Methods And Applications Of Linear Models Regression And The Analysis Of Variance 2nd Edition Ronald R. Hocking - Solutions

Consider the constrained cell means model defined by Bij hije for 1,3,1,4, and r 1,,n, subject to the constraints - =13 s -414-24 Use both the model reduction and the Lagrange multiplier methods to answer the following questions assuming Varie) 021a. Determine the estimates of p, anda. b. Determine
Referring to Example 17.2, with the design matrix X X(XX)X denote the Hat matrix, and define H, (JX2), let. H-(1/N)U. H. Use Cochran's theorem to determine the joint distribution of the quadratic forms 91= 42 = (-H,) Show that aq, N- =
Referring to Example 17.3, use the results of Section 17.1.5 to express the model in terms of the parameters aro B, a B+B and og B-B In particular, specify the transformation matrix M and the new design matrix Z. Letting, denote the expected value of the elements of y, verify that E] (B+B) = 14J
Let Ny be the numerator sum of squares, from Theorem the hypothesis H: Hh in the unconstrained linear model. is an equivalent statement of this hypothesis, say, HB Na (HB-)(HB-). Hint: See Appendix A.1.7. Applications 17.5, for testing Show that there h, such that
Consider the cell means model y = th + Cup 1 = 1,.,p./= 1, ..., n.a. Determine the test statistic for the hypothesis H = using both the model reduction and the Lagrange multiplier methods.b. Specify the distribution of this statistic.e. For p=5 and 10, determine the critical value for a = 0.05.d.
Referring to Example 17.3, assume that the response vector is written in partitioned form as ya. Confirm the expression for the inverse of XTX, and determine the estimates of the parameter vector .b. Confirm the expression for the elliptical acceptance region, and write it in algebraic form.e.
Refer to Example 17.2, where X (J | X).a. Show that the square of the sample correlation coefficient based on the vector of observations y and the vector of predicted values is equal to Rb. Show that the least squares estimate maximizes this correlation in the class of predictors of the form = xa.
Determine the distribution of QB). Show that this quadratic form is not independent of Q(B), hence (17.66) is not proportional to an F-statistic.
Establish the independence of QB) and N as follows:a. Consider the special case 0, noting that both numerator and denominator can be written as quadratic forms on yb. In general, with 0, the numerator sum of squares is the sum of the quadratic form in (a) and a linear form in t that is also
Verify:a. The expression for the likelihood ratio statistic in (17.66).b. The alternative ratio in (17.68).c. The distribution of the numerator sum of squares in (17.69).d. The independence of NH and Q(B) as defined in (17.73).
Consider the partitioned form of the linear model in Section 17.1.4, where B, and B, denote the estimates of 8, based on the full model and the model reduced by the constraint B = 0.a. Determine the expected value of B, under the full model.b. If 8, is an element of B, show that Var] Var]. Hint:
Generalize the results of Exercise 17.8 to the model y~ N(X,o). In particular, consider the distribution of zo Ay where A satisfies the conditions AA-IN- and AA=1-x(x*x)x and show that the maximum likelihood estimate of o2 based on the distribution of is the unbiased estimate given in (17.18).
Let y NJ, o) and let H, denote the Helmert matrix defined in Exercise 16.10, written in partitioned form as H.a. Determine the distribution of z Heyb. Using the marginal distribution of 24 determine the maximum likelihood estimate of 2.e. Show that this estimate is the unbiased estimate given in
Develop the model reduction transformations in Section 17.1.5 for the general case g 0, and compare with the results of Section 17.1.2.
Equation (17.48) gives the difference in the residual sums of squares for the original model and the model under the constraint, B-0. Show that this result is a special case of equation (17.39).
Show that the constrained least squares estimator is BLUE. Hint: Consider a linear function of the form ey+d, determine the condition for unbiasedness, and then determine the values of c and d to minimize the variance. Note that the scalar d can be written as Ag for some vector h.
a. Show that B and B are both unbiased estimates of in the constrained model.b. Show that the matrix BG(XX) is positive definite.e. Establish the relation between QB) and Q(3) shown in (17.37) and verify the inequality.d. Verify the relation in (17.38).
