Multiple Regression II KNNL Chapter 7 Extra Sums
- Slides: 14
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) • 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 X 1, denote: SSR(X 1), SSE(X 1) • Model Containing X 1, X 2: SSR(X 1, X 2), SSE(X 1, X 2) • Predictive contribution of X 2 above that of X 1: SSR(X 2|X 1) = SSE(X 1) – SSE(X 1, X 2) = SSR(X 1, X 2) – SSR(X 1) • Extends to any number of Predictors
Definitions and Decomposition of SSR Note that as the # of predictors increases, so does the ways of decomposing SSR
ANOVA – Sequential Sum of Squares
Extra Sums of Squares & Tests of Regression Coefficients (Single bk)
Extra Sums of Squares & Tests of Regression Coefficients (Multiple bk)
Extra Sums of Squares & Tests of Regression Coefficients (General Case)
Other Linear Tests
Coefficients of Partial Determination-I
Coefficients of Partial Determination-II
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
Standardized Regression Model - II
Standardized Regression Model - III
Multicollinearity • Consider model with 2 Predictors (this generalizes to any number of predictors) Yi = b 0+b 1 Xi 1+b 2 Xi 2+ei • When X 1 and X 2 are uncorrelated, the regression coefficients b 1 and b 2 are the same whether we fit simple regressions or a multiple regression, and: SSR(X 1) = SSR(X 1|X 2) SSR(X 2) = SSR(X 2|X 1) • When X 1 and X 2 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
- Extra sum of squares multiple regression
- Simple linear regression and multiple linear regression
- Multiple regression vs linear regression
- Logistic regression vs linear regression
- Logistic regression vs linear regression
- Logistic regression interaction interpretation
- Anova multiple regression
- Multiple regression analysis with qualitative information
- Define multiple regression
- Dataset for multiple regression
- Multiple regression analysis adalah
- Quadratic regression spss
- Linear regression with multiple variables machine learning
- Sum of squares
- Multiple nonlinear regression spss