Section 6.9 page 241-251 1 16:00 Thursday, September 20, 2001 Model: MODEL1 Model Crossproducts X'X X'Y Y'Y X'X INTERCEP X1 INTERCEP 21 1302.4 Intercept X1 1302.4 87707.94 Number of persons aged 16 or younger X2 360 22609.19 Per capita disposable personal income Y 3820 249643.35 Sales X'X X2 Y INTERCEP 360 3820 Intercept X1 22609.19 249643.35 Number of persons aged 16 or younger X2 6190.26 66072.75 Per capita disposable personal income Y 66072.75 721072.4 Sales X'X Inverse, Parameter Estimates, and SSE INTERCEP X1 INTERCEP 29.728923483 0.0721834719 Intercept X1 0.0721834719 0.0003701761 Number of persons aged 16 or younger X2 -1.992553186 -0.005549917 Per capita disposable personal income Y -68.85707315 1.4545595828 Sales X2 Y INTERCEP -1.992553186 -68.85707315 Intercept X1 -0.005549917 1.4545595828 Number of persons aged 16 or younger X2 0.1363106368 9.3655003765 Per capita disposable personal income Y 9.3655003765 2180.9274114 Sales Section 6.9 page 241-251 2 16:00 Thursday, September 20, 2001 Dependent Variable: Y Sales Analysis of Variance Sum of Mean Source DF Squares Square F Value Prob>F Model 2 24015.28211 12007.64106 99.103 0.0001 Error 18 2180.92741 121.16263 C Total 20 26196.20952 Root MSE 11.00739 R-square 0.9167 Dep Mean 181.90476 Adj R-sq 0.9075 C.V. 6.05118 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -68.857073 60.01695322 -1.147 0.2663 X1 1 1.454560 0.21178175 6.868 0.0001 X2 1 9.365500 4.06395814 2.305 0.0333 Variable Variable DF Label INTERCEP 1 Intercept X1 1 Number of persons aged 16 or younger X2 1 Per capita disposable personal income Covariance of Estimates COVB INTERCEP X1 INTERCEP 3602.0346743 8.7459395806 Intercept X1 8.7459395806 0.0448515096 Number of persons aged 16 or younger X2 -241.4229923 -0.672442604 Per capita disposable personal income COVB X2 INTERCEP -241.4229923 Intercept X1 -0.672442604 Number of persons aged 16 or younger X2 16.515755794 Per capita disposable personal income Correlation of Estimates CORRB INTERCEP X1 INTERCEP 1.0000 0.6881 Intercept X1 0.6881 1.0000 Number of persons aged 16 or younger X2 -0.9898 -0.7813 Per capita disposable personal income Section 6.9 page 241-251 3 16:00 Thursday, September 20, 2001 Correlation of Estimates CORRB X2 INTERCEP -0.9898 Intercept X1 -0.7813 Number of persons aged 16 or younger X2 1.0000 Per capita disposable personal income Section 6.9 page 241-251 4 Residual plot against yhat(predicted value) Fig 6.7 (a) 16:00 Thursday, September 20, 2001 Plot of RESID*PRED. Legend: A = 1 obs, B = 2 obs, etc. | | 20 + A | | | | 15 + | | | A | 10 + A A A | A A | | | 5 + R | A e | s | A i | A AA d 0 + A u | a | l | | -5 + A | A A | | | -10 + | | A | A A | -15 + A | | | A | -20 + | -+-----------+-----------+-----------+-----------+-----------+-----------+ 120 140 160 180 200 220 240 Predicted Value of Y Section 6.9 page 241-251 5 Residual plot against x1 Fig 6.7 (b) 16:00 Thursday, September 20, 2001 Plot of RESID*X1. Legend: A = 1 obs, B = 2 obs, etc. | | 20 + A | | | | 15 + | | | A | 10 + A A A | A A | | | 5 + R | A e | s | A i | A A A d 0 + A u | a | l | | -5 + A | A A | | | -10 + | | A | A A | -15 + A | | | A | -20 + | --+---------+---------+---------+---------+---------+---------+---------+- 30 40 50 60 70 80 90 100 Number of persons aged 16 or younger Section 6.9 page 241-251 6 Residual plot against x2 Fig 6.7 (c) 16:00 Thursday, September 20, 2001 Plot of RESID*X2. Legend: A = 1 obs, B = 2 obs, etc. | | 20 + A | | | | 15 + | | | A | 10 + A B | A A | | | 5 + R | A