Verify the properties of the matrix A defined in Lemma 3.1.
a. Show that the expressions for the constrained estimator given in (17.27) and (17.33) are identical.b. Show that the covariance matrices for in Theorems 17.2 and 17.3 are identical.e. Show that the matrix Xx defined in (17.23) has rank r.d. Show that there is no loss of generality in assuming
Verify that the solution of the likelihood equations given in (17.9) maximizes the likelihood function. (Hint: See Appendix A.II.3.)
Consider the linear model defined by, Ey] = (L, J, J.) and with Varly) +V+0V V = U, Ia, U Define the quadratic forms, q,y Ay, by the matrices A = S, U -U A = S, Ua. Determine the distribution of these quadratic forms.b. Suggest F-ratios for testing the hypotheses, H = 0 and H12:02c. Suggest an
Consider the linear model defined in Example 16.3.a. Letting 44, +4, +43, show that AV = 1 and r(4) = (4.)-b. Verify that A- an (D.).c. Determine the expected values of these quadratic forms, and note the relation of the non-centrality parameter to the algebraic expression for the quadratic forms.
Consider the linear model y Wu+e, where WI, J, and eN(0,0). Define the quadratic forms and by the matrices q, A = IS, and A = SUa. Determine the distributions of 1 and 2, and show that they are independent.b. Letting and r denote the ranks of A, and 42, determine the distribution of F
Suppose that A and B are symmetric matrices.a. Show that there always exists a non-singular matrix, M, such that MTAM and MTBM are diagonal. Hint: Let C be a non-singular matrix such that CAC I, and let P be an orthogonal matrix such that PCBCP is diagonal.b. Show that there exists an orthogonal
Let yN(, ), and let A be a symmetric matrix. Show that the distribution of the quadratic form q-Ay can be represented as a linear combination of independent, non-central chi-squared variables. In particular, where &, are the distinct eigenvalues of the matrix AV. Determine the degrees of freedom,
Let y, and y be vectors of lengths p; and p, and let A be a p Pz matrix.a. Determine the expected value of the bilinear form 4 Hint: Let () and consider the quadratic form "By for an appropriate matrix Bb. Let y have a general linear model with design matrix X=1, J. covariance matrix V = a + and
Suppose that y~N(,), and let [ 10 4,1,2,3, where 1 10 and As 00 1 2a. Use Theorem 16.4. to determine the distribution of each quadratic form.b. Use Theorem 16.5 to show the pairwise independence of the quadratic forms and then apply Cochran's theorem to establish the mutual independence.
Assume that the N-vector y NJ, 0), and let H be the Helmert matrix defined as follows: The first row of H is (1/N). The rth row, for -2,, N, is given by (F-1) T(T-1) %).a. Show that H is an orthogonal matrix.b. Determine the distribution of 2 - Hyc. Use this transformation to determine the joint
Prove Corollary 16.4.
a. Spell out the details of the development in the proof of Theorem 16.3.b. Verify the expression for the variance of q in Corollary 16.3, part 1, by differentiating the moment-generating function.c. Verify the expression for the covariance of a linear and a quadratic form in Corollary 16.3, part 3.
We have assumed that is non-singular since, if not, we can always use the linear relations in y to remove the redundancies and apply our results to the resulting quadratic form.a. Show how this can be done by writing 4 in partitioned form.b. Extend Theorem 16.4 to include the case in which is
The proof of the necessity half of Corollary 16.2, part 2 is tedious. To see the idea, consider the special case k = 2 and assume that 0,0 = 0. Equate the first two moments of both sides of the equation qaq+ags, and show that the condition d= d + d implies that a = a = 1.
Verify the first two moments of x2(N,6) given in Corollary 16.2 as follows:a. Use the density function from Theorem 16.2 to evaluate Elg] and Elg].b. Determine the first two derivatives of the moment-generating function.
Derive the distribution of the non-central chi-squared statistic as follows;a. Assume that y~N(,), and let z Ay, where A is an orthogonal matrix whose first row is (1/1) with y 26. (This rotation puts all of the non-centrality on one of the variables.)b. Let vand independent of a with density for
For the special case N = 2, determine the probability content of the sphere described by Equation (16.17). Use this result to confirm that qx2(2). Hint: Transform to polar coordinates.
Provide the details for the proof of Corollary 16.1. Section 16.3
Verify the results in Equations (16.5), (16.6), and (16.7). Section 16.2
Ostle and Malone (1988) describe an experiment designed to compare the yields of three varieties of a crop that was run as a randomized complete block design with four blocks. A covariate z, is the yield for the ith variety in the jth block from the preceding year. The data are shown in the
For models with covariates we have recommended centering the covariate about some constant prior to introducing the artificial values. Show that the sum of squares of the augmented covariates is minimized by centering about the mean of the original covariate values.
