Hypothesis tests for slopes in multiple linear regression modelPredictors of Hypertension StudySlide 3Slide 4Possible hypothesis tests for slopesSlide 6Testing all slope parameters are 0Slide 8Slide 9Testing one slope is 0, say β2 = 0Slide 11Equivalence of t-test to partial F-test for one slopeSlide 13Slide 14Testing whether two slopes are 0, say β1 = β2 = 0Slide 16Slide 17Slide 18Hypothesis tests for slopes in multiple linear regression modelPredictors of Hypertension Study•Measured mean arterial blood pressure (BP) of 20 individuals with hypertension.•Also, measured four possible predictor variables:–age (x1)–weight (x2)–body surface area (x3)–duration of hypertension (x4)Predictors of Hypertension Study12011053.2547.7597.32589.3752.1251.8751201108.2754.42553.2547.7597.32589.3752.1251.8758.2754.425BPAgeWeightBSADurationThe regression equation isBP = - 12.9 + 0.683 Age + 0.897 Weight + 4.86 BSA + 0.0665 DurationPredictor Coef SE Coef T PConstant -12.852 2.648 -4.85 0.000Age 0.68335 0.04490 15.22 0.000Weight 0.89701 0.04818 18.62 0.000BSA 4.860 1.492 3.26 0.005Duration 0.06653 0.04895 1.36 0.194Analysis of VarianceSource DF SS MS F PRegression 4 557.28 139.32 768.01 0.000Error 15 2.72 0.18Total 19 560.00Source DF Seq SSAge 1 243.27Weight 1 311.91BSA 1 1.77Duration 1 0.34Possible hypothesis tests for slopes#1. Is hypertension significantly (linearly) related to any of the four predictors?0 oneleast at :0:43210iAHH#2. Is hypertension significantly (linearly) related to weight?0:0:220AHHPossible hypothesis tests for slopes#3. Is hypertension significantly (linearly) related to weight and age?0 oneleast at :0:210iAHHTesting all slope parameters are 0Full modeliiXXXXY443322110SSEFSSE )(5ndfFReduced modeliiY0SSTORSSE )(1ndfRTesting all slope parameters are 0The general linear test statistic: FFRdfFSSEdfdfFSSERSSEF *becomes the usual overall F-test: MSEMSRnSSESSRF 54*Testing all slope parameters are 0Use overall F-test and P-value reported in ANOVA table.Analysis of VarianceSource DF SS MS F PRegression 4 557.28 139.32 768.01 0.000Error 15 2.72 0.18Total 19 560.000 oneleast at :0:43210iAHHTesting one slope is 0,say β2 = 0Full modeliiXXXXY443322110 4321,,,)( XXXXSSEFSSE 5ndfFReduced modeliiXXXY4433110 431,,)( XXXSSERSSE 4ndfRTesting one slope is 0,say β2 = 0The general linear test statistic: FFRdfFSSEdfdfFSSERSSEF *becomes a partial F-test: 5,,,1,,|43214312*nXXXXSSEXXXXSSRF 43214312*,,,),,|(XXXXMSEXXXXMSRF Equivalence of t-testto partial F-test for one slopeSince there is only one numerator degree of freedom in the partial F-test for one slope, it is equivalent to the t-test. ),1(2pnpnFtThe t-test is a test for the marginal signficance of the x2 predictor after x1, x3, and x4 have been taken into account.The regression equation isBP = - 12.9 + 0.683 Age + 4.86 BSA + 0.0665 Duration + 0.897 WeightPredictor Coef SE Coef T PConstant -12.852 2.648 -4.85 0.000Age 0.68335 0.04490 15.22 0.000BSA 4.860 1.492 3.26 0.005Duration 0.06653 0.04895 1.36 0.194Weight 0.89701 0.04818 18.62 0.000Analysis of VarianceSource DF SS MS F PRegression 4 557.28 139.32 768.01 0.000Residual Error 15 2.72 0.18Total 19 560.00Source DF Seq SSAge 1 243.27BSA 1 248.37Duration 1 2.76Weight 1 62.88The regression equation isBP = - 12.9 + 0.683 Age + 0.897 Weight + 4.86 BSA + 0.0665 DurationPredictor Coef SE Coef T PConstant -12.852 2.648 -4.85 0.000Age 0.68335 0.04490 15.22 0.000Weight 0.89701 0.04818 18.62 0.000BSA 4.860 1.492 3.26 0.005Duration 0.06653 0.04895 1.36 0.194Analysis of VarianceSource DF SS MS F PRegression 4 557.28 139.32 768.01 0.000Error 15 2.72 0.18Total 19 560.00Source DF Seq SSAge 1 243.27Weight 1 311.91BSA 1 1.77Duration 1 0.34Testing whether two slopes are 0, say β1 = β2 = 0Full model 4321,,,)( XXXXSSEFSSE 5ndfFReduced modeliiXXY44330 43,)( XXSSERSSE 3ndfRiiXXXXY443322110Testing whether two slopes are 0, say β1 = β2 = 0The general linear test statistic: FFRdfFSSEdfdfFSSERSSEF *becomes a partial F-test: 5,,,2,|,43214321*nXXXXSSEXXXXSSRF),,,(),|,(43214321*XXXXMSEXXXXMSRF The regression equation isBP = - 12.9 + 4.86 BSA + 0.0665 Duration + 0.683 Age + 0.897 WeightPredictor Coef SE Coef T PConstant -12.852 2.648 -4.85 0.000BSA 4.860 1.492 3.26 0.005Duration 0.06653 0.04895 1.36 0.194Age 0.68335 0.04490 15.22 0.000Weight 0.89701 0.04818 18.62 0.000Analysis of VarianceSource DF SS MS F PRegression 4 557.28 139.32 768.01 0.000Residual Error 15 2.72 0.18Total 19 560.00Source DF Seq SSBSA 1 419.86Duration 1 18.42Age 1 56.13Weight 1 62.88Cumulative Distribution FunctionF distribution with 2 DF in numerator and 15 DF in denominator x P( X <= x ) 330.6000
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