1 Stat 511 Exam1 February 26, 2008 Prof. Vardeman I have neither given nor received unauthorized assistance on this exam. ________________________________________________________ Name _______________________________________________________________ Name Printed21. Consider a problem of quadratic regression in one variable, x. In particular, suppose that 5n= values of a response y are related to values 0,1,2,3,4x= by a linear model =+YXβε for 112023134245510 011 1, , , and 12 413 91416yyyyyεβεβεβεε⎛⎞ ⎛⎞⎛⎞⎜⎟ ⎜⎟⎜⎟⎛⎞⎜⎟ ⎜⎟⎜⎟⎜⎟⎜⎟ ⎜⎟⎜⎟== = =⎜⎟⎜⎟ ⎜⎟⎜⎟⎜⎟⎝⎠⎜⎟ ⎜⎟⎜⎟⎜⎟⎜⎟ ⎜⎟⎝⎠⎝⎠ ⎝⎠YX βε a) A cofactor expansion of the determinant of a 33× matrix ()ija=A on its first row is ()()()()11 11 12 12 13 13det det det detaaa=−+AM M M where ijM is obtained from A by deleting its thi row and thj column. Use this fact and argue that X is of full rank. b) Define 1122 1260 1 4 so that is nonsingular and 0 1 40 0 1 001−−⎛⎞ ⎛⎞⎜⎟ ⎜⎟=− =⎜⎟ ⎜⎟⎜⎟ ⎜⎟⎝⎠ ⎝⎠FFF Let =WXF. Argue carefully that ()()CC=WX.3c) Notice that ′WW is diagonal. Suppose that ()2,0,4,2,2′=−Y . Find the OLS estimate of γ in the model =+YWγε and then OLS estimate of β in the original model. (Find numerical values.) OLSˆ=γ OLSˆ=β d) As it turns out, for the set of observations in c), 128/ 35SSE=. What then is an estimated variance-covariance matrix for OLSˆγ? What is the corresponding estimated variance-covariance matrix for OLSˆβ? (No need to simplify/do matrix multiplications.)4e) Based on your answer to d) give 95% confidence limits for the difference in mean responses for 3 and 1xx== in the normal Gauss-Markov model. (No need to simplify. If you don't have appropriate tables with you, tell me exactly what quantile(s) of what distribution you need.) f) Consider the hypothesis [][]0H : 0 and 4 have the same mean response and 1 and 3 have the same mean responsexx xx== == (Notice that in this model this is the hypothesis that the quadratic regression function has a critical point at 2x = .) Show how to test this hypothesis. (Show how to compute an appropriate test statistic and say exactly what null distribution you could use.) Be careful. Some ways of writing this hypothesis may not produce a "testable" hypothesis.52. Notice that the first characterization of the estimability of ′c β in the linear model =+YXβε can be rephrased as "′c β is estimable iff there is a linear combination of the variables 12,,,nyyy… with expected value ′c β for all β ." a) Consider the "effects model version of the 3-group/2-observations-per-group" one-way ANOVA model used repeatedly as an example in lecture early in the term (version 2) of example b)). Using the characterization of estimability above, show that ()12313μτττ+++ is estimable. b) In the context of part a) above, give formulas for both the BLUE of ()12313μτττ+++ AND an unbiased linear estimator of this quantity that is NOT the BLUE.63. Attached to this exam is an R printout made in the analysis of data from a pilot plant chemical process. Process yield is thought to depend upon condensation temperature and amount of boron employed. A "quadratic response surface" model of the form 2201 2 3 4 5ytBtBtBβββ β β β ε=++ + + + + has been fit to the yield data (and by the way, predicts a maximum yield near 1t =− and .5B=). a) Give the value of an F statistic and degrees of freedom for testing the hypothesis that ()()E||C∈Y1tB in this model. F = __________ d.f.=_____ , _____ b) Which observation (or observations) appears (appear) to be most influential in determining the character of the fitted surface? "Where" is (are) that (those) observation (observations) in (),tB-space? (Note that there is a plot on the printout.) Provide rationale for your choice.7 > chemical yield time temp Bamt 1 21.1 150 90 24.4 2 23.7 10 90 29.3 3 20.7 8 90 34.2 4 21.1 35 100 24.4 5 24.1 8 100 29.3 6 22.2 7 100 34.2 7 18.4 18 110 24.4 8 23.4 8 110 29.3 9 21.9 10 110 34.2 > t<-temp-100 > B<-Bamt-29.3 > plot(t,B) > t2<-t^2 > B2<-B^2 > tB<-t*B > quadyield<-lm(yield~t+B+t2+B2+tB) > summary(quadyield) Call: lm(formula = yield ~ t + B + t2 + B2 + tB) Residuals: 1 2 3 4 5 6 7 8 9 -0.06389 -0.02222 0.08611 0.27778 -0.25556 -0.02222 -0.21389 0.27778 -0.06389 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 24.355556 0.228499 106.589 1.82e-06 *** t -0.030000 0.012515 -2.397 0.096130 . B 0.142857 0.025542 5.593 0.011289 * t2 -0.009333 0.002168 -4.306 0.023061 * B2 -0.118006 0.009028 -13.070 0.000967 *** tB 0.019898 0.003128 6.361 0.007863 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.3066 on 3 degrees of freedom Multiple R-squared: 0.9889, Adjusted R-squared: 0.9704 F-statistic: 53.37 on 5 and 3 DF, p-value: 0.00394 > vcov(quadyield) (Intercept) t B t2 B2 tB (Intercept) 5.221193e-02 1.688218e-19 5.681659e-20 -3.132716e-04 -1.304755e-03 -6.622578e-20 t 1.688218e-19 1.566358e-04 5.186088e-21 -7.078389e-22 -9.409579e-21 -4.449178e-22 B 5.681659e-20 5.186088e-21 6.523774e-04 -6.614999e-22 -4.488606e-21 2.036626e-21 t2 -3.132716e-04 -7.078389e-22 -6.614999e-22 4.699074e-06 1.507303e-22 7.738612e-22 B2 -1.304755e-03 -9.409579e-21 -4.488606e-21 1.507303e-22 8.151321e-05 1.951499e-21 tB -6.622578e-20 -4.449178e-22 2.036626e-21 7.738612e-22 1.951499e-21 9.785660e-068 > anova(quadyield) Analysis of Variance Table Response: yield Df Sum Sq Mean Sq F value Pr(>F) t 1 0.5400 0.5400 5.7458 0.0961299 . B 1 2.9400 2.9400 31.2828 0.0112892 * t2 1 1.7422 1.7422 18.5379 0.0230607 * B2 1 16.0556 16.0556 170.8374 0.0009672 *** tB 1 3.8025 3.8025 40.4601 0.0078628 ** Residuals 3 0.2819 0.0940 ---