STP452topic4 - STAT 512 Applied Regression Analysis Topic 4...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: STAT 512: Applied Regression Analysis Topic 4 Spring 2008 Topic Overview General Linear Tests Extra Sums of Squares Partial Correlations Multicollinearity Model Selection General Linear Tests These are a di erent way to look at the comparison of models. So far we have looked at comparing/selecting models based on: model signi cance test and R 2 values t-tests for variables added last These are good things to look at, but they are ine ective in cases where explanatory variables work together in groups we want to test some hypotheses for some β i = b i rather than β i = 0 (for example, maybe we want to test H : β 1 = 3 ,β 4 = 7 against the alternative hypothesis that at least one of those is false) General Linear Tests look at the di erence between models in terms of SSE (unexplained SS ) 1 in terms of SSM (explained SS ) Because SSM + SSE = SST , these two comparisons are equivalent. The models we compare are hierarchical in the sense that one (the full model) includes all of the explanatory vari- ables of the other (the reduced model). We can compare models with di erent explanatory variables. For example: X 1 ,X 2 vs X 1 X 1 ,X 2 ,X 3 ,X 4 ,X 5 vs X 1 ,X 2 ,X 3 . Note that the rst model includes all X 's of the second model. We will get an F test that compares the two models. We are testing a null hypothesis that the regression coe cients for the extra variables are all zero. For X 1 ,X 2 ,X 3 ,X 4 vs X 1 ,X 2 ,X 3 H : β 4 = β 5 = 0 H a : β 4 ,β 5 are not both 0 . F-test The test statistic in general is F = ( SSE ( R )- SSE ( F )) / ( df E ( R )- df E ( F )) SSE ( F ) /df E ( F ) . Under the null hypothesis (reduced model) this statistic has an F-distribution where the degrees of freedom are the number of extra variables and the df E for the larger model. So we reject if the p-value for this test is ≤ . 05 and in that case conclude that at least one of the extra variables is useful for predicting Y in the linear model that already contains the variables in the reduced model. Example Suppose n = 100 and we are testing X 1 ,X 2 ,X 3 ,X 4 ,X 5 (full) vs X 1 ,X 2 ,X 3 (reduced). Our hypotheses are: H : β 4 = β 5 = 0 H a : β 4 ,β 5 are not both 0 . Since we are considering removing 2 variables ( X 4 and X 5 ), the numerator df is 2. The denominator df is n- 6 = 94 (since p = 6 for the full model). We reject if the p-value ≤ 0.05 and in that case would conclude that either X 4 or X 5 or both contain additional information that is useful for predicting Y in a linear model that also includes X 1 , X 2 and X 3 . 2 7.1: Extra Sums of Squares Notation for Extra SS Example using 5 variables: SSE ( X 1 ,X 2 ,X 3 ,X 4 ,X 5 ) is the SSE for the full model, SSE ( F ) SSE ( X 1 ,X 2 ,X 3 ) is the SSE for a reduced model, SSE ( R ) The extra sum of squares for this comparison is denoted SSM ( X 4 ,X 5 | X 1 ,X 2 ,X 3 ) This is the di erence in the SSE 's: SSM ( X 4 ,X 5 | X 1 ,X 2 ,X 3 ) = SSE ( R )- SSE ( F ) = SSE ( X 1 ,X 2 ,X 3 )- SSE ( X 1 ,X...
View Full 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