1. ETM 620 - 09U 1 Multiple regression More than one indicator variable may be responsible for the variation we see in the response. Gas mileage is a function of weight, horsepower, use of air conditioning, etc. Metal fatigue in airplanes is a function of number of takeoffs and landings, climbout speed, landing speed, etc. Incidence of heart attack is a function of age, BMI, cholesterol levels, etc. If the function that defines the relationship between the indicator variables and the response is linear, then we have multiple linear regression, i.e., If a polynomial relationship between indicators and response is the best fit, then we have polynomial regression, e.g., ETM 620 - 09U 1

2. 2 Multiple linear regression: Matrix approach The viscosity of slurry is believed to be a function of the temperature and the feed rate. A number of readings were taken with the following results: Hypothesize the relationship, Y = β 0 + β 1 x 1 + β 2 x 2 + ε and calculate the estimate, ETM 620 - 09U 2 TempFeed RateViscosity 8082256 9392340 100102426 82132293 90112330 9982368 8182250 96102409 94122364 93112379 97132440 95112364 10082404 85122317 8692309 87122328

3. ETM 620 - 09U 3 Matrix form of the equation Define the matrices: ETM 620 - 09U 3

4. 4 General Matrix Form We obtain the least squares estimates (b 0, b 1, b 2 ) of ( β 0, β 1, β 2 ) by solving the matrix equation: for b, or ETM 620 - 09U 4

5. 5 14.519-0.1327-0.229 -0.13270.00140.0002 -0.2291 0.00020.0203 From Excel, X T X = (X T X) -1 = X T Y = b = 161458165 145813356015028 165150281751 37577 3429550 387855 1560.67 7.73 8.11

6. ETM 620 - 09U 6 Or, using regression analysis on Excel Regression Statistics Multiple R0.962059425 R Square0.925558337 Adjusted R 2 0.914105774 Std. Error16.51595592 Observations16 ANOVA dfSSMSFSignificance F Regression244089.842204580.824.64306E-08 Residual133546.098272.78 Total1547635.94 CoefficientsStd Errt StatP-valueLower 95%Upper 95% Intercept1560.6678862.9320124.7992E-121424.7115361696.624225 Temp7.7281042110.62488112.3671E-086.3781302669.078078155 Feed Rate8.1135634812.3509363.45120.0043.03467602313.19245094

7. ETM 620 - 09U 7 How do we interpret these results? R 2 – the degree to which the variability of the data is accounted for in the model will naturally increase as number of regressor variables increases adjusted R 2 – adjusted to reflect how well the addition of new regressors improves the ability of the model to account for the variability in the data. adjusted R 2 > R 2 if the new term significantly decreases MS E adjusted R 2 << R 2 if the new term is not significant In our example, R 2 = _______________ ;adj R 2 = ________________ Interpretation?

8. ETM 620 - 09U 8 Confidence intervals around β values … Calculated by, Given in the regression results … Interpretation? CoefficientsStd Errt StatP-valueLower 95%Upper 95% Intercept1560.6678862.9320124.7992E-121424.71151696.6242 Temp7.7281042110.62488112.3671E-086.37813039.078078 Feed Rate8.1135634812.3509363.45120.0043.034676013.19245

9. ETM 620 - 09U 9 A trickier example… The gas mileage for a passenger automobile is believed to be a function of the weight of the car and the horsepower of the engine. Several cars were tested with the following results: ETM 620 - 09U 9 MPG, yWt., x 1 HP, x 2 263.4169 312.5106 203.8304 312.8155 243.6211 293.3140 203.3210 233.9255 244.1255 263.3164

10. ETM 620 - 09U 10 Regression results from Excel … Regression Statistics Multiple R0.84976 R Square0.72209 Adjusted R Square0.64269 Standard Error2.39433 Observations10 ANOVA dfSSMSFSignificance F Regression2104.352.149.0940.0113149 Residual740.135.733 Total9144.4 Coefficients Std Errt StatP-valueLower 95%Upper 95% Intercept36.7447.0465.2150.00120.08178553.406268 Wt., x1-0.19173.169-0.060.953-7.6862797.3029757 HP, x2-0.05430.025-2.150.069-0.1140190.0054114

11. ETM 620 - 09U 11 Let’s try it in Minitab … What do the residuals look like? What does the output of the regression tell us? What do we get if we try “Stepwise Regression”?

12. ETM 620 - 09U 12 Polynomial regression … Example: The expected yield of a crop of marigolds is hypothesized to be a function of the days after the first bloom. Yield (in number of blooms) from a given plot was counted in one growing season with the results as given in the data file. Step 1: plot the data …

13. ETM 620 - 09U 13 Plot of the data …

14. ETM 620 - 09U 14 Fitting the polynomial … Hypothesize the model, In Excel, In Minitab,

15. ETM 620 - 09U 15 Indicator variables Allows us to include qualitative factors in regression analysis … machine type grade of fuel operator Example, In addition to SAT scores, an admissions officer is concerned that whether or not a student attended private high school might affect the freshman GPA. Data from 20 students resulted is given in the data file. Conduct the analysis and interpret the results …

16. ETM 620 - 09U 16 Problems in multiple regression Multicollinearity Influential observations Autocorrelation