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Lecture_12_Mult_Reg_III-2012

Lecture_12_Mult_Reg_III-2012 - Lecture 12 Multiple...

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1 Lecture 12: Multiple Regression III Stat 102 2012 Full and Reduced Model Comparisons – Chapter 4.4 ° Testing a group of variables Prediction in multiple regressions – Chapter 4.5 ° Confidence intervals For the conditional mean, 1 ,.., K Y x x μ . ° Called “Mean Confidence Interval” in JMP. For the value of an individual Y given 1 ,.., K x x . ° Called “Individual Confidence Interval” in JMP.
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2 Testing a Group of Variables After controlling for other variables in a model In Lecture 10-11 we discussed the test that one coefficient is zero, after controlling for all others in the model. This test is given directly in JMP’s Effect Test table. Or you can use the Parameter Estimates table for the same result. It’s also possible to test for a group of coefficients, after controlling for all others in the model. The procedure is similar (with a few differences), but it is not given directly in JMP. Instead, you need to construct the test statistics from two separate JMP tables. Here’s the general theory. Then we’ll use this procedure in our Cars 2004 example.
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3 General Theory You want to test that the (linear) effect of a subset of the coefficients is 0; after controlling for the (linear) effect of all the others in the model. For notational convenience, we’ll label the tested coefficients as 1 ,.., L K β β + . But the procedure is similar if applied to any other subset of coefficients. In formal terms, the null and alternative hypotheses are 0 1 : .. 0 L K H β β + = = = vs 0 : is false a H H . To test the null hypothesis begin by examining the SSE terms in both a. The ANOVA table for the full model having all the coefficients 1 ,.., ,..., L K β β β (Call this SSE F ); and b. The ANOVA table for the “reduced” model that has only the effects corresponding to 1 ,.., L β β . (Call this SSE R .)
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4 Note that SSE R SSE F . {Why?} Then define the statistic ( ) ( ) ( ) 1 K L F n K = R F F SSE SS SE E S . Note that the term ( ) 1 n K F SSE in the denominator of this F-statistic. This is the MSE from the full model . And note the term ( ) K L that appears inside the numerator. This is the DF of the difference R F SSE SSE , since there are K-L extra parameters, 1 ,.., K L β β + .
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