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Math Help - Multiple linear regression: partial F-test

  1. #1
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    Multiple linear regression: partial F-test

    "Suppose that in a MULTIPLE linear regression analysis, it is of interest to compare a model with 3 independent variables to a model with the same response varaible and these same 3 independent variables plus 2 additional independent variables.
    As more predictors are added to the model, the coefficient of multiple determination (R^2) will increase, so the model with 5 predicator variables will have a higher R^2.
    The partial F-test for the coefficients of the 2 additional predictor variables (H_o: β_4=β_5=0) is equivalent to testing that the increase in R^2 is statistically signifcant."

    I don't understand the bolded sentence. Why are they equivalent?

    Thanks for explaining!

    [also under discussion in Talk Stats forum]
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  2. #2
    MHF Contributor matheagle's Avatar
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    Quote Originally Posted by kingwinner View Post
    "Suppose that in a MULTIPLE linear regression analysis, it is of interest to compare a model with 3 independent variables to a model with the same response varaible and these same 3 independent variables plus 2 additional independent variables.
    As more predictors are added to the model, the coefficient of multiple determination (R^2) will increase, so the model with 5 predicator variables will have a higher R^2.
    The partial F-test for the coefficients of the 2 additional predictor variables (H_o: β_4=β_5=0) is equivalent to testing that the increase in R^2 is statistically signifcant."

    I don't understand the bolded sentence. Why are they equivalent?

    Thanks for explaining!

    [also under discussion in Talk Stats forum]

    It's a test to see if the variables associated with \beta_4 and \beta_5 are of any value.
    By that we mean, do these x's help in explaining y any better than the first three terms in your model.
    Note that R^2 increases as we introduce more terms, but the mean squares may not.
    The mean squares, in particular MSE is SSE divided by n minus the number of parameters in the model.
    SSE will decrease as you add terms, but so will the denominator of MSE.
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