Hypothesis tests for slopes in multiple linear regression modelAn exampleStudy on heart attacks in rabbitsSlide 4A potential regression modelThe estimated regression functionPossible hypothesis tests for slopesSlide 8Linear regression’s general linear testThree basic stepsThe full modelSlide 12Slide 13The reduced modelSlide 15Slide 16The general linear test approachSlide 18Slide 19Slide 20How close is close?But for simple linear regression, it’s just the same F test as beforeThe formal F-test for slope parameter β1Example: Alcoholism and muscle strength?Fit the reduced modelFit the full modelThe ANOVA tableSequential (or extra) sums of squaresWhat is a sequential sum of squares?NotationSlide 31Allen Cognitive Level (ACL) StudyRegress y = ACL on x1 = VocabRegress y = ACL on x1 = Vocab and x3 = SDMTThe sequential sum of squares SSR(X3 | X1)Slide 36The sequential sum of squares SSR(X3 | X1)Regress y = ACL on x3 = SDMTRegress y = ACL on x3 = SDMT and x1 = VocabThe sequential sum of squares SSR(X1 | X3)Slide 41Slide 42More sequential sums of squares (Regress y on x3, x1, x2)Two- (or three- or more-) degree of freedom sequential sums of squaresThe hypothesis tests for the slopesSlide 46Slide 47Testing all slope parameters are 0Slide 49Slide 50Testing one slope is 0, say β1 = 0Slide 52Equivalence of t-test to partial F-test for one slopeSlide 54Equivalence of the t-test to the partial F-testSlide 56Testing whether two slopes are 0, say β2 = β3 = 0Slide 58Slide 59Testing whether β2 = β3 = 0Hypothesis tests for slopes in multiple linear regression modelUsing the general linear test and sequential sums of squaresAn exampleStudy on heart attacks in rabbits•An experiment in 32 anesthetized rabbits subjected to an infarction (“heart attack”)•Three experimental groups:–Hearts cooled to 6º C within 5 minutes of occluded artery (“early cooling”)–Hearts cooled to 6º C within 25 minutes of occluded artery (“late cooling”)–Hearts not cooled at all (“no cooling”)Study on heart attacks in rabbits•Measurements made at end of experiment:–Size of the infarct area (in grams)–Size of region at risk for infarction (in grams)•Primary research question:–Does the mean size of the infarcted area differ among the three treatment groups – no cooling, early cooling, late cooling – when controlling for the size of the region at risk for infarction?A potential regression model iiiiixxxy3322110where …• yi is size of infarcted area (in grams) of rabbit i • xi1 is size of the region at risk (in grams) of rabbit i • xi2 = 1 if early cooling of rabbit i, 0 if not• xi3 = 1 if late cooling of rabbit i, 0 if notand … the independent error terms i follow a normal distribution with mean 0 and equal variance 2.The estimated regression functionELC1.51.00.51.00.90.80.70.60.50.40.30.20.10.0Size of Area at Risk (grams)Size of Infarcted Area (grams)EarlyLateControlThe regression equation is InfSize = - 0.135 + 0.613 AreaSize - 0.243 X2 - 0.0657 X3Possible hypothesis tests for slopes#1. Is the regression model containing all three predictors useful in predicting the size of the infarct?0 oneleast at :0:3210iAHH#2. Is the size of the infarct significantly (linearly) related to the area of the region at risk?0:0:110AHHPossible hypothesis tests for slopes#3. (Primary research question) Is the size of the infarct area significantly (linearly) related to the type of treatment after controlling for the size of the region at risk for infarction?0 oneleast at :0:320iAHHLinear regression’sgeneral linear testAn asideThree 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 full model75706560432Height (inches)Grade point average 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 approach756555432Height (inches)Grade point averagexyF001.095.2ˆ015.3ˆyyR 5028.7ˆ)(2iiyyFSSE 5035.7)(2yyRSSEiThe general linear test approach504030200150100Latitude (at center of state)Mortality88.152ˆyyRxyF98.5389ˆ 536372yyRSSEi 17173ˆ2iiyyFSSEThe general linear test approach•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 the P-value is small).But for simple linear regression, it’s just the same F test as beforeFFRdfFSSEdfdfFSSERSSEF)()()(*1ndfR2ndfFSSTORSSE )(SSEFSSE )(
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