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The final exam for a statistics course, specifically for stat 8320. The exam covers topics such as mixed model analysis, hypothesis testing, and data interpretation. Students are required to write down models in mixed format, find hierarchical and marginal formulations, determine the variance-covariance matrix, perform statistical tests, and interpret fixed effects values. The exam also includes problems related to poisson distribution and newton-raphson method.
Typology: Exams
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Problem 1: Two different time release formulations for an osteoporosis drug are being considered. The goal is to keep the drug in the system for the longest time possible. To study the two proposed formulations, fifteen women are enrolled in the study. Each of the women is randomly assigned to one of three groups (the two drugs and a control). They are then given the drug and then measurements of their serum level are taken at four times. It is believed that the appropriate model for this data is
Yitk = β 0 + b 0 k + (β 1 i + b 1 k)t + ≤itk.
Here it is assumed that b 0 , b 1 , ≤ are independently distributed from normal distributions with variances σ^20 , σ^21 , σ^2 ≤ , respectively. (7 pts.) Write down this model in the mixed model format. Assume that the errors are iid N(0, σ^2 I). Show a minimum of three values of Y for full credit.
(5 pts.) Assume that the random effects are normally distributed with mean 0 and variance-covariance matrix D. Find both the hierarchical and marginal formulations of the mixed model. Be sure to define any vectors and matrices not defined in the previous part.
(7 pts.) Assume that the times are equally spaced (2, 4, 6, and 8 days). Find the form of the variance- covariance matrix D which is implied by the model. Hint: find the covariance between two observations which are correlated.
Problem 3: An avid local gardener is attempting to determine which variety of zucchini plant will grow best in his garden. There are three different varieties of plant that he can grow in his garden, and he obtains 7 of each type. He plants all 24 plants in his garden and lovingly tends them all throughout the growing season. He cuts the zucchini every Saturday for four weeks, taking all those which are at least 6” long. As data, the gardener records the number of zucchini harvested from each plant each week. He believes that these counts arise from a Poisson distribution,
pr(yi) = μy i iexp(−μi) yi!
(7 pts.) What type of model do you believe would be appropriate for this data? Be sure to include the link function which you believe to be appropriate. Additionally, describe the linear portion of the model. Be sure to indicate which factors are fixed and which are random.
(7 pts.) Look at the attached output. Interpret the fixed effects values for this model. In particular, deter- mine which of the types of plants bears the largest number of zucchinis.
(7 pts.) Show that the Poisson distribution is a member of the exponential family. Also, find the mean and variance of this distribution using the exponential form.
Problem 4: Consider data which is believed to arise from the following model:
yi = αxi^ + ≤i.
The data are the pairs (y,x) = (0.16,0.25), (1.28,0.50), (2.43,0.75), (2.24,1.00), (1.26,1.25), (1.78,1.50), (2.58,1.75), (3.94,2.00). (6 pts.) Find a starting value for α to use in an iterative approach. Describe the procedure that you would use.
(14 pts.) Assume that the starting value for α is 0.50 from the first part. Find all of the forms needed for the Newton-Raphson method. Use them to update the value of α for one iteration (perform the calculation).
Problem 6: BONUS: (5 pts.) Assume that y is an observation from the Normal distribution with mean and variance listed below:
μ =
Find the distribution of ( 2 y 1 − 3 y 3 y 2 + y 3