ST 705: Linear Models and Variance Components

Section 001
Spring 2019
3 Credit Hours
Course website: https://maityst705.wordpress.ncsu.edu/

Course Description

The course covers the theory underlying linear statistical models and provides the necessary theoretical foundation for understanding many advanced statistical methods and for doing methodological research in statistics.

Learning Outcomes

Students will learn basic techniques and results related to the theory of linear models at a rigorous level. Upon completion of the course students will learn the following topics:

General linear model
Generalized inverses; solving linear equations; projections;
Linear least squares and the normal equations;
Estimability;
Gauss-Markov Theorem;
Generalized least squares;
Multivariate normal distribution; central and non-central Chi-squared and F distributions
Distributions of quadratic forms;
General linear hypothesis;
Linear models with random effects; variance components

Course Structure

The course meets for lectures twice per week. Students are required to complete homework (about ten homework assignments), two mid-term exams, and a final exam.

Course Policies

Unexcused late homework will not be accepted. The final homework average will be computed after dropping the two lowest grades. You are permitted to work together on the homework sets, but each student is responsible for their final write-up of each assignment. When asked to solve problems using a computer, please provide well commented and neatly written computer code, relevant computer output, circle all relevant results, and give an appropriate discussion. Examinations will be a closed book and closed notes. Students may bring calculators to all exams, in addition to pen/pencil and scratch papers. No cell phones or other electronic devices should be in sight or used in any way during exams. Course policies are subject to change.

Instructors

Arnab Maity - Instructor
Email: amaity@ncsu.edu
Web: https://www.stat.ncsu.edu/people/maity/
Phone: 919-515-1937
Office location: SAS 5240
Office hours: Tuesday 3:00 - 4:30PM

Chelsea Robalino - Teaching Assistant
Email: cprobali@ncsu.edu
Office Location: 1101 SAS Hall (Statistics Tutorial center)
Office hours: Tuesday 11:30-1pm and Thursday 10-11:30am

Course Meetings

Lecture Days: Mo/We
Time: 11:45AM - 1:00PM
Location: 01108 SAS Hall
This meeting is required.

Laboratory days: Mo
Time: 10:40AM - 11:30AM
Location: 01108 SAS Hall

Requisites and Restrictions

Prerequisites: None
Co-requisites: ST 702 - Statistical Inference II
Restrictions: None

Course Materials

Textbooks: A Primer on Linear Models by John F. Monahan, Chapman & Hall/CRC Press, 2008. This textbook is required.

Expenses: None
Materials: None
Additional References: These references are not required.
Theory and Application of the Linear Model, F. A. Graybill, Wadsworth
Linear Models, S. R. Searle, Wiley
Linear Regression Analysis, G. A. F. Seber, Wiley
Matrix Algebra: Theory, Computations, and Applications in Statistics, James E. Gentle
Matrix Algebra From A Statistician’s Perspective, D. A. Harville, Springer
Matrix Algebra as a Tool, A. Hadi, Duxbury
The Matrix Cookbook, K. B. Petersen & M. S. Pedersen

General Education Program (GEP) Information

GEP Category: This course does not fulfill a General Education Program category.
GEP Co-requisites: This course does not fulfill a General Education Program co-requisite.

Transportation

This course will not require students to provide their own transportation. Non-scheduled class time for field trips or out-of-class activities is NOT required for this class.

Safety & Risk Assumptions

None.

Grading

The final numeric grade will be computed based on the following components. Midterm exam dates are subject to change.

Homeworks (25%): The final homework average will be computed after dropping the two lowest grades. As the lowest two scores are dropped, no late assignments are accepted.
Midterm Exam I (25%): Tentatively scheduled on Feb 11 (Monday) in class during lab and lecture time. The examination will be closed book and closed notes. The student must contact the instructor in advance if s/he is likely to miss the scheduled midterm exam.
Midterm Exam II (25%): Tentatively scheduled on Apr 01 (Monday) in-class during lab and lecture time. The examination will be closed book and closed notes. The student must contact the instructor in advance if s/he is likely to miss the scheduled midterm exam.
Final Exam (25%): Scheduled on Apr 29 (Monday) in class (8:00PM - 11:00PM, 1108 SAS Hall)

This Course uses Standard NCSU Letter Grading:
A+ ≥ 97 > A ≥ 93 > A– ≥ 90
B+ ≥ 87 > B ≥ 83 > B– ≥ 80
C+ ≥ 77 > C ≥ 73 > C– ≥ 70
D+ ≥ 67 > D ≥ 63 > D– ≥ 60 > F.

