STAT 513 -- EXAM 3 REVIEW SHEET 

0. Leftover Simple Linear Regression Topic    
  A. Prediction Interval for the Response of a New Observation at a particular x-value
    1. Prediction Error
    2. Interpretation of PI
    3. Comparison with CI for E(Y) at a particular x-value
I. Multiple Linear Regression (MLR)
   A. General MLR model with k predictors
    1. General Linear Model in Matrix Terms
      a. Y vector
      b. X matrix
      c. beta vector
      d. epsilon vector
    2. Estimating the betas in MLR
      a. Formula for vector of estimated coefficients
      b. Fitting a quadratic regression model
      c. variances and covariances for estimated coefficients
    3. Properties of Least-squares Estimators in MLR
      a. Unbiasedness of estimated coefficients
      b. Formulas for SSE and MSE in MLR
      c. Distribution of estimated coefficients
      d. Distribution of (n-k-1)MSE/(sigma^2)
   B. Inference in MLR
    1. Tests and CIs about a0*beta_0 + a1*beta_1 + ... + ak*beta_k
    2. Tests about an individual beta_i
      a. Tests marginal effect of individual explanatory variable
      b. "in the presence of" other predictors in the model
    3. Test statistic, rejection region for test of H_0: beta_i = 0
    4. Confidence Interval for an individual beta_j
    5. CI for the E(Y) at a particular X value
    6. Interpreting the CIs
    7. Prediction Interval for 'new' response value, Y*
   C. Reduced and Complete Models
    1. Behavior of SSE as independent variables are added to the model
    2. Writing Reduced model and Full model
    3. Testing whether some (or all) explanatory variables are useless
      a. Null and alternative hypotheses
      b. Cochran's Theorem
      c. Test statistic and Rejection region
II. Correlation Model
   A. Population correlation coefficient rho
    1. What does it measure?
    2. Its estimate, the sample correlation coefficient r
   B. Testing whether rho=0
    1. Relationship to t-test about beta_1 in SLR
   C. Testing whether rho = some nonzero number
    1. Fisher's z-transformation
    2. Finding a large-sample CI for rho using back-transformation
   D. Coefficient of Determination
    1. Formula for r^2 in SLR
    2. Interpretation of r^2
    3. Coefficient of multiple determination R^2 in MLR
III. Linear Model for ANOVA
   A. Basic Terms and Setup
    1. Factor and Levels
    2. One-way ANOVA Model
    3. Goal of ANOVA
    4. One-way ANOVA model equation
   B. ANOVA as a Linear Model
    1. Dummy variables
    2. Baseline level
    3. Comparing E(Y) across levels
    4. Testing whether betas all equal 0
    5. Structure of X matrix for an ANOVA model
    6. F-test
    7. Bonferroni approach to compare each pair of level means
IV. Introduction to Bayesian Inference
 A. Rationale behind Bayesian Methods vs. Classical Methods
   1. Treating Parameters as Random Variables
   2. Specification of Prior Distribution
 B. Finding the Posterior
   1. Writing the Likelihood and Prior
   2. Joint pdf of data and parameter
   3. Marginal distribution of data
   4. Posterior as the conditional distribution of parameter given the data
   5. Prior and Likelihood => Posterior
   6. Proportionality and role of Normalizing Constant
 C. Conjugate Priors and Examples
   1. Definition of Conjugate prior for a sampling model
   2. The Binomial likelihood & Beta prior Bayesian Model
   3. The Exponential/Gamma likelihood & Exponential/Gamma prior Bayesian Model
   4. Relationship of posterior to sufficient statistics
   5. The Normal likelihood & Normal prior Bayesian Model
 D. Bayesian point estimators
   1. Posterior mean as a combination of the sample mean/MLE and prior mean
   2. Posterior median, posterior mode
   3. Point estimator for a function of the parameter
 E. Effect of Prior on Posterior Inference
   1. Choosing prior paramters to reflect prior knowledge
   2. Noninformative priors
 F. Posterior Credible Intervals
   1. Formal Definition of Credible Interval
   2. Differences in Interpretation of Credible Interval vs. Confidence Interval
   3. Credible Intervals and Restricted Parameter Spaces