Department of Public Health Sciences

# BMTRY714

BMTRY 714 : Linear Models in Biology and Medicine

__DESCRIPTION__

The matrix representation of the general linear statistical model is studied through the implication, distribution, and partitioning of quadratic forms and their probability distributions. Estimation of parameters in the linear model by methods of maximum likelihood and least squares will be presented along with the accuracy and precision of these estimators. Estimability in both the full rank and less than full rank models is introduced. The test statistic for the general linear hypothesis is derived, and its distribution is determined under an assumption of normally distributed errors for both the null and a general alternative hypothesis. Sufficient examples are given to show its application to tests on means as well as in ANOVA and ANOCOVA. Students prepared in basic statistical methods and theory, and matrix algebra are eligible to take this course.

__TOPICS:__

- Elementary Matrix Concepts
- Kronecker Products
- Random Vectors
- Multivariate Normal Distribution Function
- Multivariate Normal Distribution Function
- Conditional Distributions of Multivariate Normal Random Vectors
- Distributions of Certain Quadratic Forms
- Quadratic Forms of Normal Random Vectors
- Independence
- The t and F Distributions
- Bhat’s Lemma
- Problems and Review
- Models That Admit Restrictions (Finite Models)
- Models That Do Not Admit Restrictions (Infinite Models)
- Sums of Squares and Covariance Matrix Algorithms
- Sums of Squares and Covariance Matrix Algorithms
- Expected Mean Squares
- Algorithm Applications
- Ordinary Least Squares Estimation
- Odds and Even-odds
- Best Linear Unbiased Estimators
- ANOVA Table for the Ordinary Lease-Squares Regression Function
- Weighted Least-Squares Regression 5.4
- Lack of Fit Tests
- Partitioning the Sum of Squares Regression
- The Model Y = Xb + E in Complete Balanced Factorials
- Maximum Likelihood Estimators
- Invariance Property, Sufficiency, and Completeness
- ANOVA Methods for Finding Maximum Likelihood Estimators
- The Likelihood Ratio Test for Hb = h
- Confidence Bands on Linear Combinations of b
- Replication Matrices
- Pattern Matrices and Missing Data
- Using Replication and Pattern Matrices Together
- General Balanced Incomplete Block Design
- Analysis of the General Case
- Matrix Derivations of Kempthorne’s Interblock and Intrablock Treatment Difference Estimators
- Model Assumptions and Examples
- The Mean Model Solution
- The Mean Model Analysis When Cov(E) = s2I 9.3
- Mean Model Analysis When Cov(E) = s2V 9.5

**PREREQUISITES AND OFFERINGS**

BMTRY 700, 701, 706, 707. Typically offered every other fall semester.