Use a standard linear model program to fit the fixed effects analog of the model in Example 15.10.Data for Exercise 15.12 Populations Obs. 1 2 3 1.40 1.61 1.67 2 1.79 1.31 1.41 31 1.72 1.12 1.73 4 1.47 1.35 1.23 5 1.26 1.29 1.49 6 1.28 1.24 1.22 7 1.34 1.39 8 1.55 1.56 9 1.57 10 1.26a. Develop the
Use the general description of the AVE computational procedure in Section 14.5 to verify that the expressions in (15.107) and (15.118) are identical. Section 15.6
Use the equations described in Example 15.4 to analyze the data shown below, assuming the one-way classification random model. Begin the iteration with 6,- = 1. Missing observations are denoted by a period. Section 15.5
Use a standard linear model program, and compute the analysis of variance tables for the unbalanced data in Examples 15.4 and 15.5. Compare these tables with Table 15.4 and 15.5. In particular, compare the estimates of the variance #!# components and the tests of the fixed effects hypotheses.
a. Verify the results in (15.98) and (15.99) for the one-way classification random model.b. Show that the test statistics in (15.92) and (15.96) are identical for the one- way classification model.
a. Establish the expected value in (15.94) by writing E[&] = E[(D)D'AD(D'")] and noting that D"= " = [ 17 ] M. 9, + DTX M+E[D] = D'Xa Var[D][M.M + D'X(X!VX.) 'X'D.b. Verify that the first term in (15.94) is zero for T. Hint: Use the procedure as in (15.86) and use (15.33) and the properties of B to
Spell out the details of the relation established in (15.90). Hint: Use results from the partitioned form of the inverse matrix in Appendix A.1.10, and note that we may write -- (mm)
Verify the simplification in (15.84) by noting that -8 and XIV M.-
Hartley (1967) proposed the following method for computing the expected value of mean squares for the mixed model with unbalanced data. For a given method of computing the 40V table, as if the model were fixed, let Q(y) = y By a mean square associated with a random effect. bea. Assuming that tr(B)
Show that the solution to the MINQUE problem (15.40), is equivalent to one iteration on the REML equations. Hint: Use the method of Lagrange multipliers, with Lagrangian written as L-over-exo)-(e)-a). where is the Nxr matrix of multipliers associated with the constraints Qx-0. In the solution of
Verify the relations in (15.34) - (15.36).
a. Verify that the second-order approximation is given by (15.19) and (15.20).b. Verify that E[h] (1/2) and hence the relation in (15.22).c. Verify that the information matrix is given by (15.24).d. Show that the method of scoring is equivalent to iterating on (15.16).
Write the linear model for the situation described by the index set T={1,2, 12, 3(12), 4(12), 3(12)4(12)).a. Describe the implied covariance structure.b. Write the AOV table, including the Kronecker product expressions for the mean squares and the expected mean squares.c. Use the relations from
Write the linear model for the situation described by the index set T (1, 2, 12, 3(1), 23(1), 4(1), 24(1), 3(1)4(1), 23(1)4(1)).a. Describe the implied covariance structure.b. Write the AOV table, including the Kronecker product expressions for the mean squares and the expected mean squares.c. Use
For the four-factor model described in Example 14.10, suggest a possible third factor that might have been considered in the Kirk (1995) example.a. Write the linear model and describe the implied covariance structure.b. Write the AOV table, including the Kronecker product expressions for the mean
Develop the analysis for the situation described in Example 14.9. In particular:a. Write the appropriate linear model and describe the implied covariance structure.b. Write the AOV table, including the Kronecker product expressions for the mean squares and the expected mean squares.c. Use the
Develop the analysis for the situation described in Example 14.8. In particular, Write the appropriate linear model and describe the implied covariance structure.b. Write the mean squares for Table 14.16 in Kronecker product form and verify that (14.70) yields the expected values and matrices in
Recall Example 14.7 with four factors such that the first two factors are crossed, the third factor is nested in the first, and the fourth is nested in the third. Assume that the first two factors are fixed with cell means 14, and that the remaining factors are random.a. Justify the choice of the
Apply the AVE method to the Brownlee, three-factor data in Exercise 13.18.b. Determine the expected values of the AVE quadratic forms under the alternative definition of the variance components. Show that the diagnostic features of AVE are also informative under this model. Section 14.6
Use the general description of Lin (14.58) to describe the relation between y and A for the four-way classification model.
