Outline

1.

The simple linear model: least squares estimation of parameters; inference on slope and intercept; use for prediction;assessment of fit-model diagnostics; relationship with correlation analysis; transformations on X and/or Y.

2.

Multiple linear regression: least squares estimation of parameters; matrix approach to estimation; how to do overall F-test; partial F and T-tests on individual coefficients; tests on subsets of coefficients; how to obtain and interpret standardized coefficients; assessment of collinearity and its effects; variance inflation factors; use and interpretation of R-squared and adjusted R-squared; polynomial models; use of indicator variables for categorical scale independent variables.

3.

Model building and variable selection; stepwise slection; backwards elimination; use of adjusted R-squared, Mallow's Cp, AIC as tools in model selection.

4.

Special problems in regression analysis; autocorrelation in time series; effects of measurement errors in independent variables.

5.

Basic features of experiments: randomization, replication, and blocking. Advantages of controlled experiments over observational studies.

6.

Factorial experiments: from single factor to multi-factor experiements: how to randomize the experiment; how to fit ANOVA models and partition total sums of squares and degrees of freedom; how to conduct F-tests for tests of significance; use multiple comparison procedures and conrasts; checking model assumptions.

7.

Randomized complete block design; when blocking is important; how to do randomization; ANOVA and post-hoc analysis.

8.

Fixed, random, and mixed models: Factorial experiments when one or more effects are random; how to determine proper F tests; method of moment estimation of variance components.

9.

Nested structures; split plots, repeated measures. Recognize these structures, be able to state an appropriate model and conduct ANOVA.

10.

2k and 3k factorials; fractional replication; confounding and aliasing of effects; estimation of effects; how to generate a design.

11.

Matrix algebra background: matrix algebtra operations; rank of a matrix, inverses and generalized inverses, positive definite matrices, quadratic forms, idempotent matrices, eigenvalues and eigenvectors.

12.

Distribution theory of random vectors, expected values, covariance and correlation matrices, distributions of linear combinations of random variables.

13.

Properties of multivariate normal distribution: marginal distributions, conditional distributions, partial correlation, distribution of linear combinations.; relationship to chi-square distribution.

14.

Distributions of quadratic forms: sums of squares as quadratic forms; expected values of quadratic forms.

15.

How to derive T, Chi-square and F distributions from multivariate normal distributions and know relationships between these distributions.

16.

Linear Regression model: matrix formulation and least squares estimation; maximum likelihhod parameter estimation for normal errors model; Gauss-Markov theorem; generalized least squares estimation; general linear hypothesis test; reduced vs. full model tests; tests and confidence intervals for coefficients and linear combinations of coefficients. Be able to derive and do these tests/CI's for normal errors model.

17.

Analyis of Variance models: how to parametrize the factor effects and cell means models; using side conditions (constraints) to achieve a full rank model; use of generalized inverses for estimation and testing in non-full rank model; estimability of parameters. Use of cell means models for analysis of unbalanced designs.