Compare Covariance Matrices using Flury's Hierarchy of Tests (V0.95) Analysis of file: bone.dat *** Test for Equality Among the Matrices *** Criterion for Equality: -127.65760 Number of parameters (Equality): 10 Likelihood ratio test chi-square for H0-equality versus: Proportionality: 0.121 (df = 1, p = 0.7282) Common PC : 7.490 (df = 4, p = 0.1121) CPC(2) : 8.664 (df = 5, p = 0.1233) CPC(1) : 12.170 (df = 7, p = 0.0951) Unrelated : 15.833 (df = 10, p = 0.1045) Eigenvalues of Best Pooled Covariance Matrix: Vector 1 Vector 2 Vector 3 Vector 4 0.12721 0.57381 1.26026 4.28134 Eigenvectors for Best Pooled Covariance Matrix: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.73046 -0.22649 -0.50973 0.3941 HumWid -0.13514 -0.61536 0.52335 0.57374 FemLen -0.66374 -0.02474 -0.63425 0.39567 FemWid 0.08728 0.7546 0.25299 0.59913 Best Pooled Covariance Matrix: HumLen HumWid FemLen FemWid HumLen 1.0897 0.69927 1.01658 0.7584 HumWid 0.69927 1.9741 0.57374 1.37059 FemLen 1.01658 0.57374 1.23364 0.79462 FemWid 0.7584 1.37059 0.79462 1.94517 *** Test for Proportionality Among the Matrices *** Criterion for Proportionality: -127.77835 Number of parameters (Proportionality): 11 Likelihood ratio test chi-square for H0-proportionality versus: Common PC : 7.370 (df = 3, p = 0.0610) CPC(2) : 8.543 (df = 4, p = 0.0736) CPC(1) : 12.049 (df = 6, p = 0.0609) Unrelated : 15.713 (df = 9, p = 0.0731) Eigenvalues for Proportional Matrix: Vector 1 Vector 2 Vector 3 Vector 4 0.12932 0.58003 1.28497 4.34019 Eigenvectors for Proportional Matrix: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.73052 -0.22787 -0.50947 0.39351 HumWid -0.13375 -0.61398 0.5257 0.57339 FemLen -0.66401 -0.02824 -0.6337 0.39587 FemWid 0.08685 0.75519 0.24999 0.59972 Proportionality constant for Males: 1.000000 Maximum likelihood estimate of Males (proportionality): HumLen HumWid FemLen FemWid HumLen 1.10475 0.70366 1.03198 0.76901 HumWid 0.70366 2.00302 0.57864 1.39088 FemLen 1.03198 0.57864 1.25368 0.80702 FemWid 0.76901 1.39088 0.80702 1.97306 Proportionality constant for Females: 0.955458 Maximum likelihood estimate of Females (proportionality): HumLen HumWid FemLen FemWid HumLen 1.05554 0.67232 0.98602 0.73475 HumWid 0.67232 1.9138 0.55287 1.32892 FemLen 0.98602 0.55287 1.19784 0.77108 FemWid 0.73475 1.32892 0.77108 1.88517 *** Test for Common Principal Components Among the Matrices *** Criterion for CPC: -135.14809 Number of parameters (CPC): 14 Likelihood ratio test chi-square for H0-CPC versus: CPC(2) : 1.173 (df = 1, p = 0.2787) CPC(1) : 4.679 (df = 3, p = 0.1969) Unrelated : 8.343 (df = 6, p = 0.2140) Eigenvalues for CPC Matrices: Males: Vector 1 Vector 2 Vector 3 Vector 4 4.54187 1.08106 0.68443 0.12285 Females: Vector 1 Vector 2 Vector 3 Vector 4 3.76411 1.59873 0.37273 0.13593 Eigenvectors for CPC Matrix: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.39141 0.48643 -0.28108 0.72882 HumWid 0.56619 -0.57566 -0.5729 -0.14082 FemLen 0.39408 0.63056 -0.08099 -0.66372 FemWid 0.60904 -0.18547 0.76565 0.09198 Maximum likelihood estimate of Males (CPC): HumLen HumWid FemLen FemWid HumLen 1.07095 0.80143 0.98832 0.84612 HumWid 0.80143 2.04131 0.66424 1.37978 FemLen 0.98832 0.66424 1.19381 0.91373 FemWid 0.84612 1.37978 0.91373 2.12414 Maximum likelihood estimate of Females (CPC): HumLen HumWid FemLen FemWid HumLen 1.0566 0.43258 1.01371 0.68197 HumWid 0.43258 1.86148 0.28955 1.30341 FemLen 1.01371 0.28955 1.28257 0.68505 FemWid 0.68197 1.30341 0.68505 1.67084 *** Test for Common Principal Components Among Matrices *** *** Test for 2-Common Principal Components *** Criterion for CPC(2): -136.32134 Number of parameters [CPC(2)]: 15 Likelihood ratio test chi-square for H0-CPC(2) versus: CPC(1) : 3.506 (df = 2, p = 0.1733) Unrelated : 7.170 (df = 5, p = 0.2083) Eigenvalues for CPC(2) Matrices: Males: Vector 1 Vector 2 