration at time x ij. R and WinBUGS Examples p. 11/27

31 PK Example This model is best fit on the log scale, i.e. where ǫ ij ind N(0,τ i ). Z ij log Y ij = log η ij (x ij ) + ǫ ij, R and WinBUGS Examples p. 12/27

32 PK Example This model is best fit on the log scale, i.e. where ǫ ij ind N(0,τ i ). Z ij log Y ij = log η ij (x ij ) + ǫ ij, The mean structure for the Z ij s thus emerges as log η ij (x ij ) = log [ 30α 1 i exp( β i x ij /α i ) ] where a i = log α i and b i = log β i. = log 30 log α i β i x ij /α i = log 30 a i exp(b i a i )x ij, R and WinBUGS Examples p. 12/27

33 PK Data no. of hours following drug administration, x patient R and WinBUGS Examples p. 13/27

34 PK Data, original scale x Y R and WinBUGS Examples p. 14/27

35 PK Data, log scale x log Y R and WinBUGS Examples p. 15/27

36 PK Example For the subject-specific random effects θ i (a i,b i ), assume θ i iid N 2 (µ, Ω), where µ = (µ a,µ b ). R and WinBUGS Examples p. 16/27

37 PK Example For the subject-specific random effects θ i (a i,b i ), assume θ i iid N 2 (µ, Ω), where µ = (µ a,µ b ). Usual conjugate prior specification: µ N 2 (λ,c) τ i Ω iid G(ν 0 /2, ν 0 τ 0 /2) Wishart((ρR) 1,ρ) R and WinBUGS Examples p. 16/27

38 PK Example For the subject-specific random effects θ i (a i,b i ), assume θ i iid N 2 (µ, Ω), where µ = (µ a,µ b ). Usual conjugate prior specification: µ N 2 (λ,c) τ i Ω iid G(ν 0 /2, ν 0 τ 0 /2) Wishart((ρR) 1,ρ) Note that the θ i full conditional distributions are: not simple conjugate forms not guaranteed to be log-concave Thus, the Metropolis capability of WinBUGS is required. R and WinBUGS Examples p. 16/27

39 PK Results (WinBUGS vs. Fortran) BUGS Sargent et al. (2000) parameter mean sd lag 1 acf mean sd lag 1 acf a a a a b b b b τ τ τ τ Y 2, Y 7, R and WinBUGS Examples p. 17/27

40 Lip Cancer Example Consider the spatial disease mapping model: Y i µ i ind Po (E i e µ i ), where Y i = observed disease count, E i = expected count (known), and µ i = x iβ + θ i + φ i R and WinBUGS Examples p. 18/27

41 Lip Cancer Example Consider the spatial disease mapping model: Y i µ i ind Po (E i e µ i ), where Y i = observed disease count, E i = expected count (known), and µ i = x iβ + θ i + φ i The x i are explanatory spatial covariates; typically β is assigned a flat prior. R and WinBUGS Examples p. 18/27

42 Lip Cancer Example Consider the spatial disease mapping model: Y i µ i ind Po (E i e µ i ), where Y i = observed disease count, E i = expected count (known), and µ i = x iβ + θ i + φ i The x i are explanatory spatial covariates; typically β is assigned a flat prior. Note the mean structure also contains two sets of random effects! The first, θ i, capture heterogeneity among the regions via θ i iid N(0, 1/τ h ). R and WinBUGS Examples p. 18/27

43 Lip Cancer Example The second set, φ i, capture regional clustering via a conditionally autoregressive (CAR) prior, φ i φ j i N( φ i, 1/(τ c m i )), where m i is the number of neighbors of region i, and φ i = m 1 i Σ j i φ j. R and WinBUGS Examples p. 19/27

44 Lip Cancer Example The second set, φ i, capture regional clustering via a conditionally autoregressive (CAR) prior, φ i φ j i N( φ i, 1/(τ c m i )), where m i is the number of neighbors of region i, and φ i = m 1 i Σ j i φ j. The CAR prior is translation invariant, so typically we insist Σ I i=1 φ i = 0 (imposed numerically after each MCMC iteration). Still, Y i cannot inform about θ i or φ i, but only about their sum η i = θ i + φ i. R and WinBUGS Examples p. 19/27

45 Lip Cancer Example The second set, φ i, capture regional clustering via a conditionally autoregressive (CAR) prior, φ i φ j i N( φ i, 1/(τ c m i )), where m i is the number of neighbors of region i, and φ i = m 1 i Σ j i φ j. The CAR prior is translation invariant, so typically we insist Σ I i=1 φ i = 0 (imposed numerically after each MCMC iteration). Still, Y i cannot inform about θ i or φ i, but only about their sum η i = θ i + φ i. Making the reparametrization from (θ,φ) to (θ,η), we have the joint posterior p(θ, η y) L(η; y)p(θ)p(η θ). R and WinBUGS Examples p. 19/27