Requirements for Credit-Only (S/U) Grading

In order to receive a grade of S, students are required to take all exams and quizzes, complete all assignments, and earn a grade of C- or better. Conversion from letter grading to credit only (S/U) grading is subject to university deadlines. Refer to the Registration and Records calendar for deadlines related to grading. For more details refer to http://policies.ncsu.edu/regulation/reg-02-20-15.

Requirements for Auditors (AU)

Information about and requirements for auditing a course can be found at http://policies.ncsu.edu/regulation/reg-02-20-04. Auditors are expected to attend class regularly and submit homework on the same schedule as the other students. The final grade for auditors (AU or NR) will be based on their final homework average (final homework grade will be calculated by dropping the two lowest grades). A final homework score of at least 70% is required for an AU.

Policies on Incomplete Grades

If an extended deadline is not authorized by the Graduate School, an unfinished incomplete grade will automatically change to an F after either (a) the end of the next regular semester in which the student is enrolled (not including summer sessions), or (b) by the end of 12 months if the student is not enrolled, whichever is shorter. Incompletes that change to F will count as an attempted course on transcripts. The burden of fulfilling an incomplete grade is the responsibility of the student. The university policy on incomplete grades is located at http://policies.ncsu.edu/regulation/reg-02-50-03

Additional information relative to incomplete grades for graduate students can be found in the Graduate Administrative Handbook in Section 3.18.F at http://www.fis.ncsu.edu/grad_publicns/handbook/

Late Assignments

Homework is due in class on the due date. No late assignments are accepted. If an emergency arises that prevents you from completing your work on time, please email the instructor as soon as possible so that arrangements can be made for you to keep up in the class.

Attendance Policy

For complete attendance and excused absence policies, please see http://policies.ncsu.edu/regulation/reg02-20-03. Students are expected to attend all lectures and exams.
Absences Policy: None.
Makeup Work Policy: No late HW submissions are accepted as two lowest scores will be dropped. Students who are unable to attend an exam for a legitimate unavoidable reason may take a make-up exam only if the student provides suitable documentation of the delay and they are able to take the make-up in a very timely manner. If a make-up can’t be taken then one of the midterm exams will be reweighted for the missing midterm exams. This may only be done with one of the exams. Students are required to take at least one of the two midterm exams on scheduled time otherwise a grade of F will be assigned.

Academic Integrity

Students are required to comply with the university policy on academic integrity found in the Code of Student Conduct found at http://policies.ncsu.edu/policy/pol-11-35-01
Academic Honesty: See http://policies.ncsu.edu/policy/pol-11-35-01 for a detailed explanation of academic honesty.
Honor Pledge: Your signature on any test or assignment indicates ”I have neither given nor received unauthorized aid on this test or assignment.”
While students are allowed to work in groups on homework, no one should copy directly from someone else’s work (this includes present or past students). It is strongly urged that everyone work on their own as much as possible.

Electronically-Hosted Course Components

Students may be required to disclose personally identifiable information to other students in the course, via electronic tools like email or web-postings, where relevant to the course. Examples include online discussions of class topics and posting of student coursework. All students are expected to respect the privacy of each other by not sharing or using such information outside the course.

Electronically-hosted Components: https://maityst501.wordpress.ncsu.edu/

Accommodations for Disabilities

Reasonable accommodations will be made for students with verifiable disabilities. In order to take advantage of available accommodations, students must register with the Disability Services Office at Suite 2221, Student Health Center, Campus Box 7509, 919-515-7653. For more information on NC State’s policy on working with students with disabilities, please see the Academic Accommodations for Students with Disabilities Regulation (REG 02.20.01).

