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# Chp2 - Inference in Regression Analysis Inference...

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Inference in Regression Analysis Inference Concerning β 0 and 1 Estimate E ( Y h ) and Predict New Observations ConFdence Band for Regression Line Analysis of Variance Approach to Linear Regression Association Between X in Regression Model Regression Model Assumptions we assume the normal error regression model throughout the lecture: i = + ± ; =1 , ··· ,n, (1) where: 1. are parameters 2. are known constants 3. are independent N (0 2 ).

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Remarks If ± i are not independent, the Y are correlated. E.g. the autocorre- lation in time series data or repeated measurements in longitudinal data analysis. The least squares method is not appropriate for these data, the maximum likelihood principle can be used to model the dependence. do not have constant variance σ 2 , the model has heteroscedastic errors. In this case weighted regression methods could be employed (it can be derived through maximum likelihood principle). In applications X could also be random. E.g. in the measurement error model, the are assume to be measured with errors. 3 Inference Concerning β 0 and 1 Point Estimation: b ConFdence Intervals Hypothesis Testing Some Considerations in Inference 4
Interpretation of β 0 and 1 is the slope of the regression line. It indicates the change in E ( Y ) per unit increase in X . If =0 , i = + ± There is no linear association between and the means of are all equal. For the normal error regression model (1), when = 0 the probability distribution of are identical. So there is no relation of any type between is the intercept of the regression line. When the scope of the model includes gives | = 0), the mean of the probability distribution of at = 0. When the scope of the model doesnt cover doesnt have any particular meaning as a separate term in the regression model. 5 Point Estimation of b ¯ )( ) 2 ,b (2) As we discussed in Chapter 1, both have normal distribution: N ² n ³ σ ´ ; (3) 6

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Point Estimation of β 0 and 1 :Con td . The distributions of b refer to their diFerent values that would be obtained with repeated sampling when the predictor vari- able X are held constant from sample to sample. Or you can think of as functions of random variables Y i in (2), where are ±xed constants. So they are also random variables and have distributions. has minimum variance among all unbiased linear estimators of the form: ˆ = c .Why? Cov( ,b )=0 Note: read the textbook pp.41-44 on the detailed mathematical deductions. 7 ConFdence Intervals for In the distribution of in (3), there is an extra unknown pa- rameter σ 2 except .W ee s t ima te by the unbiased MSE: ( ) n , 2)ˆ χ 2) (4) By replacing with ˆ in (3), we have the unbiased estimators for Var( )andVar( ): )= ¯ ,s ± + ² (5) 8
Confdence Intervals For β 0 and 1 :Con td . Also we can prove that Cov( b , ˆ σ 2 )=Cov( ) = 0 and hence ( j ) /s t n 2); =0 (6) So we can make the following probability statements: Pr ± α/ ,n 2) s (1 ² =1 α ; Moreover, since 2) = 2), we have the conFdence limits for : ); 9 Hypothesis Testing: Two-sided Test Test the following hypotheses H = ,H a 6 Test statistic: ,t |

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Chp2 - Inference in Regression Analysis Inference...

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