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Multiple Regression II

Multiple Regression II. KNNL – Chapter 7. Extra Sums of Squares. For a given dataset, the total sum of squares remains the same, no matter what predictors are included (when no missing values exist among variables)

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Multiple Regression II

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  1. Multiple Regression II KNNL – Chapter 7

  2. Extra Sums of Squares • For a given dataset, the total sum of squares remains the same, no matter what predictors are included (when no missing values exist among variables) • As we include more predictors, the regression sum of squares (SSR) increases (technically does not decrease), and the error sum of squares (SSE) decreases • SSR + SSE = SSTO, regardless of predictors in model • When a model contains just X1, denote: SSR(X1), SSE(X1) • Model Containing X1, X2: SSR(X1,X2), SSE(X1,X2) • Predictive contribution of X2 above that of X1: SSR(X2|X1) = SSE(X1) – SSE(X1,X2) = SSR(X1,X2) – SSR(X1) • Extends to any number of Predictors

  3. Definitions and Decomposition of SSR Note that as the # of predictors increases, so does the ways of decomposing SSR

  4. ANOVA – Sequential Sum of Squares

  5. Extra Sums of Squares & Tests of Regression Coefficients (Single bk)

  6. Extra Sums of Squares & Tests of Regression Coefficients (Multiple bk)

  7. Extra Sums of Squares & Tests of Regression Coefficients (General Case)

  8. Other Linear Tests

  9. Coefficients of Partial Determination-I Switch 1 and 2

  10. Coefficients of Partial Determination-II

  11. Standardized Regression Model - I • Useful in removing round-off errors in computing (X’X)-1 • Makes easier comparison of magnitude of effects of predictors measured on different measurement scales • Coefficients represent changes in Y (in standard deviation units) as each predictor increases 1 SD (holding all others constant) • Since all variables are centered, no intercept term

  12. Standardized Regression Model - II

  13. Standardized Regression Model - III

  14. Multicollinearity • Consider model with 2 Predictors (this generalizes to any number of predictors) Yi = b0+b1Xi1+b2Xi2+ei • When X1 and X2 are uncorrelated, the regression coefficients b1 and b2 are the same whether we fit simple regressions or a multiple regression, and: SSR(X1) = SSR(X1|X2) SSR(X2) = SSR(X2|X1) • When X1 and X2 are highly correlated, their regression coefficients become unstable, and their standard errors become larger (smaller t-statistics, wider CIs), leading to strange inferences when comparing simple and partial effects of each predictor • Estimated means and Predicted values are not affected

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