Matakuliah I 0174 Analisis Regresi Tahun

Matakuliah : I 0174 – Analisis Regresi Tahun : Ganjil 2007/2008 Regresi Linear Ganda dengan Peubah Boneka Pertemuan 07 Bina Nusantara

Regresi Linier Ganda Dengan Peubah Boneka Dua katagori Peubah Boneka Lebih Dari Dua Katagori Bina Nusantara

Chapter Topics • Dummy-Variables and Interaction Terms Bina Nusantara

The Multiple Regression Model Relationship between 1 dependent & 2 or more independent variables is a linear function Population Y-intercept Dependent (Response) variable Bina Nusantara Population slopes Independent (Explanatory) variables Random error

Multiple Regression Model Bivariate model Bina Nusantara

Multiple Regression Equation Bivariate model Bina Nusantara Multiple Regression Equation

Multiple Regression Equation Too complicated by hand! Bina Nusantara Ouch!

Interpretation of Estimated Coefficients • Slope (bj ) – Estimated that the average value of Y changes by bj for each 1 unit increase in Xj , holding all other variables constant (ceterus paribus) – Example: If b 1 = -2, then fuel oil usage (Y) is expected to decrease by an estimated 2 gallons for each 1 degree increase in temperature (X 1), given the inches of insulation (X 2) • Y-Intercept (b 0) – The estimated average value of Y when all Xj = 0 Bina Nusantara

Multiple Regression Model: Example (0 F) Develop a model for estimating heating oil used for a single family home in the month of January, based on average temperature and amount of insulation in inches. Bina Nusantara

Multiple Regression Equation: Example Excel Output For each degree increase in temperature, the estimated average amount of heating oil used is decreased by 5. 437 gallons, holding insulation constant. Bina Nusantara For each increase in one inch of insulation, the estimated average use of heating oil is decreased by 20. 012 gallons, holding temperature constant.

Dummy-Variable Models • • Bina Nusantara Categorical Explanatory Variable with 2 or More Levels Yes or No, On or Off, Male or Female, Use Dummy-Variables (Coded as 0 or 1) Only Intercepts are Different Assumes Equal Slopes Across Categories The Number of Dummy-Variables Needed is (# of Levels - 1) Regression Model Has Same Form:

Dummy-Variable Models (with 2 Levels) Given: Y = Assessed Value of House X 1 = Square Footage of House X 2 = Desirability of Neighborhood = Desirable (X 2 = 1) Undesirable (X 2 = 0) Bina Nusantara 0 if undesirable 1 if desirable Same slopes

Dummy-Variable Models (with 2 Levels) (continued) Y (Assessed Value) rable Desi b 0 + b 2 Intercepts different Bina Nusantara b 0 ation Loc Same slopes able esir Und X 1 (Square footage)

Interpretation of the Dummy-Variable Coefficient (with 2 Levels) Example: : Annual salary of college graduate in thousand $ : GPA : 0 non-business degree 1 business degree With the same GPA, college graduates with a business degree are making an estimated 6 thousand dollars more than graduates with a non-business degree, on average. Bina Nusantara

Dummy-Variable Models (with 3 Levels) Bina Nusantara

Interpretation of the Dummy-Variable Coefficients (with 3 Levels) With the same footage, a Splitlevel will have an estimated average assessed value of 18. 84 thousand dollars more than a Condo. With the same footage, a Ranch will have an estimated average assessed value of 23. 53 thousand dollars more than a Condo. Bina Nusantara

Regression Model Containing an Interaction Term • Hypothesizes Interaction between a Pair of X Variables – Response to one X variable varies at different levels of another X variable • Contains a Cross-Product Term – • Can Be Combined with Other Models – E. g. , Dummy-Variable Model Bina Nusantara

Effect of Interaction • Given: – • Without Interaction Term, Effect of X 1 on Y is Measured by 1 • With Interaction Term, Effect of X 1 on Y is Measured by 1 + 3 X 2 • Effect Changes as X 2 Changes Bina Nusantara

Interaction Example Y Y = 1 + 2 X 1 + 3 X 2 + 4 X 1 X 2 Y = 1 + 2 X 1 + 3(1) + 4 X 1(1) = 4 + 6 X 1 12 8 Y = 1 + 2 X 1 + 3(0) + 4 X 1(0) = 1 + 2 X 1 4 0 0 0. 5 1 1. 5 X 1 Effect (slope) of X 1 on Y depends on X 2 value Bina Nusantara

Interaction Regression Model Worksheet Case, i Yi X 1 i X 2 i 1 1 1 3 3 2 3 4 1 8 3 5 2 40 6 4 3 5 6 30 : : : Multiply X 1 by X 2 to get X 1 X 2 Run regression with Y, X 1, X 2 , X 1 X 2 Bina Nusantara

Interpretation When There Are 3+ Levels MALE = 0 if female and 1 if male MARRIED = 1 if married; 0 if not DIVORCED = 1 if divorced; 0 if not MALE • MARRIED = 1 if male married; 0 otherwise = (MALE times MARRIED) MALE • DIVORCED = 1 if male divorced; 0 otherwise = (MALE times DIVORCED) Bina Nusantara

Interpretation When There Are 3+ Levels (continued) Bina Nusantara

Interpreting Results FEMALE Single: Married: Divorced: Difference Main Effects : MALE, MARRIED and DIVORCED Interaction Effects : MALE • MARRIED and MALE • DIVORCED Bina Nusantara

Evaluating the Presence of Interaction with Dummy. Variable • Suppose X 1 and X 2 are Numerical Variables and X 3 is a Dummy-Variable • To Test if the Slope of Y with X 1 and/or X 2 are the Same for the Two Levels of X 3 • Model: • Hypotheses: – H 0: 4 = 5 = 0 (No Interaction between X 1 and X 3 or X 2 and X 3 ) – H 1: 4 and/or 5 0 (X 1 and/or X 2 Interacts with X 3) • Perform a Partial F Test Bina Nusantara

Evaluating the Presence of Interaction with Numerical Variables • Suppose X 1, X 2 and X 3 are Numerical Variables • To Test If the Independent Variables Interact with Each Other • Model: • Hypotheses: – H 0: 4 = 5 = 6 = 0 (no interaction among X 1, X 2 and X 3 ) – H 1: at least one of 4, 5, 6 0 (at least one pair of X 1, X 2, X 3 interact with each other) • Perform a Partial F Test Bina Nusantara