Formulas Useful for Linear Regression Analysis and Related Matrix ...

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126 Indexdefinition 51constraints 39contrast 52in hypothesis testing 37, 41of ˇ 52of ˇk 52of ı in the extended model 46of ı in the extended model 44of K 0ˇ 51of X 2ˇ2 51of c 0 52extended (mean-shift) model 44, 46, 47Ffactor analysis 79factor scores 80finiteness matters 24fitted values 12frequency table 23Frisch–Waugh–Lovell theorem 15, 22generalized 61Frobenius inequality 84full rank decomposition 92, 111GGalton’s discovery of regression vGauss–Markov model see linear modelGauss–Markov theorem 53generalized inversethe 4 Moore–Penrose conditions 90Banachiewicz–Schur form 93least-squares g-inverse 94minimum norm g-inverse 94partitioned nnd matrix 93reflexive g-inverse 91, 93through SVD 92generalized variance 10, 11generated observations from N 2 .0; †/ 121HHadi, Ali S. vHartwig–Spindelböck decomposition 112Haslett, Stephen J. vihat matrix H 12Henderson’s mixed model equations 73Hotelling’s T 2 28, 29, 77when n 1 D 1 30hypothesisˇ2 D 0 38ˇx D 0 31, 40ı D 0 in outlier testing 46I D 0 29 1 D 2 30, 77ˇ1 D D ˇk D 0 40ˇi D 0 40k 0ˇ D 0 40ı D 0 in outlier testing 44 1 D D g 41 1 D 2 42K 0ˇ D d 39, 40K 0ˇ D d when cov.y/ D 2 V 40K 0ˇ D 0 37K 0 B D D 77overall-F -value 40testable 41under intraclass correlation 59idempotent matrixeigenvalues 97full rank decomposition 97rank D trace 97several conditions 97illustration of SST = SSR + SSE 121independencelinear 9statistical 24Nu and T D U 0 .I J/U 29observations, rows 28, 29of dichotomous variables 24quadratic forms 27, 28inequalitych i .AA 0 / ch i .AA 0 C BB 0 / 103 n x 0 Ax=x 0 x 1 104kAxk kAx C Byk 93kAxk 2 kAk 2 kxk 2 110tr.AB/ ch 1 .A/ tr.B/ 103Bloomfield–Watson–Knott 62Cauchy–Schwarz 116Frobenius 84Kantorovich 116Samuelson’s 118Sylvester’s 84Wielandt 117inertia 115intercept 14interlacing theorem 103intersectionC .A/ \ C .B/ 84C .A/ \ C .B/, several properties 96intraclass correlationF -test 59OLSE D BLUE 59eigenvalues 101, 102
Index 127invarianceof C .AB C/ 91of AB C 91of X 0 .V C XUX 0 / X 50of r.AB C/ 91inverseof A nn 86intraclass correlation 101of A D fa ij g D fmin.i; j /g 36of R 89of R xx 17, 90of X 0 X 16, 88of † 22 25of a partitioned block matrix 87of autocorrelation matrix 35of partitioned pd matrix 88Schur complement 87irreducible matrix 112KKantorovich inequality 116Kronecker product 119in multivariate model 76in multivariate sample 29LLagrangian multiplier 39Laplace expansion of determinant 86L A TEX vileast squares see OLSELee, Alan J. vLehmann–Scheffé theorem 68leverage h ii 45likelihood function 30likelihood ratio 30linear hypothesis see hypothesislinear independence 9collinearity 46linear model 5multivariate 76linear prediction sufficiency 72linear zero function 54, 70linearly complete 68linearly sufficient see sufficiencyLöwner ordering see minimizingdefinition 113Albert’s theorem 114Y 0 MY L .Y XB/ 0 .Y XB/ 22, 117Mmagic square 113Magnus, Jan R. vMahalanobis distanceMHLN 2 .Ny 1 ; Ny 2 ; S / 30, 77, 79MHLN 2 .u .i/ ; Nu; S/ 8MHLN 2 .x; ; †/ 9, 27MHLN 2 .x ; Nx; S xx / 18MHLN 2 .x .i/ ; Nx; S xx / 45, 118MHLN 2 . 1 ; 2 ; †/ 79major/minor axis of the ellipse 25matrixirreducible 112nonnegative 112permutation 112reducible 112shorted 115spanning C .A/ \ C .B/ 84stochastic 113matrix RMP VPMP V 48P WPMP W D RM W 49matrix PMM.MVM/ M 48SSE.V / D y 0 PMymatrix PM 156, 57M 1 .M 1 VM 1 / M 1 57, 99matrix PM 1WM 1 .M 1 W 2 M 1 / M 1 57matrix angle 65matrix of corrected sums of squaresT D Xy 0 CX y 7T xx D X 0 0 CX 0 7maximizingtr.G 0 AG/ subject to G 0 G D I k 104.a 0 x/ 2 =x 0 Bx 106.x 0 Ay/ 2 =.x 0 Bx y 0 Cy/ 110.x 0 Ay/ 2 =.x 0 x y 0 y/ 110Œa 0 . 1 2 / 2 =a 0 †a 79Œa 0 .Nu 0 / 2 =a 0 Sa 29Œa 0 .Nu 1 Nu 2 / 2 =a 0 S a 79Œa 0 .u .i/ Nu/ 2 =a 0 Sa 9†.Vz; z/ 65cor.a 0 x; b 0 y/ 80cor.y; b 0 x/ 33cor d .y; Xˇ/ 20cor d .y; X 0 b/ 33cos 2 .u; v/; u 2 A; v 2 B 81.˛0A 0 Bˇ/ 2˛0A 0 A˛ ˇ0B 0 Bˇ 81kHVMk 2 F66u 0 u subject to u 0 † 1 u D c 2 25var.b 0 z/ 78x 0 Ax=x 0 Bx 106x 0 Ax=x 0 Bx, B nnd 108x 0 Hx=x 0 Vx 108
Page 3: Index 125always nonnegative definit
Page 8 and 9: 130 Indexof det.R/ 89of det.S/ 34of

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