EXST7034 - Regression Techniques Page 1General Linear Hypothesis Test Approach (GLHT)Given a Regression Model with all variables of interest Y = b b X e = Y b (X - X) e––3! "3 3 3 "3 3 we will call this the FULL model Given a second Regression Model with parameters of interest which are a subsetof the FULL model; for the SLR we are testing the hypothesis that 0,""œthen Y = b = Y–3!3 33%%we will call this the REDUCED modelBoth models have a total of “n" observations The reduced model has “p" parameters including the intercept (=1 for SLR) The full model has “p q" parameters i*including the intercept (=2 forSLR)Our objective is to perform a test of the DIFFERENCE between the two models. 1) If the models are DIFFERENT, then the full model is BETTER since ithas more information. 2) If the models are NOT DIFFERENT, then the reduced model is better,because it fits equally as well with fewer degrees of freedom.FIT BOTH MODELS from each take the d.f. Error and SSErrorEXST7034 - Regression Techniques Page 2Perform the General Linear Hypothesis test as follows Model d.f SS MS F Reduced (Error) p+q SSEReg Full (Error) p SSEFull Difference q SS MS Diff DiffMSMSEDiffFull Full (Error) n-p-q SSE MSEFull Full1) The Reduced Model will always have more degrees of freedom in the errorbecause it has fewer terms and fewer d.f. in the model2) The Reduced Model will always have a larger (or equal) Sum of Squares Errorbecause it cannot fit as well with fewer parameters3) Therefore, the d.f. and SS will always be positive (or zero)Diff Diff4) To test the DIFFERENCE, what do we use as an error term? a) Either there is no difference between the models, and it makes nodifference which error term is used, OR ... b) There is a difference in the models, in which case the FULL model isbetter SO WE USE THE FULL MODEL ERROR TERM For the SLR, the SSE(full) is simply the SSE for the SLR the SSE(reduced) is the CORRECTED SSTotalThe difference then has one degree of freedom, and is equal to the SSRegression Model d.f SS MS F Reduced (Error) n-1 SSTotal Full (Error) n-2 SSE Difference 1 SS =SS MSReg .300 V/1MSRegMSError Full (Error) n-2 SSE MSE Which is the same as the test of the model we saw in our ANOVA table. This isanother way of viewing the test that we have done. However, this test isreadily generalized to any full and reduced model, where the reducedmodel has a subset of the parameter estimates of the full
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