 # Linear Models

Linear Models
Theory of estimation and testing in linear models. Analysis of full-rank model, over-parameterized model, cell-means model, unequal subclass frequencies, and missing and fused cells. Estimability issues, diagnostics.
STAT
535
 Hours 3.0 Credit, 3.0 Lecture, 0.0 Lab Prerequisites MATH 313 & STAT 340 Taught Fall
Course Outcomes:

### Course Outcomes

Upon successful completion of this course, the student will be able to:

### Understand Derivation

Understand derivation and distribution of linear and quadratic forms.

### Understand Definitions

Understand definitions of non-central chi-square, t, and F distributions.

### Derive Maximum Likelihood

Be able to derive maximum likelihood estimates of parameters in a linear model with normal, independent errors.

### BLUE and MVUE

Understand Best Linear Unbiased Estimation (BLUE) and Minimum Variance Unbiased Estimation (MVUE) in linear models.

### Estimation

Know how to estimate in both the unconstrained and constrained model.

### Hypothesis Tests

Know how to implement hypothesis tests in the normal linear model.

### Cell Means Model

Be able to implement the cell means model in one-way and multiway fixed designs.

### Multiple Comparison

Know how to test in a multiple comparison setting.

### Lack of Fit

Be able to derive and use measures of lack of fit and importance.

### Sums of Squares

Understand the difference and compute Type I and Type III sums of squares.

### Missing Cells

Understand and be able to compute tests and estimates when a design has data missing in some cells.

### Gaussian Linear Models

Demonstrate the application of Gaussian Linear Models for observational studies and designed experiments.

### Solve problems

Solve problems using random vectors.

### Understand derivation

Understand derivation and distribution of linear and quadratic forms.

### Understand definitions

Understand definitions and properties of multivariate normal, non-central chi-square, t, and F distributions.

### Derive maximum likelihood

Derive maximum likelihood estimates of parameters in a linear model with normal, independent errors.

### Linear models estimates

Derive the properties of linear models estimates (Gauss-Markov Theorem, Wald tests).

### Unconstrained and con-strained models

Derive tests on linear hypotheses by estimation of both the unconstrained and con-strained model (full and reduced LRT/ANOVA).

### Cell means model

Apply the cell means model in one-way and multiway fixed designs, interpret parame- ters from alternative model reparameterizations, estimability.

### Regression

Explore consequences of model assumption violations and use regression diagnostics to identify possible model violations.

### Theoretical consequences

Derive theoretical consequences of overfitting and underfitting in model selection.