Statistics General Linear Model Midterm Exam

Q1. There are a series of papers by de Souza et al. on the generalized linear models in astronomy. These articles can be found either from the links provided below, or from the Blackboard where three pdf files were uploaded. In this question, we focus on the second paper. You may read other papers if you are interested in.

• R.S. de Souza, E. Cameron, M. Killedar, J. Hilbe, R. Vilalta, U. Maio, V. Biffi, B. Ciardi, J.D. Riggs (2015). The overlooked potential of generalized linear models in astronomy, I: binomial regression, Astronomy and Computing, 12, 21-32.
• J. Elliott, R.S. de Souza, A. Krone-Martins, E. Cameron, E.E.O. Ishida, J. Hilbe (2015). The overlooked potential of generalized linear models in astronomy, II: gamma regression and photometric redshifts. Astronomy and Computing, 10, 61-72.
• R. S. de Souza, J. M. Hilbe, B. Buelens, J. D. Riggs, E. Cameron, E. E. O. Ishida, A. L. Chies-Santos, M. Killedar (2015). The overlooked potential of generalized linear models in astronomy, III: Bayesian negative binomial regression and globular cluster populations. Monthly Notices of the Royal Astronomical Society, 453, 1928-1940.

(a) Read the second paper by Elliott et al. on gamma regression and photometric redshifts. Write a brief summary of section 2 overview of regression methods, page 62-64.

(b) Appendix A (page 68-69) provides instructions to perform the photometric redshift estimation using the R package. Run these R codes line by line, and explain the purpose and output of each command line. Elliott et al. also provide python codes in Appendix B. If you prefer python, you can run and explain the python codes. Note, you only need to choose either R or python.

Q2. Consider the data from All Time World Rankings. We use man's 100 meter dash records and woman's 100 meter dash records.

First, summarize these records by using a table with columns Time Record (second), Age (year), and Gender (Female or Male).

For the time record in each age and gender group, you should use the fastest times without wind assistance. Based on Rule 260.14(c) of IAAF Competition Rules 2016-2017, if a tail wind exceeds 2 meters per second the result cannot be registered as a record on any level. So you should use the fastest times among the wind speed less than or equal to +2 m/s.

For age, use the lower bound of each age group. For instance, the age for age group M35-39 is 35, the age for age group W90-94 is 90.

(a) Summarize the record of each age and gender group and form an R-readable table. For example, the first several rows of the table may be

(b) Consider time as the response variable (y) and age as the explanatory variable (x). For female students, use woman's record; for male students, use man's record. Fit the models

y = β₁₀ + β₁₁x

and

y = β₂₀ + β₂₁x + β₂₂x².

Include your R codes and report your estimates. Does the extra quadratic term appear necessary?

(c) Denote the estimates in part (a) of the intercept of model y = β₁₀ + β₂₀x as b_0F in woman's record model, and as b_0M in man's record model.

Include gender as an additional explanatory variable (v), and v = 1 corresponds to woman's record, and v = 0 corresponds to man's record. Consider the model

y = β₃₀ + β₃₁x + β₃₂v.

Include your R codes and report your estimates. How does gender appear to affect the records?

(d) For female students, compare β^ˆ₃₀ + β^ˆ₃₂ and b_0F. For male students, compare β^ˆ₃₀ and b_0M. Explain the difference.

(e) For female students, use woman's record; for male students, use man's record. Using the data fit a Gamma generalized linear model. Interpret your findings and compare with part (b). Include your R codes, and write down the link function you choose, and the equation of your fitted model.

(f) Show that the density of inverse Gaussian distribution lies in the exponential family, and write the distribution in the canonical form of a generalized linear model. Then repeat part (e) using an inverse Gaussian generalized linear model.

Q3. Two items A and B are weighed on a balance, first separately and then together, to yield observations y₁, y₂, and y₃. Say, suppose the true weights of A and B are α_A and α_B, we have

y₁ = α_A + ε₁

y₂ = α_B + ε₂

y₃ = α_A + α_B + ε₃

(a) If ε_i ∼ N(0, σ²_ε), i = 1, 2, 3, find the reasonable estimates of α_A and α_B. Show your work.

(b) If ε_i ∼ N(0, σ²_ε) for i = 1, 2, and ε₃ ∼ N(0, k²σ²_ε), where constant k > 1, find the reasonable estimates of α_A and α_B. Show your work.

(c) Let y₁ = 41, y₂ = 53, y₃ = 97, k = 1.2. Choose a suitable function in R, and find the estimates of α_A and α_B in (a) and (b). Include your R codes, and highlight the key R function you use. Compare the estimates of α_A and α_B in (a) and (b) and explain the differences.

Attachment:- Assignment File.rar