46 Lip Cancer Example This means that p(θ i θ j i,η,y) p(θ i ) p(η i θ i {η j θ j } j i ). R and WinBUGS Examples p. 20/27

47 Lip Cancer Example This means that p(θ i θ j i,η,y) p(θ i ) p(η i θ i {η j θ j } j i ). Since this distribution is free of the data y, the θ i are Bayesianly unidentified (and so are the φ i ). R and WinBUGS Examples p. 20/27

48 Lip Cancer Example This means that p(θ i θ j i,η,y) p(θ i ) p(η i θ i {η j θ j } j i ). Since this distribution is free of the data y, the θ i are Bayesianly unidentified (and so are the φ i ). BUT this does not preclude Bayesian learning about θ i ; this would instead require p(θ i y) = p(θ i ). [Stronger condition: data have no impact on the marginal (not conditional) posterior.] R and WinBUGS Examples p. 20/27

49 Lip Cancer Example Dilemma: Though unidentified, the θ i and φ i are interesting in their own right, as is ψ = sd(φ) sd(θ) + sd(φ), where sd( ) is the empirical marginal standard deviation function. Can we specify vague but proper prior values τ h and τ c that R and WinBUGS Examples p. 21/27

50 Lip Cancer Example Dilemma: Though unidentified, the θ i and φ i are interesting in their own right, as is ψ = sd(φ) sd(θ) + sd(φ), where sd( ) is the empirical marginal standard deviation function. Can we specify vague but proper prior values τ h and τ c that lead to acceptable convergence behavior, and R and WinBUGS Examples p. 21/27

51 Lip Cancer Example Dilemma: Though unidentified, the θ i and φ i are interesting in their own right, as is ψ = sd(φ) sd(θ) + sd(φ), where sd( ) is the empirical marginal standard deviation function. Can we specify vague but proper prior values τ h and τ c that lead to acceptable convergence behavior, and still allow Bayesian learning? R and WinBUGS Examples p. 21/27

52 Lip Cancer Example Dilemma: Though unidentified, the θ i and φ i are interesting in their own right, as is ψ = sd(φ) sd(θ) + sd(φ), where sd( ) is the empirical marginal standard deviation function. Can we specify vague but proper prior values τ h and τ c that lead to acceptable convergence behavior, and still allow Bayesian learning? Tricky to specify a fair prior balance between heterogeneity and clustering (e.g., one for which ψ 1/2) since θ i prior is specified marginally while the φ i prior is specified conditionally. R and WinBUGS Examples p. 21/27

53 Dataset: Scottish lip cancer data a) b) > < 50 > N Miles a) SMR i = 100Y i /E i, standardized mortality ratio for lip cancer in I = 56 districts, R and WinBUGS Examples p. 22/27

54 Dataset: Scottish lip cancer data a) b) > < 50 > N Miles a) SMR i = 100Y i /E i, standardized mortality ratio for lip cancer in I = 56 districts, b) one covariate, x i = percentage of the population engaged in agriculture, fishing or forestry (AFF) R and WinBUGS Examples p. 22/27

55 WinBUGS code to fit this model model { for (i in 1 : regions) { O[i] dpois(mu[i]) log(mu[i]) <- log(e[i]) + beta*aff[i]/10 + phi[i] + theta[i] theta[i] dnorm(0.0,tau.h) eta[i] <- theta[i] + phi[i] } phi[1:regions] {\red car.normal}(adj[], weights[], num[], tau.c) beta dnorm(0.0, 1.0E-5) # vague prior on covariate effect tau.h dgamma(1.0e-3,1.0e-3) # fair prior from Best et al. tau.c dgamma(1.0e-1,1.0e-1) # (1999, Bayesian Statistics 6) } sd.h <- sd(theta[]) # marginal SD of heterogeneity effects sd.c <- sd(phi[]) # marginal SD of clustering (spatial) effects psi <- sd.c / (sd.h + sd.c) (See WinBUGS Map manual for DATA and INITS for this example) R and WinBUGS Examples p. 23/27

56 Lip Cancer Results posterior for ψ posterior for β priors for τ c, τ h mean sd l1acf mean sd l1acf G(1.0, 1.0), G(3.2761, 1.81) G(.1,.1), G(.32761,.181) G(.1,.1), G(.001,.001) posterior for η 1 posterior for η 56 priors for τ c, τ h mean sd l1acf mean sd l1acf G(1.0, 1.0), G(3.2761, 1.81) G(.1,.1), G(.32761,.181) G(.1,.1), G(.001,.001) AFF covariate is significantly 0 under all 3 priors R and WinBUGS Examples p. 24/27