a. Verify that (14.54) provides the relation between the AOV expected mean squares and the expected values of the AVE quadratic forms.b. Use the Kronecker product definition of the mean squares in the AOV table to verify that (14.54) yields the quadratic forms as described in Table 14.10. Data for
Brownlee (1960) described an experiment designed to study two different annealing methods used in making of cans. Three coils of material were selected from the populations of coils made by each of the two methods. From each coil two samples were taken from each of two locations on the coil. The
a. Apply the AVE diagnostic analysis to the Littell, nested-factorial data in Exercise 13.7. In particular, compute the estimate of for each of the four modes.b. Repeat the analysis if factor two, the position on the wafer, is assumed to be random.
a. Verify algebraically, using the AOV mean squares and expected values, that (14.38) is the estimate of (1)b. Write the test statistics for the marginal means hypotheses for main effects for Example 14.4 in terms of the AVE estimates for the random model using Table 14.8 and verify (14.43).
Write the AOV table for the two-fold nested model and compare the quadratic forms and the expected values with the AVE in Table 14.4. Section 14.4
Apply the AVE method to examine the variance component estimates in the Thompson-Moore, two-factor data in Exercise 13.5. In particular, note how W, explains the source of the negative estimate. Note also the unusual structure of W. Section 14.3
a. In Example 14.1 determine the matrix W, as if the fertilizer factor was random. Examine this matrix and the associated scatter-plot matrix for any unusual features.b. Describe the average of the diagonal elements of W, and determine the expected value under the mixed model with factor one
a. Verify the relation in (14.9) and the expression for , in (14.7).b. Verify the algebraic expression for d in (14.10).c. Show that T12 +12 is given by (+12)/na. Verify the expressions for 12 in (14.11) is given by (+2)/na.d. Verify the expressions for the individual elements of Win (14.12) and
Develop the analysis for Example 13.7 assuming that the term (ed), is included in the model. Write the AOV table and compute the estimates of the variance components Data for Exercise 13.18 Tube ABC ABC Sample Bottle Bottle II 2 3 4 6 7 8 9 111 13 2 3 4 2 2 1 3 324 3 2 4 3 36 2 41100 2 1 3 2 4 6
Verify the expected mean squares in Table 13.8. What changes occur if we include the term (ed) N(0,a) in the model?#!# 13.18. Brownlee (1960) describes an experiment in bacteriological testing of milk. Twelve milk samples were examined in all six combinations of two types of bottles and three types
Prove Theorem 13.1. The sufficiency follows as in the proof in Section 13.5.2.2. For the necessity, show that the condition of the theorem is satisfied with 13.15 Verify the expression for Vin (13.81) using the general results in Section 13.5. 13.16 Write the analysis of variance table for the
Verify that the coefficient of 4; in (13.54) is given by (13.56). Hint: From (13.54), the matrices in 4 have coefficients A, and A for s 11. It is sufficient to verify that = As for s 1.b. Verify the expression for the inverse of in (13.59) and the analogous expression for the nested-factorial
a. Verify the relation in (13.41) for the one- and two-way classification models.b. Suggest a general proof by mathematical induction.
Consider the three-way, cross-classification model with factors one and two and their interaction fixed and all other effects random.a. Describe the covariance structure algebraically and then in matrix form as in (13.36) and (13.48).b. Write the AOV table with expected mean squares.e. Describe the
To illustrate the differences in the various methods of writing confidence intervals, consider the one-way classification model described in Section 13.2 with mean squares given in Table 13.1.a. Determine the estimates of o, and, and compute exact 90% confidence intervals for, and the ratio /c. Use
Verify the confidence intervals for Example 13.2 given in Section 13.5. Note that you may use Table C-1, noting that x(a;d) da;d, oo) and F((-a); d,d)=1/F(a;d, ds).
