PubH 7440 – Spring 2010 Midterm 2 – April Problem 1a: Because \theta^t_i is a linear combination of normal random variables, it will also be normal. Thus the mean and variance completely characterize the distribution. We also use that the Z and \theta^{t-1}_i are independent. ()()()()()()()()()()()()0. covariance their Hence, versa. viceand Y up dependnot do VAR(X) and EX :Note1,0Y and ,X Where,),(21,1,0,1,0),(12222222222222222221=−===+−+=−+−+=−+−+=−+−+=−ασσαμσμασσαμασσααμμαμασαμσααμμασμθαμθiiiiiiiiiiiiiiiiiiiitiiiNNNYXCOVNNNNNZ Problem 1b: While it is true that Alder’s provides better MCMC convergence in some cases, the model presented here does not present any autocorrelation problems with ordinary Gibbs sampling. Therefore, the overrelaxation of Adler’s is compensating for a problem that does not exist, creating a new problem of large *negative* lag-1 autocorrelation (and then large positive, continuing in this alternating manner). In the figures below, this effect is quite clear. First, the full conditionals: Full Bayes Model:()() ()()()()()⎟⎠⎞⎜⎝⎛⎭⎬⎫⎩⎨⎧−−=⎭⎬⎫⎩⎨⎧+−−∝⎭⎬⎫⎩⎨⎧−−∝⎥⎦⎤⎢⎣⎡∝==⎟⎠⎞⎜⎝⎛+++⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡+⎟⎠⎞⎜⎝⎛++−⎟⎠⎞⎜⎝⎛+−=⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛++⎟⎠⎞⎜⎝⎛+−−=⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛++⎟⎠⎞⎜⎝⎛+−−=⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−+⎟⎠⎞⎜⎝⎛+⎟⎠⎞⎜⎝⎛−−∝⎭⎬⎫⎩⎨⎧⎥⎦⎤⎢⎣⎡+−++−−∝⎭⎬⎫⎩⎨⎧−−⎭⎬⎫⎩⎨⎧−−=∝==∑∑∏ntNtntctPtPtvtvvttvvtyNtvvtyvttvvttvvtvtytvtvyttvvytyyvttyvvtPvyftviiiiiiiiii,~2exp221exp121exp)(),|() and (where of lConditiona Full,~221exp221exp11221exp121221exp212121exp21exp2121exp21),|(,,|) and (where of lConditiona Full21012i2i1012i101i22ii2ii2ii2ii2ii2ii22ii22i2iii22iθθμμμθθμθμμθτσμμθμθθμθθμθθμθθθθθμμθθμθπθπμθμθτσθProblem 1c: Adding the additional modeling uncertainty of priors on \tau^2 and \sigma^2 shows essentially the same result. Large alternating-sign autocorrelation in Adlers versus very good results for Gibbs. Notice that we cannot use Adler’s on the variance parameter updates, since their full conditionals are not Gaussian. So these use Gibbs steps in either method. ()()()()()()()()()()()()⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛+−+⎭⎬⎫⎩⎨⎧+−−⎟⎠⎞⎜⎝⎛∝⎭⎬⎫⎩⎨⎧−−−⎟⎠⎞⎜⎝⎛=⎭⎬⎫⎩⎨⎧−⎟⎠⎞⎜⎝⎛⎭⎬⎫⎩⎨⎧−−⎟⎠⎞⎜⎝⎛∝⎥⎦⎤⎢⎣⎡∝=⎟⎟⎟⎟⎠⎞⎜⎜⎜⎜⎝⎛+−−⎭⎬⎫⎩⎨⎧+−−⎟⎠⎞⎜⎝⎛=⎭⎬⎫⎩⎨⎧−−−⎟⎠⎞⎜⎝⎛=⎭⎬⎫⎩⎨⎧−⎟⎠⎞⎜⎝⎛⎭⎬⎫⎩⎨⎧−−⎟⎠⎞⎜⎝⎛∝⎥⎦⎤⎢⎣⎡∝=∑∑∑∑∏∑∑∑∑∑∏+++1012101222/101222/210122/1012101210122/5101222/101210122/101222,12~121exp121exp11exp121exp1,|),,,|(: tof lConditiona Full22,125~21exp121exp11exp121exp1,|),,,|(: of lConditiona Fullbefore. as and of lConditiona FullμθμθμθμθμθμθτθθθθθμθσμθiininniniiiiiiiniinnniiniiiiinIGttvnvvtttttPtPvytPnynIGnyvvvnyvvvvyvvvPvyftyvPvProblem 2b: The first problem we notice is high autocorrelation and cross correlation in \beta_0 and \beta_1. The easiest fix, as in nearly all linear