Adding additional independent variables to the regression equation will always increase the value of R² - Even though not all independent variables are worth including in the regression model
o Showing It In Action ▪ Add 3 rd variable that affects car price, seller age ▪ Compute multiple regression analysis in PHStat2 - Same procedure as in earlier examples o Results ▪ Regression Statistics – R Square and Adjusted R Square - Adjusted = with third variable included - R² increases and adjusted R² decreases ▪ Why R² increases - Adding 3 rd IV increases SSR - Adding new IV to model will ALWAYS increase R² ▪ MSR decreases - Harder to reject null hypothesis when test for significance o Evaluation ▪ If added IV causes reduction in adjusted R² - evidence that the new variable might not be worth keeping in model - Benefit of increasing SSR offset by having cost of additional DF ▪ IN our example – Adjusted R² decreases - Shouldn’t include this 3 rd variable in the regression analysis o Other Notes ▪ Adjusted R² will always be ≤ R² ▪ If R² is very small and large # of IV - Adjusted R² COULD be negative - → report value as zero ➢ 15.3 – INFERENCES ABOUT THE INDEPENDENT VARIABLES ▪ Examine e/independent variable separately to make sure that it belongs in the final regression model ▪ Confidence intervals for the regression coefficients ● A Significance Test for the Regression Coefficients o The Basics
▪ Obtain regression coefficients for all 3 IVs - PHStat2 procedures - Coefficients Column under last table o Step 1 – Hypothesis Statement ▪ H0: M = 0 - No relationship exists ▪ H1: M ≠ 0 - A relationship does exist ▪ Rejecting the null hypothesis - Support of a significant relationship b/w mileage / price - Want to reject null hypothesis b/c included IVs in model that we think belong there ▪ Set ∞ = 0.05 o Step 2 – Test Statistic T = (b – B) ÷ Sb - Sb = standard error of the regression coefficient - b = regression coefficient - B = population regression coefficient ▪ Standard error of the regression coefficient (Sb) – measures the variation in the regression coefficient, bi, among several samples - Larger variation → larger value of Sb - In our example, Sb = 0.0415 coefficient for mileage ▪ In our example (mileage variable) - b = -0.1434 - B = 0 b/c null hypothesis assumes no relationship (= 0) - Sb = 0.0415 - → T = -3.46 o Step 3 – Critical T-Score ▪ Follows T-distribution ▪ DF = (n – k – 1) - In our example, 20 – 3 – 1 = 16
▪ Two-tail test at ∞ = 0.05 → ±2.120 ▪ FUNCTION in Excel (two-tail) - = TINV (∞, DF) - for one-tail, double value of ∞ o Step 4 – Comparing T and T-Score ▪ -3.46 < -2.120 → REJECT o Step 5 – Conclusions ▪ Reject null hypothesis ▪ Conclude that mileage population coefficient is NOT equal to zero ▪ Relationship b/w mileage and asking price is significant o 3 rd Variable – Seller Age ▪ P-value (0.778) > ∞ = 0.05 → DO NOT REJECT - No relationship b/w the variables - The seller’s age is not a good predictor of car price ▪ Should remove this independent variable from regression model o When to Use F and T-Test ▪ F-Test - Det. if overall regression model (all IVs included) is statistically signficiant ▪ T-Test - Examine difference of individual independent variables ● Confidence Intervals for the Regression Coefficients ▪
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- Fall '12
- Normal Distribution, Null hypothesis, Hypothesis testing, Statistical hypothesis testing