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 : β 1 = β 2 =…= β k = 0 H 1 : β 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 – α ; k, n – k – 1): terima H 0 Jika F 0 > F( 1 – α ; 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 : β j = 0 v.s. H 1 : β j  0 Let C jj be the j-th diagonal element of (X’X) -1. The test statistic: Kesimpulan: Jika |t 0 | ≤ t(1 – α /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 β 2 given β 1 This is called the extra sum of squares due to β 2 and the degree of freedom is p – (p – r) = r The test statistic

12 If β 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 Mallow’s 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 don’t take this as evidence that the full model is good unless all the other Cp’s are bigger.

16 Mallow’s 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

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