Statistics 333 Assignment 6 Due Nov. 17, 20031. The data shown belowrelate to a study on the reaction of formaldehyde with cotton cellu-lose, and were givenbyDev ore (1991). The data consist of measurements on three predic-tor variables and the response Y with n = 30 cases. The variables are:X1= HCHO (formaldehyde) concentration,X2= catalyst ratio,X3= curing temperature,Y = durable press rating, a quantitative measure of wrinkle resistance.X1X2X3YX1X2X3Y84100 3.4 410160 4.624180 3.2 413100 4.374180 4.6 10 10 120 4.910 7 120 4.9 54100 2.974180 4.6 813140 4.677180 4.7 10 1 180 3.6713140 4.6 213140 3.154160 4.5 613180 4.747140 4.8 71120 3.451100 2.4 513140 4.5810140 4.7 81160 3.124100 2.6 41180 2.8410180 4.5 61160 2.567120 4.7 41100 2.310 13 180 4.8 710100 4.6These data are stored in the file: /u/r/e/reinsel/stat333/press-rating.dati) Provide simple scatter plots of Y versus X1, X2,and X3,respectively,and makeany com-ments that seem suitable.ii) Fit a full second-order polynomial (response) model to the data, of the general formY =β0+β1X1+β2X2+β3X3+β11X21+β22X22+β33X23+β12X1X2+β13X1X3+β23X2X3+εFollowbyperforming a complete analysis, leading to selection of a ‘reasonable’ finalreduced model (one that satisfies the origin shift criterion, e.g., see pp. 267−268 in text-book). A ‘complete’ analysis should provide details and include formal justifications (e.g.,consideration of F-tests, R2,adjusted-R2,and S2values) for the selection of a final reducedmodel, and should include the usual plots and examination of (various types of) residualsfrom the final fitted model for checking assumptions and adequacyofthe model.iii) As an additional exercise for these data, for your final fitted model obtain and examineboth the standardized and studentized residuals, and find the Cook statistic value corre-sponding to each data case. Check to assess whether anyofthe cases seem ‘unusual’ interms of outlier behavior or extreme influence, and discuss.2. i) For the simple linear regression model Yi=β0+β1Xi+εi, i = 1, . . . , n,showthat the ithdiagonal of the ‘hat’ matrix H, hii≡ XXi′ ( X ′ X )− 1XXitakes the form hii= (1/n) + (Xi− X )2/Sxx,where Sxx=Σni=1(Xi− X )2.(Hint: It is convenient to express the model in ‘centered’ formas Yi=β*0+β1(Xi− X) +εi,sothat XXi= [1, Xi− X ]′ and X′X = Diag{ n, Sxx}.)ii) For the model Yi=β0+β1Xi1+β2Xi2+εi, i = 1, . . . , n,determine a condition or cir-cumstance under which the hiivalues will takethe explicit form hii= (1/n) +(Xi1− X1)2/Sx1x1+ (Xi2− X2)2/Sx2x2,where Sx2x2=Σni=1(Xi2− X2)2.Verify your result underthe stated condition. (Again, consider the ‘centered’ form Yi=β*0+β1(Xi1− X1) +β2(Xi2− X2) +εi,and consider circumstances involving orthogonality.)iii) When the special results for the model in (ii) do hold, also give simple expressions forthe LSE b2ofβ2in the model, for Var(b2),and for SSR(b2|b0, b1),inv o lving Sx2x2.3. Suppose a response variable Y is fitted by LS using the straight line modelE(Y ) =β*0+β*1X,based on a sample of n = 7 observations with X-values equal to−5, − 3, − 1, 0, 1, 3, 5.Howev e r, itisfeared that there may be some additional quadraticeffect and that Y may actually followthe quadratic model Y =β0+β1X +β2X2+ε.i) Under the assumption that the quadratic model was actually the true model, determinethe biases of the LS estimates b0and b1in estimatingβ0andβ1when the straight linemodel is estimated. [Note: You need to calculate the ‘bias’ matrix (column vector in thiscase), A = (X1′X1)−1X1′XX2,where X1= [11, XX ] with XX = ( − 5, − 3, − 1, 0, 1, 3, 5 )′,andXX2= (25, 9, 1, 0, 1, 9, 25 )′ is the column of X2-values.]ii) Use the result from Problem 4(ii) to find the expected value of the MSE2from fitting the‘reduced’ straight line model E(Y) =β*0+β*1X,when the quadratic model is actually thetrue model. [Note: You need to find the values of˜XX2= XX2− X1( X1′ X1)− 1X1′ XX2.]4. Consider the linear regression model YY = X1ββ1+ X2ββ2+εε,where X = [ X1, X2] is n × p andX1is n × p1, p1< p.The ‘extra’ regression sum of squares due to inclusion of the X2terms, after the X1terms, isSSR( b2| b1) = S1− S2= b2′˜X2′YY = YY ′˜X2(˜X2′˜X2)− 1˜X2′ YY ≡ YY ′˜H2YY ,say,where b2= (˜X2′˜X2)−1˜X2′YY ,˜H2=˜X2(˜X2′˜X2)− 1˜X2′ ,and˜X2= X2− X1(X1′X1)−1X1′X2=(I − H1)X2. SSR( b2| b1) is also the ‘hypothesis’ sum of squares for testing H0:ββ2= 00 .Also recall the useful result that if YY is a n × 1 random vector with mean vectorµµ= E ( YY )and covariance matrix Σ=Cov(YY ) ,and A is a n × n symmetric matrix of constants, then therandom variable Q = YY ′ A YY (a quadratic form in YY )has mean or expected value equal toE( Q ) = E( YY ′ A YY ) = tr( AΣ ) +µµ′ Aµµ,where tr(B) denotes the trace of a matrix B,the sum of its diagonal elements.i) Use the result above todetermine the expected value of SSR( b2| b1),i.e., determineE[SSR( b2| b1)];hence also give the expected value of the mean square MSR( b2| b1) =SSR( b2| b1)/( p − p1).Express your results in simplest terms by noting that X1′˜X2= 00 andalso X2′˜X2= X2′(I − H1)X2= X2′(I − H1)(I − H1)X2≡˜X2′˜X2,showing in particular that theexpected values do not involvethe parametersββ1,only the parametersββ2andσ2.ii) When the ‘reduced’ model E(YY ) = X1ββ*1is estimated by LS, we have the LS estimateb*1= (X1′X1)−1X1′YY and we knowthat the residual SS isSSE2= YY ′ ( I − X1( X1′ X1)− 1X1′ ) YY = YY ′ YY − b*1′ X1′ YY = SSE1+ SSR( b2| b1),where SSE1is the residual SS from fitting the ‘full’ model. Using the known fact (e.g., seeProblem 2 of Assignment 4) that E[SSE1] = (n − p)σ2and the results from (i), determineE[SSE2] and hence also E[MSE2],where MSE2= SSE2/(n − p1),under the assumption thatthe ‘full’ model is the true model, i.e.,ββ2≠ 00