Regression.pdf

# C h β 1 β 2 β 3 0 vs h 1 β 1 β 2 β 3 not all

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c) H 0 : β 1 = β 2 = β 3 = 0 vs. H 1 : β 1 , β 2 , β 3 not all zero. Note that if H 0 is rejected, we do not know which one or how many of the β k ’s are nonzero; we only know that at least one is. PAGE 25

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2.1 Sampling Distributions and Inference for the parameters c circlecopyrt HYON-JUNG KIM, 2017 SUMS OF SQUARES AS QUADRATIC FORMS Recall that in page 8 the corrected sums of squares were partitioned and expressed in terms of random vectors. To enhance the understanding of these sums of squares, we often express them in terms of quadratic forms. SST = Y ( I 1 n 11 ) Y = Y ( I 1 n J ) Y SSR = Y ( H 1 n J ) Y SSE = Y ( I H ) Y Recall that any n × n real symmetric matrix A determines a quadratic form such that Y A Y = n summationdisplay i =1 n summationdisplay j =1 a ij Y i Y j . Def. The trace of a n × n square matrix A is tr ( A ) = n i =1 A ii Lemma: E ( Y A Y ) = tr ( A Σ) + µ A µ , where E ( Y ) = µ and V ar ( Y ) = Σ . EXPECTED MEAN SQUARES We have seen that MSE is unbiased for σ 2 and this holds whether or not H 0 : β 0 = . . . = β p 1 = 0 is true. On the other hand, E (SSR) = E [ Y ( H n 1 11 ) Y ] = tr[( H n 1 11 ) σ 2 I ] + ( ) ( H n 1 11 ) = ( p 1) σ 2 , only when H 0 is true. E (MSR) = E parenleftbigg SSR p 1 parenrightbigg = σ 2 , only when H 0 is true. NOTE. When H 0 is true, both MSR and MSE are estimating the same quantity, and the F statistic should be close to one. When H 0 is not true, the ( ) ( H n 1 11 ) > 0, and hence, MSR is estimating something larger than σ 2 . In this case, we would expect F to be larger than one. This gives an intuitive explanation of why F statistic is computed large when H 0 is not true. Also, the name of ‘Analysis of Variance’ is coined with this kind of study because we are conducting hypothesis tests by comparing different estimators for the variance. The extra term ( ) ( H n 1 11 ) is called a noncentrality parameter. PAGE 26
2.1 Sampling Distributions and Inference for the parameters c circlecopyrt HYON-JUNG KIM, 2017 Source degrees of freedom Sum of Squares Mean Square Regression p 1 ˆ β X Y 1 n Y 11 Y SSR p 1 Error n p Y Y ˆ β X Y SSE n p Total n 1 Y Y 1 n Y 11 Y Example (cheese data continued). Considering the full model, d) Find a 95 percent confidence interval for the mean taste rating when concentration of acetic acid, hydrogen sulfide, lactic acid are 5,6,1, respectively. e) Find a 95 percent prediction interval for a particular taste rating score when concen- tration of acetic acid, hydrogen sulfide, lactic acid are 5,6,1, respectively. Coefficient of Determination, R-squared, and Adjusted R-squared As in simple linear regression, R 2 =SSR/SSE, represents the proportion of variation in Y (about its mean) ‘explained’ by the multiple linear regression model with predictors.

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