F19-MLR-InferencesPartII

F19-MLR-InferencesPartII - PubH 7405: REGRESSION ANALYSIS...

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PubH 7405: REGRESSION ANALYSIS MLR: INFERENCES, Part II
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TESTING HYPOTHESES Once we have fitted a multiple linear regression model and obtained estimates for the various parameters of interest, we want to answer questions about the contributions of factor or factors to the prediction of the dependent variable Y. There are three types of tests : (1) An overall test (2) Test for the value of a single factor (3) Test for contribution of a group of factors
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MARGINAL CONTRIBUTION The change, for example, from the model containing only X 1 to the model containing all three variables X 1 ,X 2 3 represent “marginal contribution” by the addition of X 2 3 . The marginal contribution represent the part (of SSE) that is further explained by X 2 and X 3 (in addition to what already explained by X 1 ): SSR(X 2 ,X 3 |X 1 ) = SSR(X 1 ,X 2 ,X 3 ) - SSR(X 1 )
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A TYPICAL STRATEGY Two models: a larger one & a smaller one; larger model has more terms and larger SSR The difference in SSR is accountable for by extra terms in the regression model; the group of terms under investigation. Decompose the SSR and the df(R); then calculating the MR and the F ratio Numerator of F is the MS due to the extra terms; denominator is MSE of larger model. Use the F ratio to test difference of 2 models.
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COEFFICIENT OF PARTIAL DETERMINATION #a Suppose we have a multiple regression model with 2 independent variables (the Full Model) and suppose we are interested in the marginal contribution of X 2 : The coefficient of partial determination , between Y and X 2 measures the marginal reduction in the variation of Y associated with the addition of X 2 : ) σ N(0, ε ε x β x β β Y 2 2 2 1 1 0 ) ( ) | ( 1 1 2 2 1 | 2 X SSE X X SSR R Y
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COEFFICIENTS OF PARTIAL CORRELATION The square root of a coefficient of partial determination is called a coefficient of partial correlation . It is given the same sign as that of the corresponding regression coefficient in the fitted regression model. Coefficients of partial correlation are sometimes used in practice, although they do not have a clear meaning as coefficients of partial determination nor the (single) coefficient of correlation.
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PARTIAL CORRELATION & PARTIAL DETERMINATION Let both the response variable Y and the predictor under investigation (say, X 1 ) be both regressed against the other predictor variables already in the regression model and the residuals are obtained for each. These two sets of residuals reflect the part of each (Y and X 1 ) that is not linearly associated with the other predictor variables. Result: The correlation coefficient between the above two sets of residuals is equal to the Coefficient of Partial Correlation between Y & X1, which is the square root of the Coefficient of Partial Determination (of X 1 on Y).
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In the decomposition of the sums of squares; the “extra sums of squares” are very useful for testing/measuring for the marginal contribution of individual
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This note was uploaded on 11/21/2011 for the course PUBH 7405 taught by Professor Staff during the Fall '08 term at Minnesota.

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F19-MLR-InferencesPartII - PubH 7405: REGRESSION ANALYSIS...

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