Reference no: EM131051466
The number of occurences of a rare, nongenetic birth effect in a five-year period for six neighboring counties is y = (1, 3, 2, 12, 1, 1). The counties have populations of x = (33, 14, 27, 90, 12, 17), given in thousands. The second county has higher rates of toxic chemicals (PCBs) present in the soil samples, and it is of interest to know if this town has a high disease rate as well. We will use the following hierarchial model to analyze these data:
• Yi|?i , xi ~ Po(?ixi);
• ?1, . . . , ?6|a, b ~ G(a, b);
• a ~ G(1, 1) ; b ~ G(10, 1).
(a) Describe in words what the various components of the hierarchical model represent in terms of the observed and expected disease rates.
(b) Identify the form of the conditional distribution of p(?1, . . . , ?6|a, b, x, D), and from this identify the full conditional distribution of the rate for each county p(?i |?-i, a, b, x, D).
(c) Write out the ratio of the posterior densities comparing a set of proposal values (a*, b*, ?) to values (a, b, ?). Note that the value of ?, the vector of county-specific rates, is unchanged. (d) Construct a Metropolis-Hastings algorithm which generates samples of (a, b, ?) from the posterior. Do this by iterating the following steps:
1. Given the current value (a, b, ?), generate a proposal (a * , b* , ?) by sampling a * and b * from a symmetric proposal distribution centered around a and b, but making sure all proposals are positive. Accept the proposal with the appropriate probability.
2. Sample new values of ?j 's from their full conditional distributions. Perform diagnostic tests on your chain and modify if necessary.
(e) Make posterior inference on the infection rates using the samples from the Markov chain. In particular,
(i) Compute marginal posterior distributions of ?1, . . . , ?6 and compare them to y1/x1, . . . , y6/x6.
(ii) Examine the posterior distribution of a/b, and compare it to the corresponding prior distribution as well as to the average of yi/xi across the six counties.
(iii) Plot samples of ?2 versus ?j for j ? 2, and draw a 45 degree line on the plot as well. Also, estimate P(?2 > ?j |x, D) for each j ?2 and P(?2 = max{?1, . . . , ?6}|x, D). Interpret the results of these calculations, and compare them to the conclusions one might obtain if they just examined yj/xj for each county j.
|
Benefits and challenges of business mobility
: 1. What are some of the benefits and challenges of business mobility? 2. What types of mobile business applications do you have personal experience with? Describe a positive and a negative experiences with mobile business applications.
|
|
Use the internet to compare goods and prices
: How often do you buy online? If you don't actually buy, do you use the Internet to compare goods and prices? Do you or your friends take advantage of low prices on used goods on eBay or other online sites like Craigslist?
|
|
United states consumer product safety commission website
: Visit the United States Consumer Product Safety Commission website. Click on "Recalls." Choose one product that has been recalled. Describe the product subject to recall, including the recall date, recall number, and the reason for the recall.
|
|
Communicate with environmentalists
: What kinds of problems can emerge if a firm doesn't communicate with environmentalists, the news media, and the local community? Do you know of any firms that aren't responsive to your community? What are the consequences?
|
|
Examine the posterior distribution of a/b
: Examine the posterior distribution of a/b, and compare it to the corresponding prior distribution as well as to the average of yi/xi across the six counties.
|
|
Type of corporate compliance processes
: If you were the CFO of the facility, what type of corporate compliance processes would you have in place? Which federal statutes would you include in your compliance process?
|
|
How capital gains arising from property investment are treat
: BULAW5916 Taxation Law & Practice Assignment. In your response you must explain what is meant by negative gearing and how capital gains arising from property investment are treated. You should refer to sections of legislation, tax rulings and case..
|
|
Find the collapsing load
: Find the collapsing load.
|
|
Describe how e-commerce began
: Describe how e-commerce began. Define the drivers that made e-commerce successful. And how they correlate each other? Therefore, the APA rules for formatting, quoting, paraphrasing, citing, and listing of sources are to be followed.
|