e | s | A i | A A A d 0 + A u | a | l | | -5 + A | A A | | | -10 + | | A | A A | -15 + A | | | A | -20 + | -+--------+--------+--------+--------+--------+--------+--------+--------+ 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 Per capita disposable personal income Section 6.9 page 241-251 7 Residual plot against x1*x2 Fig 6.7 (d) 16:00 Thursday, September 20, 2001 Plot of RESID*X12. Legend: A = 1 obs, B = 2 obs, etc. | | 20 + A | | | | 15 + | | | A | 10 + A A A | A A | | | 5 + R | A e | s | A i | A B d 0 + A u | a | l | | -5 + A | A A | | | -10 + | | A | A A | -15 + A | | | A | -20 + | -+-----------+-----------+-----------+-----------+-----------+-----------+ 600 800 1000 1200 1400 1600 1800 X12 Section 6.9 page 241-251 8 Absolute residual plot against predicted value Fig 6.8 (a) 16:00 Thursday, September 20, 2001 Plot of ABSRESID*PRED. Legend: A = 1 obs, B = 2 obs, etc. ABSRESID | | 20.0 + A | | | A | 17.5 + | | | | 15.0 + A | | | | A A 12.5 + A | | A | | 10.0 + A A A | A A | | | 7.5 + | | | A A | 5.0 + | A | A | | 2.5 + | | A A | | A A A 0.0 + | --+----------+----------+----------+----------+----------+----------+-- 120 140 160 180 200 220 240 Predicted Value of Y Section 6.9 page 241-251 9 Normal probability plot Fig 6.8 (b) 16:00 Thursday, September 20, 2001 -------------------------------------------------------------------- 30 + | | | | | | | | | | | | | | | | | 20 + + | | | | | | | | | | | | | | | | + | 10 + + + + | R | + + | e | | s | | i | | d | | u | + | a | | l | + ++ + | 0 + + | | | | | | | | + | | + | | + | | | | | -10 + | | | | + | | + + | | | | + | | | | | | + | -20 + | +---------+----+--------+---------+--------+--------+----+---------+ 1 5 10 25 50 75 90 95 99 Normal Percentiles Section 6.9 page 241-251 10 To show that CORR(YHAT,Y)=SQRT(R^2) Comment 2 page 230 16:00 Thursday, September 20, 2001 Correlation Analysis 2 'VAR' Variables: Y PRED Simple Statistics Variable N Mean Std Dev Sum Y 21 181.904762 36.191304 3820.000000 PRED 21 181.904762 34.652043 3820.000000 Simple Statistics Variable Minimum Maximum Label Y 137.200000 244.200000 Sales PRED 136.846021 234.396324 Predicted Value of Y Pearson Correlation Coefficients / Prob > |R| under Ho: Rho=0 / N = 21 Y PRED Y 1.00000 0.95747 Sales 0.0 0.0001 PRED 0.95747 1.00000 Predicted Value of Y 0.0001 0.0 Section 6.9 page 241-251 11 Test for normality-Shapiro-Wilks test etc 16:00 Thursday, September 20, 2001 Univariate Procedure Variable=RESID Residual Moments N 21 Sum Wgts 21 Mean 0 Sum 0 Std Dev 10.44253 Variance 109.0464 Skewness -0.09705 Kurtosis -0.79427 USS 2180.927 CSS 2180.927 CV . Std Mean 2.278746 T:Mean=0 0 Pr>|T| 1.0000 Num ^= 0 21 Num > 0 13 M(Sign) 2.5 Pr>=|M| 0.3833 Sgn Rank 1.5 Pr>=|S| 0.9599 W:Normal 0.955277 PrF Model 3 24023.71574 8007.90525 62.663 0.0001 Error 17 2172.49378 127.79375 C Total 20 26196.20952 Root MSE 11.30459 R-square 0.9171 Dep Mean 181.90476 Adj R-sq 0.9024 C.V. 6.21457 Parameter Estimates Parameter Standard T for H0: Variable DF Estimate Error Parameter=0 Prob > |T| INTERCEP 1 -12.476266 227.96283849 -0.055 0.9570 X1 1 0.531948 3.59800075 0.148 0.8842 X2 1 6.093302 13.40394510 0.455 0.6552 X12 1 0.052881 0.20584943 0.257 0.8003 Variable Variable DF Label INTERCEP 1 Intercept X1 1 Number of persons aged 16 or younger X2 1 Per capita disposable personal income X12 1 Section 6.9 page 241-251 13 For joint inferences we need to find t & F-percentiles not in the tables 16:00 Thursday, September 20, 2001 OBS A B C D 1 2.44501 2.63914 3.15991 3.55456