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

Info icon This preview shows pages 25–28. Sign up to view the full content.

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
Image of page 25

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

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
Image of 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.
Image of page 27

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Image of page 28
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern