The general linear test approach to regression analysisThree basic stepsThe full modelSlide 4The reduced modelSlide 6Slide 7The general linear test approachThe general linear test approach (cont’d)Slide 10Slide 11How close is close?But for simple linear regression, it’s just the same F test as before …The formal F-test for slope parameter β1Example: Alcoholism and muscle strength?Slide 16Slide 17The ANOVA tableThe general linear test approach to regression analysisThree basic steps•Define a (larger) full model.•Define a (smaller) reduced model.•Use an F statistic to decide whether or not to reject the smaller reduced model in favor of the larger full model.The full modelFor simple linear regression, the full model is:iiiXY10The full model (or unrestricted model) is the model thought to be most appropriate for the data.The full model54321221814106High school gpaCollege entrance test score xYEY 10 iixY10The reduced modelThe reduced model (or restricted model) is the model described by the null hypothesis H0.For simple linear regression, the null hypothesis is H0: β1 = 0. Therefore, the reduced model is:iiY0The reduced model5432125155High school gpaCollege entrance test score 0 YEYiiY0The reduced model756555432Height (inches)Grade point average 0 YEYiiY0The general linear test approach•“Fit the full model” to the data.–Obtain least squares estimates of β0 and β1.–Determine error sum of squares – “SSE(F).”•“Fit the reduced model” to the data.–Obtain least squares estimate of β0.–Determine error sum of squares – “SSE(R).”The general linear test approach (cont’d)756555432Height (inches)Grade point averagexy 001.095.2ˆ015.3ˆyy5028.7)( FSSE5035.7)( RSSEThe general linear test approach (cont’d)504030200150100Latitude (at center of state)Mortality88.152ˆyyxy 98.5389ˆ 53637RSSE 17173FSSEThe general linear test approach (cont’d)•Compare SSE(R) and SSE(F). •SSE(R) is always larger than (or same as) SSE(F).–If SSE(F) is close to SSE(R), then variation around fitted full model regression function is almost as large as variation around fitted reduced model regression function. –If SSE(F) and SSE(R) differ greatly, then the additional parameter(s) in the full model substantially reduce the variation around the fitted regression function.How close is close?The test statistic is a function of SSE(R)-SSE(F):FFRdfFSSEdfdfFSSERSSEF)()()(*The degrees of freedom (dfR and dfF) are those associated with the reduced and full model error sum of squares, respectively.Reject H0 if F* is large (or if P-value is small).But for simple linear regression, it’s just the same F test as before …FFRdfFSSEdfdfFSSERSSEF)()()(*1ndfR2ndfFSSTORSSE )(SSEFSSE )( MSEMSRnSSEnnSSESSTOF 221*The formal F-test for slope parameter β1Null hypothesis H0: β1 = 0Alternative hypothesis HA: β1 ≠ 0Test statisticMSEMSRF *P-value = What is the probability that we’d get an F* statistic as large as we did, if the null hypothesis is true? (One-tailed test!)The P-value is determined by comparing F* to an F distribution with 1 numerator degree of freedom and n-2 denominator degrees of freedom.Example: Alcoholism and muscle strength?•Report on strength tests for a sample of 50 alcoholic men–X = total lifetime dose of alcohol (kg per kg of body weight)–Y = strength of deltoid muscle in man’s non-dominant arm0 10 20 30 40102030alcoholstrengthReduced Model Fit 32.1224)(21niiYYRSSE164.20ˆyy0 10 20 30 40102030alcoholstrengthFull Model Fit 27.720ˆ)(21niiiYYFSSExy 3.037.26ˆThe ANOVA tableAnalysis of VarianceSource DF SS MS F PRegression 1 504.04 504.040 33.5899 0.000Error 48 720.27 15.006 Total 49 1224.32 SSE(R)=SSTO SSE(F)=SSEThere is a statistically significant linear association between alcoholism and arm
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