Vector 3 Vector 4 4.54048 1.0849 0.68272 0.1221 Females: Vector 1 Vector 2 Vector 3 Vector 4 3.76537 1.59549 0.37843 0.1322 Eigenvectors of Males [CPC(2)]: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.39018 0.48381 -0.29305 0.7265 HumWid 0.56358 -0.56622 -0.57993 -0.15954 FemLen 0.39376 0.63756 -0.05836 -0.6596 FemWid 0.61244 -0.19709 0.75789 0.10804 Eigenvectors of Females [CPC(2)]: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.39018 0.48381 -0.16942 0.76484 HumWid 0.56358 -0.56622 -0.59828 -0.06187 FemLen 0.39376 0.63756 -0.16618 -0.64098 FemWid 0.61244 -0.19709 0.76533 -0.01823 Maximum likelihood estimate of Males [CPC(2)]: HumLen HumWid FemLen FemWid HumLen 1.06828 0.80313 0.9854 0.83951 HumWid 0.80313 2.02272 0.65192 1.3861 FemLen 0.9854 0.65192 1.20042 0.91973 FemWid 0.83951 1.3861 0.91973 2.13878 Maximum likelihood estimate of Females [CPC(2)]: HumLen HumWid FemLen FemWid HumLen 1.03491 0.42304 1.01649 0.69674 HumWid 0.42304 1.84347 0.3025 1.30459 FemLen 1.01649 0.3025 1.2971 0.66096 FemWid 0.69674 1.30459 0.66096 1.69602 *** Test for 1-Common Principal Component *** Criterion for CPC(1): -139.82719 Number of parameters [CPC(1)]: 17 Likelihood ratio test chi-square for H0-CPC(1) versus: Unrelated : 3.664 (df = 3, p = 0.3001) Eigenvalues for CPC(1) Matrices: Males: Vector 1 Vector 2 Vector 3 Vector 4 4.53988 1.11692 0.65129 0.1221 Females: Vector 1 Vector 2 Vector 3 Vector 4 3.76643 1.6047 0.372 0.12837 Eigenvectors of Males [CPC(1)]: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.38689 0.52726 -0.16648 0.73797 HumWid 0.56425 -0.39233 -0.70514 -0.17458 FemLen 0.39668 0.64513 0.11752 -0.64237 FemWid 0.61204 -0.38973 0.67915 0.11079 Eigenvectors of Females [CPC(1)]: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.38689 0.50166 -0.21676 0.74274 HumWid 0.56425 -0.61049 -0.55412 -0.04328 FemLen 0.39668 0.59647 -0.20104 -0.66817 FemWid 0.61204 -0.14089 0.77817 0.00345 Maximum likelihood estimate of Males [CPC(1)]: HumLen HumWid FemLen FemWid HumLen 1.07459 0.82074 1.00604 0.78183 HumWid 0.82074 1.94486 0.69317 1.42431 FemLen 1.00604 0.69317 1.23861 0.86468 FemWid 0.78183 1.42431 0.86468 2.17214 Maximum likelihood estimate of Females [CPC(1)]: HumLen HumWid FemLen FemWid HumLen 1.05591 0.3713 1.01072 0.71601 HumWid 0.3713 1.91167 0.30383 1.27829 FemLen 1.01072 0.30383 1.23594 0.72108 FemWid 0.71601 1.27829 0.72108 1.66798 *** Principal Component Analysis of Each Matrix Separately *** Eigenstructure of Males: Eigenvalues: Vector 1 Vector 2 Vector 3 Vector 4 0.12092 0.64475 1.11635 4.54818 Eigenvectors: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.73272 -0.14839 -0.52083 0.41212 HumWid -0.18062 -0.6809 0.40248 0.5846 FemLen -0.6489 0.12754 -0.64109 0.38944 FemWid 0.09707 0.70575 0.39465 0.58029 Eigenstructure of Females: Eigenvalues: Vector 1 Vector 2 Vector 3 Vector 4 0.124 0.36791 1.60469 3.7749 Eigenvectors: Vector 1 Vector 2 Vector 3 Vector 4 HumLen 0.75938 -0.21664 0.5025 -0.352 HumWid -0.01438 -0.57376 -0.60907 -0.54739 FemLen -0.65039 -0.2266 0.59767 -0.41041 FemWid 0.01174 0.75665 -0.13898 -0.63877 *** Flury Decomposition of Chi Square *** (Step-up & model building approaches) _______Model______ Higher Lower Chi Sqr df p-val CS/df AIC -------------------------------------------------------- Equality Proport 0.121 1 0.7282 0.121 15.833 Proport CPC 7.370 3 0.0610 2.457 17.713 CPC CPC(2) 1.173 1 0.2787 1.173 16.343 CPC(2) CPC(1) 3.506 2 0.1733 1.753 17.170 CPC(1) Unrelated 3.664 3 0.3001 1.221 17.664 Unrelated --- 20.000 *** Test against Unrelated/Arbitrary Structure *** (Jump-up approach) _______Model______ Higher Lower Chi Sqr df p-val ----------------------------------------- Equality Unrelated 15.833 10 0.1045 Proport Unrelated 15.713 9 0.0731 CPC Unrelated 8.343 6 0.2140 CPC(2) Unrelated 7.170 5 0.2083 CPC(1) Unrelated 3.664 3 0.3001