Verify the expression for V in (13.36).
a. For the nested-factorial model in (13.27), verify the covariance structure in (13.28) and the matrix expressions in (13.29) and (13.31).b. Verify the expressions for the expected mean squares in Table#!# 13.7.c. Verify the distributions in (13.34). 13.7 Littell (1987) examined data, shown below,
Thompson and Moore (1963) described a study that examined the muzzle velocity of a type of ammunition as a function of propelling charges and projectiles. The objective of the study was to examine the sources of variability. In the production process the manufacturer groups the charges and
Using (13.26) write 100(1-a) % separate-t and Scheffe confidence intervals for machine differences as estimated by the marginal means.
Using the expression for V in (13.20), verify the expected mean squares in Table 13.4.
For the two-way classification random model, verify the covariance structure in (13.14) and the matrix expression in (13.15) and (13.29). Use the expressions for the sums of squares in Table 11.4 to determine the expected values of the mean squares.
a. For the model in (13.2), verify the covariance structure in (13.3) and the matrix expressions in (13.4) and (13.7).b. Verify the expressions for the expected values of the quadratic forms in (13.8) and the distributions in (13.10). Section 13.3
For the nested factorial models defined by and Ta. Develop the hypothesis matrices for the usual preliminary tests.b. Develop the associated reparameterized model.c. Develop the AOV table including the sums of squares in matrix and algebraic form and the degrees of freedom.d. Relate the sums of
a. Verify the relation in (12.94) by considering the various possibilities if rm. The examples and I will be useful.b. Verify the general expression for the matrix of the numerator sum of squares given in (12.95).e. Verify the relations, Na-Na + Nia anda) = a + for the models in Section 12.1 and
Verify that the model for the nested factorial in Section 12.5 is given by the general notation in Section 12.6.
a. Write the transformation matrix for the nested factorial model associated with the hypotheses in Table 12.9 and determine the inverse of that matrix.b. Determine the parameter matrix and the basic design matrix. Section 12.6
Hocking (1985) described an example for which the nested factorial model is appropriate. Observations are taken at laundromats using different types of washing machines and different detergents. For the study we have seven laundromats using three types of machines and four brands of detergent. Each
In the two-fold nested model, with unequal but non-zero cell frequencies, the parameterization in (12.46) is often replaced by either of the following definitions of the parameters. Hip-Hab = - -- wherea, is defined for i=1, (a-1) and is defined for i 1, and J-1,(6-1).a. Determine the parameter
Determine the transformation matrix and its inverse for the parameters as defined in (12.63).
Describe the extension of the second form of reparameterization in (11.45) to the two-fold nested model.
Using the terminology of Example 12.2 for the unbalanced, two-fold nested model;a. Show that the sum of squares for testing the college-effect hypothesis defined by (12.60) is as given in Ne" in Table 12.6. Hint: Note that this is identical to the hypothesis H as given in (11.70) for the
Using the terminology of Example 12.2 for the unbalanced, two-fold nested model;a. Write the college-effect hypothesis in the form, He: in matrix form.b. Using the Lagrange multiplier method, or any other method, show that the estimate of the cell means, when constrained by the hypothesis are given
For the two-fold nested model, using the terminology of Example 12.2, verify that the numerator sums of squares for testing the hypothesis He and Hoc are given by R(a) and R(T). Also show that. R(a) = R(a.).
Extend the analysis of Example 12.2 to include simultaneous confidence intervals and ellipses.
a. Verify that the hypotheses for the three-factor factorial model may be written in matrix form as in (12.13).b. Use the general form for the numerator sum of squares and the matrix expression for the hypotheses for factorial models to verify the expression for the numerator sums of squares in
Consider the three-factor, cross-classification model with -a-3 and except 112 121 123 = #221 = Assume that the three-factor interaction constraints, (12.5), are satisfied. a = 2. 2 = 0.a. Determine the effective constraints.b. Are all cell means estimable?e. Determine the effective hypothesis
Describe the extension, to the three-factor model, of the parameterization in (11.45)
Describe the transformation, basic design, and parameter matrices for the three-factor, cross-classification model using the parameter definitions in (12.9).
Expand on the preliminary analysis of Example 12.1, shown in Table 12.3 by preparing interaction plots, developing simultaneous confidence intervals and simultaneous confidence ellipses based on the main effect hypotheses.
Develop the algebraic expressions for the sums of squares and the associated expected mean squares in Table 12.1.

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Methods And Applications Of Linear Models Regression And The Analysis Of Variance 2nd Edition Ronald R. Hocking - Solutions

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