presented by: Regresi Linier Berganda (Cont.) (RLB) Dudi Barmana, M.Si. 55 1 Agenda Pengujian parameter model: overall F-Test individual t-Test sequential test (partial F-Test) Ukuran penilaian kemampuan/kesesuaian model: Adjusted R-square (R 2 ) Cp Mallow Statistic 2 Today Quote Jika anda terlahir dalam kemiskinan itu bukanlah kesalahan anda, tapi jika anda mati dalam kemiskinan itu adalah kesalahan anda ---Bill Gates--- 3 Pengujian parameter model: overall F-Test individual t-Test sequential test (partial F-Test) 4 Hypothesis Testing in Multiple Linear Regression Questions: What is the overall adequacy of the model? Which specific regressors seem important? Assume the errors are independent and follow a normal distribution with mean 0 and variance 2 5 6 Overall F-Test (Test for Significance of Regression) Determine if there is a linear relationship between y and x j, j = 1,2,,k. The hypotheses are H 0 : 1 = 2 == k = 0 H 1 : j 0 for at least one j ANOVA SS T = SS R + SS Res SS R / 2 ~ 2 k, SS Res / 2 ~ 2 n-k-1, and SS R and SS Res are independent 7 ANOVA table Sumber Variasi SSdfMSF0F0 RegresiSS R kMS R MS R /MS RES ResidualSS RES n k 1MS RES TotalSS T n 1 Kesimpulan: Jika F 0 F( 1 ; k, n k 1): terima H 0 Jika F 0 > F( 1 ; k, n k 1): tolak H 0 8 Under H 1, F 0 follows F distribution with k and n-k-1 and a noncentrality parameter of 9 Tests on Individual Regression Coefficients (Individual t-Test) For the individual regression coefficient: H 0 : j = 0 v.s. H 1 : j 0 Let C jj be the j-th diagonal element of (XX) -1. The test statistic: Kesimpulan: Jika |t 0 | t(1 /2; n p): terima H 0 This is a partial or marginal test because any estimate of the regression coefficient depends on all of the other regression variables. This test is a test of contribution of x j given the other regressors in the model 10 Sequential Test (Partial F-Test) The subset of regressors: 11 For the full model, the regression sum of square Under the null hypothesis, the regression sum of squares for the reduce model The degree of freedom is p r for the reduce model. The regression sum of square due to 2 given 1 This is called the extra sum of squares due to 2 and the degree of freedom is p (p r) = r The test statistic 12 If 2 0, F 0 follows a noncentral F distribution with Multicollinearity: this test actually has no power! This test has maximal power when X 1 and X 2 are orthogonal to one another! Partial F test: Given the regressors in X 1, measure the contribution of the regressors in X 2. Ukuran penilaian kemampuan/kesesuaian model: Adjusted R-square (R 2 ) Cp Mallow Statistic 13 14 R 2 and Adjusted R 2 R 2 always increase when a regressor is added to the model, regardless of the value of the contribution of that variable. An adjusted R 2 : The adjusted R 2 will only increase on adding a variable to the model if the addition of the variable reduces the residual mean squares. 15 Mallows Cp Suppose we have a model with p regression coefficients. Mallows C p provides an estimate of how well the model predicts new data, and is given by 15 The subscript FULL refers to the full model with k variables. Small values of Cp with Cp about p are good. Warning: C k+1 =k+1 always, so dont take this as evidence that the full model is good unless all the other Cps are bigger. 16 Mallows Cp If the p-coefficient model contains all the important explanatory variables, then SS RES(p) is about the same as (n- p) 2. Moreover, MS RES(FULL) will also be about the same as 2. Thus pertanyaan 17