Texas Weather Example Multiple Linear Regression

Data Response (Y) – Average January High Temp Predictors: –Latitude –Elevation –Longitude Units – n=16 County Weather Stations CountyTempLatElevLong Harris Dallas Kennedy Midland Deaf Smith Knox Maverick Nolan El Paso Collington Pecos Sherman Travis Zapata Lasalle Cameron

Estimating the Full Model Temp =     LAT   ELEV   LONG  CoefficientsStandard Errort StatP-value Intercept E-05 Lat E-09 Elev Long

Testing the Full Model H 0 :        0 H A : Not all  i = 0 TS: F obs = MSR/MSE = P-Value: P(F≥ )  0 ANOVA dfSSMSFSignificance F Regression E-13 Residual Total

Testing Individual Partial Coefficients H 0 :  i = 0 H A :  i ≠ 0 TS: t obs = b i /SE(b i ) Latitude: t obs = P-value  0 Elevation: t obs = P-value =.1182 Longitude: t obs = P-value =.1182 CoefficientsStandard Errort StatP-value Intercept E-05 Lat E-09 Elev Long

Comparing Regression Models Note: Controlling for ELEV and LAT, LONG does not appear significant (at  =.10 level) and same result holds for LONG. Test whether after controlling for LAT, neither ELEV or LONG related to TEMP H 0 :      H A :    and/or   ≠ 0 Complete Model: –Temp =     LAT   ELEV   LONG  Reduced Model –Temp =     LAT 

Complete and Reduced Models Complete ANOVA (n=16, k=3) dfSSMS Regression Residual Total Reduced ANOVA (g=1) dfSSMS Regression Residual Total

Test of H 0 :      SSR c = , SSE c = SSR r = N=16, k=3, g=1

Model with Latitude and Elevation Temp =     LAT   ELEV  Coefficie nts Standard Errort StatP-value Intercept E-14 Lat E-10 Elev E-06 ANOVA dfSSMSFSignificance F Regression E-14 Residual Total