Properties of the LS Estimates Inference for Individual Coefficients

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Presentation transcript:

Properties of the LS Estimates Inference for Individual Coefficients Lecture 8 Review of Lecture 7 Properties of the LS Estimates Inference for Individual Coefficients Procedure for Solving Problems An Example 11/16/2018 ST3131, Lecture 8

Review: Interpret the Regression Equation 1 Straight line 2 Plane >= 3 Hyper-plane Interpret the Regression Coefficients Interpretation 1: from the view of response change 11/16/2018 ST3131, Lecture 8

Interpretation 2: The MLR coefficients and the associated p SLR coefficients are not the same unless all predictor variables are uncorrelated. For example, 11/16/2018 ST3131, Lecture 8

Properties of LS Estimators We first show that the LS estimators are Linear combinations of Y1, Y2, …, Yn First note that b=(n-1)Cov(X,Y) is linear combination of Y1,Y2,…Yn. 11/16/2018 ST3131, Lecture 8

3). BLUE estimators (Best Linear Unbiased Estimators) It follows that Properties: 1). Linearity 2). Unbiased 3). BLUE estimators (Best Linear Unbiased Estimators) 11/16/2018 ST3131, Lecture 8

Variances of the Estimates In particular, when p=1, we have 11/16/2018 ST3131, Lecture 8

Noise variance estimator and Standard Errors 4) Normality 11/16/2018 ST3131, Lecture 8

Inferences for Individual Coefficients Hypothesis Testing Confidence Interval 11/16/2018 ST3131, Lecture 8

Procedure for Solving Problems Step 1. List the given conditions from the question statement. Step 2. Express the question in mathematical formula. Step 3. Plug-in the known conditions or results from other parts of the problem. Step 4. Simplify the results. Step 5. State your conclusion in ordinary languages. 11/16/2018 ST3131, Lecture 8

n=30. Answer the following questions: Example For the Supervisor Performance Data, you are given the regression coefficient table for the MLR model: n=30. Answer the following questions: (1). Find SSE and its degrees of freedom (2). Find the proportion of the variability in Y explained by Regression 11/16/2018 ST3131, Lecture 8

Regression Analysis: Y versus X1, X2, X3, X4, X5, X6 The regression equation is Y = 10.8 + 0.613 X1 - 0.073 X2 + 0.320 X3 + 0.082 X4 + 0.038 X5 - 0.217 X6 Predictor Coef SE Coef T P Constant 10.79 11.59 0.93 0.362 X1 0.6132 0.1610 3.81 0.001 X2 -0.0731 0.1357 -0.54 0.596 X3 0.3203 0.1685 1.90 0.070 X4 0.0817 0.2215 0.37 0.715 X5 0.0384 0.1470 0.26 0.796 X6 -0.2171 0.1782 -1.22 0.236 S = 7.068 R-Sq = 73.3% R-Sq(adj) = 66.3% 11/16/2018 ST3131, Lecture 8

(4) Find the effect of X1 and its estimated variance (3). Find the total variability of Y and the variability explained by Regression (4) Find the effect of X1 and its estimated variance 11/16/2018 ST3131, Lecture 8

(6). Without any calculation, can you give the intercept of the fitted line? What is it? 11/16/2018 ST3131, Lecture 8

11/16/2018 ST3131, Lecture 8

11/16/2018 ST3131, Lecture 8

11/16/2018 ST3131, Lecture 8

been explained by the regression? (9e). Based on the reduced model, how much of the total variability in Y has not been explained by the regression? 11/16/2018 ST3131, Lecture 8

After-class Questions: Can you interpret the exact meaning of the un-biasedness of a LS estimate? Why can we say the LS estimates are best linear unbiased estimates? 11/16/2018 ST3131, Lecture 8