STAT 701 -- EXAM 1 REVIEW SHEET

I. Simple Linear Regression (SLR)
 A. Basic Ideas
  1. What Is a Model?
  2. SLR Model and Terminology
  3. Deterministic and Random Component
  4. Model Assumptions
 B. Estimation of Regression Line
  1. Least Squares Method
    a. Idea behind least squares
    b. Nice properties of least-squares estimators
  2. Interpretion of estimates
  3. Fitted Values and Residuals
 C. Estimating the Error Variance
  1. Why is this necessary?
  2. What is our unbiased estimator of sigma^2?
    a. Corresponding estimate of sigma
 D. Assumption of Normal Errors
  1. What does this allow us to do?
  2. Implication for the least-squares estimates
 E. Inference in SLR
  1. Inference about the true slope
    a. Sampling distribution of b_1
    b. CI for the true slope beta_1
    c. Testing whether beta_1 = 0
    d. Implication about relationship between Y and X
  2. CI for the mean response, E(Y_h)
    a. Where is this CI narrowest?
    b. Confidence band for entire regression line
 F. ANOVA Approach to Regression
  1. Partitioning the Total Variation in Y
    a. SSTO, SSR, SSE
    b. What does each SS measure?
    c. Degrees of freedom for each SS in SLR
    d. MSR and MSE
    e. Summarizing this in the ANOVA table
  2. Overall (ANOVA) F-test
    a. What is the test statistic F*?
    b. The intuition behind this test
    c. General Linear Test (concept of reduced vs. full model)
 G. Measures of Linear Fit & Association
  1. Coefficient of Determination, R^2
    a. Possible Values of R^2
    b. Interpretion of R^2
  2. Correlation Coefficient r
    a. Possible Values of r
    b. Interpretion of r
  3. Limitations of R^2 and r
 H. Cautions about Regression
  1. Extrapolation, Future Values, Causation, Multiple Inferences

II. Multiple Linear Regression (MLR)
 A. Basics of the MLR model
  1. Form of the Model
  2. Interpretation of Regression Coefficients
  3. The General Linear Model
  4. Writing a General Linear Model in Matrix Terms
    a. Y vector
    b. X matrix
    c. beta vector
    d. epsilon vector
    e. vector of estimated coefficients
    f. vector of fitted values
    g. vector of residuals
  5. Interpretations of regression coefficients in MLR
 B. Analysis of Variance in MLR
  1. SSTO, SSR, SSE
  2. Degrees of Freedom for each SS
  3. Overall ANOVA F-test
    a. Null and alternative hypotheses
    b. Test statistic value
  4. Coefficient of Multiple Determination R^2
  5. Adjusted R^2
 C. Inference about Individual Regression Coefficients
  1. CI for an individual beta
  2. Test for whether an individual beta = 0
    a. Tests marginal effect of individual predictor
    b. "in the presence of" other predictors in the model
  3. CI for the mean response, E(Y_h)
    a. Confidence region for entire regression surface
 D. Checking Assumptions through Residual Plots
  1. What values are plotted on the axes of a residual plot?
  2. Checking for model misspecification
  3. Checking for non-constant error variance
  4. Checking for departures from normality
 E. Extra SS and F-tests
  1. Behavior of SSE as predictors are added to the model
  2. Reduced model vs. Full model
  3. Testing whether some (but not all) predictors can be dropped
    a. Null and alternative hypotheses
    b. Test statistic value
  4. Coefficients of Partial Determination
    a. How are they related to Extra SS?
 F. Multicollinearity
  1.  What is multicollinearity?
  2.  Common Problems Caused by Multicollinearity
  3.  Detecting Multicollinearity with VIFs
  4.  Possible Remedies for Multicollinearity