1. [[From Speigelhalter et al. (2004) book, Ex. 3.4 page 63)]] Suppose we are interested in the long-term
systolic blood pressure (SBP) in mmHg of a particular 60-year old patient named Marion. We take twoindependent readings 6 weeks apart, and their average is 130. We assume that SBP is measured with astandard deviation (SD) of σ = 5 . We also have considerable additional information about SBP’s that wecan incorporate in a prior distribution. Suppose that a survey in the same population revealed thatfemales aged 60 had a mean long-term SBP of 120 with SD 10.
(a) What is the posterior distribution for Marion’s long-term SBP?
(b) Draw a picture containing 3 graphs stacked on top of each other [use ♣❛r✭♠❢r♦✇ ❂ ❝✭✸✱✶✮✮ in R
• The bottom graph is the posterior density
Make all 3 graphs have the same horizontal axis, running from 100 to 140. You should be able tosee in your plot how, compared to the likelihood, the posterior density is slightly “shrunken” intoward the mean of the prior distribution.
(c) What is the posterior probability that Marion’s long-term SBP is greater than or equal to 135?
It turns out that in this problem standard nonBayesian confidence intervals are simply what we would get by letting the priorstandard deviation (which we took to be 10 here) grow to infinity. So the Bayesian methods can reproduce classical confidenceintervals, but can also incorporate actual prior information.
The remaining 3 problems concern the Helsinki Heart Study (New England Journal of Medicine, 1987 Nov12, 317(20):1237-45). Here is the abstract from an article that summarizes the findings.
In a randomized, double-blind five-year trial, we tested the efficacy of simultaneously elevating serum
levels of high-density lipoprotein (HDL) cholesterol and lowering levels of non-HDL cholesterol with
gemfibrozil in reducing the risk of coronary heart disease in 4081 asymptomatic middle-aged men (40 to
55 years of age) with primary dyslipidemia (non-HDL cholesterol greater than or equal to 200 mg per
deciliter [5.2 mmol per liter] in two consecutive pretreatment measurements). One group (2051 men)
received 600 mg of gemfibrozil twice daily, and the other (2030 men) received placebo. Gemfibrozil
caused a marked increase in HDL cholesterol and persistent reductions in serum levels of total,
low-density lipoprotein (LDL), and non-HDL cholesterol and triglycerides. There were minimal
changes in serum lipid levels in the placebo group. The cumulative rate of cardiac end points at five
years was 27.3 per 1,000 in the gemfibrozil group and 41.4 per 1,000 in the placebo group–a reduction
of 34.0 percent in the incidence of coronary heart disease (95 percent confidence interval, 8.2 to 52.6; P
less than 0.02; two-tailed test). The decline in incidence in the gemfibrozil group became evident in the
second year and continued throughout the study. There was no difference between the groups in the total
death rate, nor did the treatment influence the cancer rates. The results are in accord with two previous
trials with different pharmacologic agents and indicate that modification of lipoprotein levels with
gemfibrozil reduces the incidence of coronary heart disease in men with dyslipidemia.
In other words, among the 2051 men in the treatment (gemfibrozil) group, the number of heart attacks(“cardiac endpoints”) was 27.3 × 2051 = 56, and among the 2030 men in the control (placebo) group, the
number of heart attacks was 41.4 × 2030 = 84 . You are interested in two parameters θ
heart attack probabilities under treatment and the placebo. Suppose you use independent U [0, 1] priors forthese.
2. [[Metropolis sampling from scratch in R]] Use Metropolis sampling to find approximate posterior
distributions for θtreatment and θcontrol. Find 95% posterior probability intervals for θtreatment and θcontrol.
Also find a 95% interval for the percentage reduction in the rate of heart attacks, defined as
100. See if you get an answer anything like the 8.2 to 52.6 given in the
How do we find a confidence interval for the percentage reduction? Once you have done the MCMC, it’s really easyForexample, if you have generated 10,000 samples 1
θtreatment, . . . , θtreatment and θcontrol, . . . , θcontrol, you can then get a sample of
10,000 percentage reduction values in the obvious way: form
(θcontrol − θtreatment)/θcontrol for i = 1, . . . , 10000. Then you can
get the confidence intervals by finding the 2.5 and 97.5 quantiles of these 10000 values.
3. Actually we don’t really need Markov chain Monte Carlo for this study (and for most of it we don’t need
Monte Carlo at all), since this is a simple conjugate prior scenario: the Uniform prior is in the Betafamily, so the posterior distributions for θtreatment and θcontrol are also in the Beta family.
(a) Find 95% intervals for θtreatment and θcontrol without simulation (hint: use the R function qbeta).
(b) For a 95% interval for the percentage reduction, let’s use Monte Carlo simulation but without the
“Markov chain” part. You start by getting samples of θtreatment and θcontrol from the appropriate Betadensities (using rbeta). Then, you can use your two samples to form a confidence interval for thepercentage reduction by the simple transformation method described above.
4. Repeat the analysis using JAGS to do the Monte Carlo sampling. [[My model file for this problem has 4
lines inside the ♠♦❞❡❧④✳✳✳⑥ thing; 2 lines for the likelihood and 2 lines for the prior.]]
∗ By the way, if we were making a standard nonBayesian confidence interval for the percentage reduction, we would need to
know a special formula for it. That is, for each different transformation of the unknown parameters (for example, reduction in risk,or odds ratio, or whatever) we will have to go though a slightly complicated procedure for making a confidence interval, and to top itoff, the formula is an approximation, not exact. Here we do something I think is a lot more straightforward: just take our samples ofthe parameters, do whatever transformations we want to them, and then find the 2.5 and 97.5 percentiles to get a confidence interval.
MS Bjerneld worked as a nurse in a camp for Nambian refuges in Angola for the Africa Groups of Sweden. She is telling us about the conditions in the camp as well as the cooperation with SWAPO. She was later the administrator in Sweden for the Medical Aid for SWAPO project. A position the interviewer had held at the beginning of the project. She was lso a board member of the Africa Groups of Swed
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