Non-Discrimination Policy

NC State University provides equality of opportunity in education and employment for all students and employees. Accordingly, NC State affirms its commitment to maintain a work environment for all employees and an academic environment for all students that is free from all forms of discrimination. Discrimination based on race, color, religion, creed, sex, national origin, age, disability, veteran status, or sexual orientation is a violation of state and federal law and/or NC State University policy and will not be tolerated. Harassment of any person (either in the form of quid pro quo or creation of a hostile environment) based on race, color, religion, creed, sex, national origin, age, disability, veteran status, or sexual orientation also is a violation of state and federal law and/or NC State University policy and will not be tolerated. Retaliation against any person who complains about discrimination is also prohibited. NC State’s policies and regulations covering discrimination, harassment, and retaliation may be accessed at http://policies.ncsu.edu/policy/pol-04-25-05 or http://www.ncsu.edu/equal_op/. Any person who feels that he or she has been the subject of prohibited discrimination, harassment, or retaliation should contact the Office for Equal Opportunity (OEO) at 919-515-3148.

N.C. State University Policies, Regulations, and Rules (PRR)

Students are responsible for reviewing the PRRs which pertain to their course rights and responsibilities. These include http://policies.ncsu.edu/policy/pol-04-25-05 (Equal Opportunity and Non-Discrimination Policy Statement), http://oied.ncsu.edu/oied/policies.php (Office for Institutional Equity and Diversity), http://policies.ncsu.edu/policy/pol-11-35-01 (Code of Student Conduct), and http://policies.ncsu.edu/regulation/reg-02-50-03 (Grades and Grade Point Average).

Miscellany

Attendance is expected at all lectures.
Disputes about homework/exam grading must be brought to the instructor’s attention within one week after the graded paper is returned.
Students may discuss the homework problems with others. However, each student must submit their own independent write-up of the solutions. Copying someone else’s work—including online resources—is not acceptable and may result in disciplinary action. The instructor is committed to upholding the university’s policy on academic integrity, as described in the Code of Student Conduct. http://policies.ncsu.edu/policy/pol-11-35-01
Students are responsible for reading, understanding, and adhering to the university’s policies, regulations, and rules. https://policies.ncsu.edu/

Course Schedule

NOTE: The course schedule is subject to change.

Basic linear algebra: Vector spaces, Null space, and Column space, Idempotent matrix, the Projection matrix, Matrix decompositions (QR, Cholesky, eigenvalues and eigenvectors, SVD) (2 lectures)
Solving a system of linear equations: the existence of solutions, generalized inverse (2 lectures)
General linear models: different classes of linear models and related assumptions (1 lecture)
Least squares method: normal equations and their general solutions (2 lectures)
Reparameterization (1 lectures)
Identifiability and Estimability: linearly estimable functions of model parameters, least squares estimators of estimable functions (3 lectures)
The Gauss-Markov model: best linear unbiased estimators, the Gauss-Markov theorem, variance estimation, model misspecification, underfitting and overfitting (3 lectures)
The Aitken model: generalized least squares, estimability, Aitken’s theorem (2 lectures)
Imposing constraints: estimation of model parameters with additional constraints, restricted normal equations and their solution, best estimation in a constrained model (2 lectures)
Distributional theory: Multivariate normal, Chi-squared, F and t distributions, distribution of quadratic forms, Cochran’s theorem, maximum likelihood estimation in Gaussian linear models (2 lectures)
Hypothesis testing: General linear hypotheses, Testable hypotheses, General linear test, Likelihood ratio test (2 lectures)
Confidence intervals and multiple comparisons: confidence region, simultaneous confidence intervals (Bonferroni method, Scheffe’s method, Tukey’s method), Multiple testing (2 lectures)
Linear mixed effects model: definition of general mixed effects models, maximum likelihood and restricted maximum likelihood estimation, analysis of variance in mixed effects models, best linear unbiased prediction (3 lectures)