The predicted or fitted values hatwide y i ˆ β ˆ

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The predicted (or fitted) values : hatwide Y i = ˆ β 0 ˆ β 1 X i for i = 1 , ..., n The residuals for data point i : e i = Y i hatwide Y i The squared prediction error for data point i : e 2 i = ( Y i hatwide Y i ) 2 Sum of Squared Error (SSE): SSE = n summationdisplay i =1 ( Y i hatwide Y ) 2 cf. MSE = SSE n 2 : Mean Squared Error Because the formulas for ˆ β 0 and ˆ β 1 are derived using the least squares criterion, the resulting equation often referred to as the “least squares regression line”, or simply the “least squares line”. It is also sometimes called the estimated regression equation. We need a few assumptions in order to justify using the least squares regression: (a) the expected value of the errors is zero. i.e. E ( ǫ i ) = 0 for all i . (or the mean of the response E ( Y i ) at each value of the predictor X i is a linear function of X i .) (b) errors are uncorrelated with each other (independent). (c) errors are normally distributed (confidence/prediction intervals or hypothesis tests). (d) the variance of the errors is constant (equal). i.e. Var( ǫ i ) = σ 2 for all i . Gauss-Markov theorem : under the above conditions of the simple linear model (SLM) the least squares estimators are unbiased and have minimum variance among all unbiased linear estimators. PAGE 3
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c circlecopyrt HYON-JUNG KIM, 2017 Example. Effects of ozone pollution on soybean yield Four dose levels of ozone and the resulting mean seed yield of soybeans are given. The dose of ozone is the average concentration (parts per million, ppm) during the growing season. Yield is reported in grams per plant. X Y Ozone (ppm) Yield (gm/plt) .02 242 .07 237 .11 231 .15 201 Y i ˆ Y i e i e 2 i 242 237 231 201 PAGE 4
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1.1 INTERVAL ESTIMATES and HYPOTHESES TESTS c circlecopyrt HYON-JUNG KIM, 2017 1.1 INTERVAL ESTIMATES and HYPOTHESES TESTS In order to construct confidence intervals or test hypotheses it is necessary to make some assumptions about the distribution of ǫ i , i = 1 , ..., n . The usual assumption is that ǫ i Normal(0 , σ 2 ) , i.e. ǫ i is normally distributed, independent of all other ǫ j and all have the same variance, σ 2 , and mean zero. Since the random errors are unknown, the residuals can be used to get an estimate of the error variance: 1 n p n summationdisplay i =1 e i 2 = 1 n p n summationdisplay i =1 ( Y i hatwide Y ) 2 = MSE Properties of least squares estimates We consider the simple linear regression model Y i = β 0 + β 1 X i + ǫ i , for i = 1 , ..., n , where ǫ i i.i.d. Normal (0 , σ 2 ). (a) E ( ˆ β 0 ) = β 0 and E ( ˆ β 1 ) = β 1 ; unbiased (b) Var( ˆ β 0 ) = ( 1 n + X 2 S xx ) σ 2 (c) Var( ˆ β 1 ) = σ 2 S xx (d) Cov( ˆ β 0 , ˆ β 1 ) = σ 2 X S xx (e) E(MSE) = σ 2 , where SSE σ 2 χ 2 n 2 (f) Both ˆ β 0 and ˆ β 1 are normally distributed. (MSE is independent of both ˆ β 1 and ˆ β 1 .) Since ˆ β 1 Normal( β 1 , σ 2 /S xx ) , t = ( ˆ β 1 β 1 ) S xx radicalBig ( n 2) MSE σ 2 / ( n 2) t n 2 .
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