# Note the divisor n 2 only applies to simple

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Note: The divisor n - 2 only applies to simple regression. The general rule is that the divisor is n - p , where p =number of parameters in the regression equation. STAT331 Applied Linear Models, Chapter2, Part 1 Chapter2: Review of Simple Linear Regression Part 1
An Example about the Simple Regression Model Example: The simple regression equation relating height ( y ) and distance between fingertips (DF, x ) is average height = β 0 + β 1 · DF . Suppose the sample is: DF (cm) 156 176 167 155 180 178 145 177 189 Height (cm) 153 171 163 150 180 188 142 182 180 DF (cm) 165 176 178 182 158 163 171 150 188 Height (cm) 160 173 176 185 158 165 169 154 186 STAT331 Applied Linear Models, Chapter2, Part 1 Chapter2: Review of Simple Linear Regression Part 1
An Example about the Simple Regression Model Calculate ¯ x = 169 . 668, ¯ y = 168 . 611, S xx = 2850, S xy = 2856 . 667 Then the sample intercept is ˆ β 0 = - 1 . 452 and the sample slope is ˆ β 1 = 1 . 002. Interpret the parameters: ˆ β 0 : the estimated average height at DF=0 is -1.452 cm. ˆ β 1 : For one unit increase in DF, the estimated average height increases by 1.002 cm. Note: It would be wrong, for example, to write the regression equation average height = - 1 . 452 + 1 . 002 · DF . But we can write \ height = - 1 . 452 + 1 . 002 · DF . STAT331 Applied Linear Models, Chapter2, Part 1 Chapter2: Review of Simple Linear Regression Part 1
Properties of Least Squares Estimators We have following properties of LSEs Property 1. LSEs are unbiased: E ( ˆ β 1 ) = β 1 , E ( ˆ β 0 ) = β 0 Property 2. The theoretical variances of ˆ β 0 and ˆ β 1 are: Var ( ˆ β 1 ) = σ 2 ( x i - ¯ x ) 2 Var ( ˆ β 0 ) = σ 2 [ 1 n + ¯ x 2 ( x i - ¯ x ) 2 ] Property 3. cov( ˆ β 0 , ˆ β 1 ) = - σ 2 ¯ x ( x i - ¯ x ) 2 STAT331 Applied Linear Models, Chapter2, Part 1 Chapter2: Review of Simple Linear Regression Part 1
Property 4. Under the normality assumption (iv), we have: ˆ β 1 N ( β 1 , σ 2 ( x i - ¯ x ) 2 ) ˆ β 0 N ( β 0 , σ 2 [ 1 n + ¯ x 2 ( x i - ¯ x ) 2 ]) STAT331 Applied Linear Models, Chapter2, Part 1 Chapter2: Review of Simple Linear Regression Part 1
Proof STAT331 Applied Linear Models, Chapter2, Part 1 Chapter2: Review of Simple Linear Regression Part 1
Chapter2: Review of Simple Linear Regression Part 2 Yeying Zhu Fall, 2013 STAT331 Applied Linear Regression, Chapter2, Part 2 Chapter2: Review of Simple Linear Regression Part 2
Statistical Inference Basically, there are two main statistical inference tools: Confidence interval Hypothesis testing They are equivalent in some sense. STAT331 Applied Linear Regression, Chapter2, Part 2 Chapter2: Review of Simple Linear Regression Part 2
Distribution of ˆ β 1 , ˆ β 0 Under the normality assumption i N (0 , σ 2 ) for i = 1 , . . . , n , we have ˆ β 1 and ˆ β 0 are normally distributed and: ˆ β 1 N ( β 1 , σ 2 ( x i - ¯ x ) 2 ) ˆ β 0 N ( β 0 , σ 2 [ 1 n + ¯ x 2 ( x i - ¯ x ) 2 ]) STAT331 Applied Linear Regression, Chapter2, Part 2 Chapter2: Review of Simple Linear Regression Part 2
Estimated Variance Recall: s 2 (MSE) is an unbiased estimator of σ 2 . Replace the unknown parameter σ 2 with s 2 , we obtain: se ( ˆ β 1 ) = s s 2 ( x i - ¯ x ) 2 se ( ˆ β 0 ) = s s 2 · [ 1 n + ¯ x 2 ( x i - ¯ x ) 2 ] STAT331 Applied Linear Regression, Chapter2, Part 2 Chapter2: Review of Simple Linear Regression Part 2