65 percent of the variability in the taste data is

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65 percent of the variability in the taste data is explained by X 1 , X 2 , and X 3 assuming the model is true. PAGE 23
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2.1 Sampling Distributions and Inference for the parameters c circlecopyrt HYON-JUNG KIM, 2017 2.1 Sampling Distributions and Inference for the parameters Sampling Distribution of ˆ β Since ˆ β is a linear combination of Y , ˆ β has a multivariate Normal distribution as ǫ and Y . ˆ β MVN ( β , σ 2 ( X X ) 1 ) . Now ( X X ) 1 is a p × p matrix. So is Var( ˆ β ), the variance-covariance matrix of ˆ β . IMPLICATIONS: (a) E ( ˆ β k ) = β k , for k = 0 , 1 , ..., p 1 . ; unbiased. (b) Var( ˆ β k ) = ( X X ) 1 kk σ 2 , for k = 0 , 1 , ..., p 1 . The value ( X X ) 1 kk represents the k th diagonal element of the ( X X ) 1 matrix. (c) Cov( ˆ β k , ˆ β l )) = ( X X ) 1 kl σ 2 , for k negationslash = l . The value ( X X ) 1 kl is the entry in the k th row and l th column of the ( X X ) 1 matrix. (d) Marginally, ˆ β k N ( β k , ( X X ) 1 kk σ 2 ) , for k = 0 , 1 , ..., p 1 . (e) E(MSE) = σ 2 where SSE σ 2 = MSE( n p ) σ 2 χ 2 n p . CONFIDENCE INTERVALS: The 100(1 α )% confidence interval for β k is ˆ β k ± t 1 α 2 ,n p s( ˆ β k ) , k = 0 , . . . , p 1 where s( ˆ β k ) is the standard error of ˆ β k , i.e., radicalBig ( X X ) 1 kk MSE . Since ( ˆ β β ) X X ( ˆ β β ) σ 2 χ 2 p is independent of MSE , ( ˆ β β ) X X ( ˆ β β ) p MSE χ 2 p /p χ 2 n p / ( n p ) F p,n p Thus, a 100(1 α )% (simultaneous) confidence region for β can be formed as ( ˆ β β ) X X ( ˆ β β ) p MSE F 1 α, ( p,n p ) . These regions are ellisoidally shaped and cannot be easily visualized due to high-dimensionality. PAGE 24
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2.1 Sampling Distributions and Inference for the parameters c circlecopyrt HYON-JUNG KIM, 2017 CONFIDENCE INTERVAL FOR THE MEAN RESPONSE: Given a new set of predic- tors X 0 , what is the estimated mean response? We call it ˆ Y 0 , which is X 0 ˆ β . Var( X 0 ˆ β ) = X 0 ( X X ) 1 X 0 σ 2 . So a 100(1 α )% confidence interval for the average of the responses with given X 0 is ˆ Y 0 ± t 1 α 2 ,n p radicalBig MSE X 0 ( X X ) 1 X 0 . PREDICTION INTERVAL OF A SINGLE FUTURE RESPONSE FOR X 0 : Recall that a future observation is predicted to be ˆ Y 0 = X 0 ˆ β (where we don’t know what the future response will turn out to be.) Then, a 100(1 α )% prediction interval for a single future response is ˆ Y 0 ± t 1 α 2 ,n p radicalBig MSE [1 + X 0 ( X X ) 1 X 0 ] . HYPOTHESIS TESTS: For testing H 0 : β k = β k, 0 vs. H 1 : β k negationslash = β k, 0 , we use | t | = ˆ β k β k, 0 radicalBig ( X X ) 1 kk MSE > t 1 α 2 ,n p . to reject H 0 ; Otherwise, do not reject H 0 . Example (cheese data continued). Considering the full model, a) Find the estimate of the variance-covariance matrix of ˆ β . We first compute ( X X ) 1 = 3 . 795 0 . 760 0 . 087 0 . 071 0 . 760 0 . 194 0 . 020 0 . 128 0 . 087 0 . 020 0 . 015 0 . 046 0 . 071 0 . 128 0 . 046 0 . 726 b) To assess the importance of the hydrogen sulfide concentration and its influence on taste of cheese, we can test H 0 : β 2 = 0 vs. H 1 : β 2 negationslash = 0.
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