models, is to center the screening covariate. Doing do fixes the convergence problem. beta1 chains 1:3 sample: 30003 -0.02 -0.01 0.0 0.01 0.0 50.0 100.0 150.0beta0 -1.0 -0.5 0.0beta1 -0.01 0.0 0.01 0.02alpha chains 1:3 sample: 30003 0.2 0.4 0.6 0.8 0.0 2.0 4.0 6.0node mean sd 2.50% median 97.50% start sample alpha 0.714 0.082 0.539 0.721 0.849 1000 30003 beta0 -0.007 0.020 -0.046 -0.007 0.032 1000 30003 beta1 0.002 0.003 -0.004 0.002 0.007 1000 30003 The estimated mean effect of \alpha—the proportion of spatial variability among the clusters—is 0.712 (95% Credible Interval: 0.538, 0.848). We do not find any significant effect of the proportion of screening in a county upon late detection rates (exp(\beta_1) = 1.002; 95% Credible Interval: 0.996, 1.007). Problem 2c: As \alpha indicates, there is a lot of spatial variability compared to non-spatial variability. This can be seen through the predicted Standardized Mortality Ratios (\hat{SMR}) and spatial residuals for each county. The lower third of Minnesota has higher mortality rate than the middle third swatch of Minnesota. Not surprisingly, the spatial residuals are clustered similarly to the \hat{SMR}s. The \phi’s represent the non-spatial variability and, unsurprisingly, show no spatial pattern. Examining the intervals estimates for individual \phi’s we notice that they nearly all overlap zero, indicating that we might want to remove the heterogeneity effect from the model. \hat{\theta}: darker blue (lower tercile), lighter blue (middle tercile), red (upper tercile)\hat{\phi}: darker blue (lower tercile); lighter blue (middle tercile); red (upper tercile)\hat{SMR}: dark blue (lower tercile); light blue (middle tercile); red (upper tercile) Problem 2e: Incorporating survey measurement error did not change the affect of \beta_1. We alter the previous model by supposing that the rates, x_i | T_i, follow a N(T_i, 100) distribution. The “true” screening rates, T_i, have a N(50, 100) prior distribution for simplicity. Such a prior is somewhat vague but centered at approximately the overall screening rate. Fitting this model, we do not find any significant effect of the proportion of screening in a county upon late detection rates (exp(\beta_1) = 1.002; 95% Credible Interval: 0.997, 1.007). Incorporating the additional uncertainty about the screening covariate did not expand the interval for \beta_1 much, but if we had not fixed \mu_0, \delta, and \lambda, we would have seen a more substantial increase in the \beta_1 interval. beta1 chains 1:3 sample: 36000 -0.01 0.0 0.01 0.0 50.0 100.0 150.0 node mean sd 2.50% median 97.50% start sample alpha 0.708 0.082 0.534 0.714 0.847 1001 66000beta0 -0.007 0.020 -0.046 -0.007 0.032 1001 66000 beta1 0.002 0.003 -0.003 0.002 0.007 1001 66000 Problem 2f: It is simple to replace a fixed precision \delta for the