OMIS 327 Lecture 13. Linear Regression 2 Final

OMIS 327 Lecture 13. Linear Regression 2 Final - OMIS 327...

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1 OMIS 327 Lecture 13: Linear Regression 2 Instructor: Dr. Lee, Jung Young
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2 Announcement SurveyMonkey for Case Competition day!!!
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3 Today’s Learning Goals Testing the Model’s Significance (F-test) Testing the Linear Relationship (t-test) Model Building Cautions and Pitfalls in Regression Analysis
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Testing the Model for Significance When the sample size is too small, you can get good values for R 2 even if there is no relationship between the variables. Testing the model for significance helps determine if the above values are statistically meaningful. We do this by performing a statistical hypothesis test. 4
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5 F-Test for the General Regression Model The hypotheses for this test are: H 0 : 1 = 2 =…= r =0 (There is no linear relationship between the dependent and any of the independent variables; i.e. the regression model has no explanatory power ) H 1 : not all s are equal to zero The alternate hypothesis says that there is is a linear relationship between Y and at least one of Xs ( i ≠ 0). If the null hypothesis can be rejected, we have proven it is useful to use this specific linear regression model. We use the F statistic for this test.
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6 F-Test for the Overall Regression Model In order to conduct the overall F-test for the regression model we could use an ANOVA table ANOVA Table for Regression
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Testing the Model for Significance The F statistic is based on the MSE and MSR: k SSR MSR where k = number of independent variables in the model The F statistic is : MSE MSR calculated F This describes an F distribution with: degrees of freedom for the numerator = df 1 = k degrees of freedom for the denominator = df 2 = n k – 1 7 1 k n SSE MSE
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8 Rejection Region The rejection region of a statistical hypothesis test is the range of numbers that will lead us to reject the null hypothesis in case the test statistic falls within this range. The rejection region, also called the critical region , is defined by the critical points . The rejection region is defined so that, before the sampling takes place, our test statistic will have a probability of falling within the rejection region if the null hypothesis is true.
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Testing the Model for Significance If there is very little error, the MSE would be small and the F -statistic (of F-cal) would be large indicating the model is useful. So when the F - value is large , we can reject the null hypothesis and accept that there is a linear relationship between X and Y and the value of R 2 are meaningful. If the F -cal is large , the significance level ( p -value) will be low , indicating it is unlikely this would have occurred by chance.
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