57 Lip Cancer Results posterior for ψ posterior for β priors for τ c, τ h mean sd l1acf mean sd l1acf G(1.0, 1.0), G(3.2761, 1.81) G(.1,.1), G(.32761,.181) G(.1,.1), G(.001,.001) posterior for η 1 posterior for η 56 priors for τ c, τ h mean sd l1acf mean sd l1acf G(1.0, 1.0), G(3.2761, 1.81) G(.1,.1), G(.32761,.181) G(.1,.1), G(.001,.001) AFF covariate is significantly 0 under all 3 priors convergence is slow for ψ and β, but rapid for η i (and µ i ) R and WinBUGS Examples p. 24/27

58 Lip Cancer Results posterior for ψ posterior for β priors for τ c, τ h mean sd l1acf mean sd l1acf G(1.0, 1.0), G(3.2761, 1.81) G(.1,.1), G(.32761,.181) G(.1,.1), G(.001,.001) posterior for η 1 posterior for η 56 priors for τ c, τ h mean sd l1acf mean sd l1acf G(1.0, 1.0), G(3.2761, 1.81) G(.1,.1), G(.32761,.181) G(.1,.1), G(.001,.001) AFF covariate is significantly 0 under all 3 priors convergence is slow for ψ and β, but rapid for η i (and µ i ) Excess variability in the data is mostly due to clustering (E(ψ y) >.50), but the posterior distribution for ψ does not seem robust to changes in the prior. R and WinBUGS Examples p. 24/27

59 InterStim Example Device that uses electrical stimulation of the brain to prevent urinary incontinences. For patients i = 1,...,49: X 1i = number of incontinences per week at baseline X 2i = number of incontinences per week at 3 months X 3i = number of incontinences per week at 6 months X 4i = number of incontinences per week at 12 months patient X 1i X 2i X 3i X 4i NA NA NA NA R and WinBUGS Examples p. 25/27

60 InterStim Example Goal 1: Obtain full predictive inference for all missing X values (point and interval estimates) R and WinBUGS Examples p. 26/27

61 InterStim Example Goal 1: Obtain full predictive inference for all missing X values (point and interval estimates) Goal 2: Obtain measure of percent improvement (relative to baseline) due to InterStim at 6 and 12 months R and WinBUGS Examples p. 26/27

62 InterStim Example Goal 1: Obtain full predictive inference for all missing X values (point and interval estimates) Goal 2: Obtain measure of percent improvement (relative to baseline) due to InterStim at 6 and 12 months Model: Let X i = (X 1i,X 2i,X 3i,X 4i ) and θ = (θ 1,θ 2,θ 3,θ 4 ). Clearly the X ij s are correlated, but an ordinary longitudinal model does not seem appropriate (we can t just use a linear model here). So instead, maintain the generality: X i θ, Υ iid N 4 (θ, Υ 1 ) θ N 4 (µ, Ω 1 ) Υ Wishart 4 (R,ρ) R and WinBUGS Examples p. 26/27

63 InterStim Example WinBUGS will generate all missing X s ( NA s in the dataset) from their full conditional distributions as part of the Gibbs algorithm. Thus we will obtain samples from p(x ij X obs ) for all missing X ij (achieving Goal 1). R and WinBUGS Examples p. 27/27

64 InterStim Example WinBUGS will generate all missing X s ( NA s in the dataset) from their full conditional distributions as part of the Gibbs algorithm. Thus we will obtain samples from p(x ij X obs ) for all missing X ij (achieving Goal 1). Re: Goal 2 (percent improvement from baseline), let α = θ 1 θ 3 θ 1 and β = θ 1 θ 4 θ 1 Then p(α X obs ) and p(β X obs ) address this issue! R and WinBUGS Examples p. 27/27

65 InterStim Example WinBUGS will generate all missing X s ( NA s in the dataset) from their full conditional distributions as part of the Gibbs algorithm. Thus we will obtain samples from p(x ij X obs ) for all missing X ij (achieving Goal 1). Re: Goal 2 (percent improvement from baseline), let α = θ 1 θ 3 θ 1 and β = θ 1 θ 4 θ 1 Then p(α X obs ) and p(β X obs ) address this issue! Hyperparameters: all vague: Ω = Diag(10 6,...,10 6 ), µ = 0, ρ = 4 (the smallest value for which the Wishart prior for Υ is proper) and R = Diag(10 1,...,10 1 ). R and WinBUGS Examples p. 27/27

66 InterStim Example WinBUGS will generate all missing X s ( NA s in the dataset) from their full conditional distributions as part of the Gibbs algorithm. Thus we will obtain samples from p(x ij X obs ) for all missing X ij (achieving Goal 1). Re: Goal 2 (percent improvement from baseline), let α = θ 1 θ 3 θ 1 and β = θ 1 θ 4 θ 1 Then p(α X obs ) and p(β X obs ) address this issue! Hyperparameters: all vague: Ω = Diag(10 6,...,10 6 ), µ = 0, ρ = 4 (the smallest value for which the Wishart prior for Υ is proper) and R = Diag(10 1,...,10 1 ). Code and Data: brad/data/interstim.odc R and WinBUGS Examples p. 27/27