Note that the anova sums of squares are all quadratic

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Note that the ANOVA sums of squares are all quadratic forms. i) SST = Y Y 1 n Y 11 Y = Y [ I 1 n 11 ] Y ii) SSR = ˆ β X Y 1 n Y 11 Y = Y [ H 1 n 11 ] Y iii) SSE = Y Y ˆ β X Y = Y [ I H ] Y PAGE 14
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1.3 Matrix approach to Simple Linear Models c circlecopyrt HYON-JUNG KIM, 2017 We can also test a hypothesis about a linear combination of parameters, l β . l ˆ β Normal ( l β , l (X X) 1 l σ 2 ) . A two sided test: H 0 : l β = M vs. H a : l β negationslash = M can be done as follows. Reject H 0 if | t | = vextendsingle vextendsingle vextendsingle vextendsingle l ˆ β M vextendsingle vextendsingle vextendsingle vextendsingle s ( l ˆ β ) > t 1 α 2 ,n p , s( l ˆ β ) = radicalbig l (X X) 1 l MSE. (Recap) Under the normality assumption, the 100(1 α )% confidence interval for β k is ˆ β k ± t 1 α 2 s( ˆ β k ) , k = 1 , 2 s( ˆ β k ) = radicalBig (X X) 1 kk MSE . Simple example. DATA (X,Y): (4,5), (7,7), (5,6), (4,6) Assume that σ 2 = 0 . 25 known. 5 7 6 6 = 1 4 1 7 1 5 1 4 bracketleftBigg β 0 β 1 bracketrightBigg + ǫ 1 ǫ 2 . . . ǫ n We find the estimated parameters using the formula ˆ β = (X X) 1 X Y = (3 . 5 , 0 . 5) (Verify!) The variance-covariance matrix of parameter estimates is bracketleftBigg 4 . 4167 0 . 8333 0 . 8333 0 . 1667 bracketrightBigg (0 . 25) = bracketleftBigg 1 . 1042 0 . 2083 0 . 2083 0 . 0417 bracketrightBigg (1) Test that the slope is 0: t= 0 . 5 / 0 . 0417 = 0 . 5 / 0 . 2041 = 2 . 45 (2 degrees of freedom) (2) Test that the intercept is 0: (2 degrees of freedom) (3) Estimate the mean value of Y at X = 6 and give a 95% confidence interval. We are estimating: β 0 + 6 β 1 = [1 , 6] bracketleftBigg β 0 β 1 bracketrightBigg and l ˆ β = hatwidest Var( l ˆ β ) = l hatwidest Var( ˆ β ) l = [1 , 6] bracketleftBigg 1 . 1042 0 . 2083 0 . 2083 0 . 0417 bracketrightBigg bracketleftBigg 1 6 bracketrightBigg = 0 . 1042 . The 95% confidence interval for our little example is Note that the 97.5% quantile of the t distribution with 2 d.f. = 4 . 30. PAGE 15
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1.4 DIAGNOSTICS and MODEL EVALUATION c circlecopyrt HYON-JUNG KIM, 2017 1.4 DIAGNOSTICS and MODEL EVALUATION When a simple linear model is selected for an application, one cannot be certain in advance that the model is appropriate to the dataset in question. Note that all the tests, intervals, predictions, etc., are based on believing that the assumptions of the regression hold. To assess whether the assumptions underlying the model seem reasonable, we study the residuals with graphical analyses. That is, we analyze the residuals to see if they support the assumptions of linearity, independence, normality and equal variances. When conducting a residual analysis, we make the following plots; (1) plot of the residuals versus the fitted values.
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