Linear Statistical Models

Department: 
MATH
Course Number: 
6266
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every fall semester

Basic unifying theory underlying techniques of regression, analysis of variance and covariance, from a geometric point of view. Modern computational capabilities are exploited fully. Students apply the theory to real data through canned and coded programs.

Prerequisites: 

MATH 4261, MATH 4262 or equivalent and MATH 6241

Course Text: 

At the level of Graybill, Theory and Application of the Linear Model

Topic Outline: 
  • Introduction: Examples of statistical problems covered by the general linear model including regression, ANOVA, ANCOVA, and time series
  • Geometric preliminaries: elementary analysis of simple linear regression through likelihood ratio tests and a geometric view of the result
  • The Moore-Penrose inverse: Elementary review of finite dimensional vector and inner product spaces including basis extension theorems, linear mappings, ranges, null spaces, projections, careful mathematical development of the Moore-Penrose inverse through projections
  • The Gauss-Markov theorem: estimability, proof of the Gauss-Markov theorem, and development of the covariance matrix of the least squares estimators. All are done in the most general (non-full rank) setting using the Moore-Penrose inverse
  • Testing the general linear hypothesis: distributional results for the likelihood ratio and the Wald statistic through simultaneous diagonalization of projections, application of results to data sets through canned programs like SAS and through Matlab or other specialized programs
  • Traditional formulas: Derivation of some well known formulas from applied regression and one-way ANOVA
  • Multiway ANOVA: proof of Cochran's theorem and derivation of formulas for the traditional models through Cochran's theorem
  • Individual comparisons: proof of Scheffe's simultaneous confidence intervals using the Wald statistic
  • log-linear models: analogous results for logistic regression through asymptotic theory of likelihood ratio tests, applications to real data using SAS