Total and Effect Deviation Matrices for General Linear Hypothesis
## Default S3 method: glhHmat(x,A,C,...) ## S3 method for class 'data.frame' glhHmat(x,A,C,...) ## S3 method for class 'formula' glhHmat(formula,C,data=NULL,...)
x |
A matrix or data frame containing the variables for which the SSCP matrix is to be computed. |
A |
A matrix or data frame containing a design matrix specifying a linear model in which x is the response. |
C |
A matrix or vector containing the coefficients of the reference hypothesis. |
formula |
A formula of the form |
data |
Data frame from which variables specified in 'formula' are preferentially to be taken. |
... |
further arguments for the method. |
Consider a multivariate linear model x = A Ψ + U and a
reference hypothesis H0: C Ψ = 0, with Ψ being a
matrix of unknown
parameters and C a known coefficient matrix with rank r. It is well
known that, under classical Gaussian assumptions, H_0 can be tested by
several increasing functions of the r positive
eigenvalues of a product T^{-1} H, where T and H are
total and effect
matrices of SSCP deviations associated with H_0. Furthermore, whether
or not the classical assumptions hold, the same eigenvalues can be
used to define descriptive indices that measure an "effect"
characterized by the violation of H_0 (see reference [1] for further
details).
Those SSCP matrices are given by T = x'(I - P_{ω}) x and H =
x'(P_{Ω} - P_{ω}) x, where I is an identity matrix and
P_{Ω} = A(A'A)^-A' ,
P_{ω} = A(A'A)^-A' - A(A'A)^-C'[C(A'A)^-C']^-C(A'A)^-A'
are projection matrices on the spaces spanned by the columns of A
(space Ω) and by the linear combinations of these columns that
satisfy the reference hypothesis (space ω). In these
formulae M' denotes the transpose of M and M^- a
generalized inverse. glhHmat computes the T and H
matrices which then can be used as input to the
search routines anneal, genetic
improve and eleaps that try to select
subsets of x according to their contribution to the violation of H_0.
A list with four items:
mat |
The total SSCP matrix |
H |
The effect SSCP matrix |
r |
The expected rank of the H matrix which equals the rank of C. The true rank of H can be different from r if the x variables are linearly dependent. |
call |
The function call which generated the output. |
[1] Duarte Silva. A.P. (2001). Efficient Variable Screening for Multivariate Analysis, Journal of Multivariate Analysis, Vol. 76, 35-62.
##---------------------------------------------------------------------------- ## The following examples create T and H matrices for different analysis ## of the MASS data set "crabs". This data records physical measurements ## on 200 specimens of Leptograpsus variegatus crabs observed on the shores ## of Western Australia. The crabs are classified by two factors, sex and sp ## (crab species as defined by its colour: blue or orange), with two levels ## each. The measurement variables include the carapace length (CL), ## the carapace width (CW), the size of the frontal lobe (FL) and the size of ## the rear width (RW). In the analysis provided, we assume that there is ## an interest in comparing the subsets of these variables measured in their ## original and logarithmic scales. library(MASS) data(crabs) lFL <- log(crabs$FL) lRW <- log(crabs$RW) lCL <- log(crabs$CL) lCW <- log(crabs$CW) # 1) Create the T and H matrices associated with a linear # discriminant analysis on the groups defined by the sp factor. # This call is equivalent to ldaHmat(sp ~ FL + RW + CL + CW + lFL + # lRW + lCL + lCW,crabs) Hmat1 <- glhHmat(cbind(FL,RW,CL,CW,lFL,lRW,lCL,lCW) ~ sp,c(0,1),crabs) Hmat1 ##$mat ## FL RW CL CW lFL lRW lCL ##FL 2431.2422 1623.4509 4846.9787 5283.6093 162.718609 133.360397 158.865134 ##RW 1623.4509 1317.7935 3254.5776 3629.6883 109.877182 107.287243 108.335721 ##CL 4846.9787 3254.5776 10085.3040 11096.5141 326.243285 269.564742 330.912570 ##CW 5283.6093 3629.6883 11096.5141 12331.5680 356.317934 300.786770 364.620761 ##lFL 162.7186 109.8772 326.2433 356.3179 11.114733 9.188391 10.910730 ##lRW 133.3604 107.2872 269.5647 300.7868 9.188391 8.906350 9.130692 ##lCL 158.8651 108.3357 330.9126 364.6208 10.910730 9.130692 11.088706 ##lCW 152.7872 106.4277 321.0253 357.0051 10.503303 8.970570 10.765175 ## lCW ##FL 152.78716 ##RW 106.42775 ##CL 321.02534 ##CW 357.00510 ##lFL 10.50330 ##lRW 8.97057 ##lCL 10.76517 ##lCW 10.54334 ##$H ## FL RW CL CW lFL lRW lCL ##FL 466.34580 247.526700 625.30650 518.41650 30.7408809 19.4543206 20.5494907 ##RW 247.52670 131.382050 331.89975 275.16475 16.3166234 10.3259508 10.9072444 ##CL 625.30650 331.899750 838.45125 695.12625 41.2193540 26.0856066 27.5540813 ##CW 518.41650 275.164750 695.12625 576.30125 34.1733106 21.6265286 22.8439819 ##lFL 30.74088 16.316623 41.21935 34.17331 2.0263971 1.2824024 1.3545945 ##lRW 19.45432 10.325951 26.08561 21.62653 1.2824024 0.8115664 0.8572531 ##lCL 20.54949 10.907244 27.55408 22.84398 1.3545945 0.8572531 0.9055117 ##lCW 15.16136 8.047335 20.32933 16.85423 0.9994161 0.6324790 0.6680840 ## lCW ##FL 15.1613582 ##RW 8.0473352 ##CL 20.3293260 ##CW 16.8542276 ##lFL 0.9994161 ##lRW 0.6324790 ##lCL 0.6680840 ##lCW 0.4929106 ##$r ##[1] 1 ##$call ##glhHmat.formula(formula = cbind(FL, RW, CL, CW, lFL, lRW, lCL, ## lCW) ~ sp, C = c(0, 1), data = crabs) # 2) Create the T and H matrices associated with an analysis # of the interactions between the sp and sex factors Hmat2 <- glhHmat(cbind(FL,RW,CL,CW,lFL,lRW,lCL,lCW) ~ sp*sex,c(0,0,0,1),crabs) Hmat2 ##$mat ## FL RW CL CW lFL lRW lCL ##FL 1960.3362 1398.52890 4199.1581 4747.5409 131.651804 115.607172 137.663744 ##RW 1398.5289 1074.36105 3034.2793 3442.0233 95.176151 88.529040 100.659912 ##CL 4199.1581 3034.27925 9135.6987 10314.2389 283.414814 251.877591 300.140005 ##CW 4747.5409 3442.02325 10314.2389 11686.9387 320.883015 285.744945 339.253367 ##lFL 131.6518 95.17615 283.4148 320.8830 9.065041 8.027569 9.509543 ##lRW 115.6072 88.52904 251.8776 285.7449 8.027569 7.460222 8.516618 ##lCL 137.6637 100.65991 300.1400 339.2534 9.509543 8.516618 10.090003 ##lCW 137.2059 100.46203 298.6227 338.5254 9.473873 8.494741 10.037059 ## lCW ##FL 137.205863 ##RW 100.462028 ##CL 298.622747 ##CW 338.525352 ##lFL 9.473873 ##lRW 8.494741 ##lCL 10.037059 ##lCW 10.011755 ##$H ## FL RW CL CW lFL lRW lCL ##FL 80.645000 68.389500 153.73350 191.57950 5.4708199 5.1596883 5.2140868 ##RW 68.389500 57.996450 130.37085 162.46545 4.6394276 4.3755782 4.4217098 ##CL 153.733500 130.370850 293.06205 365.20785 10.4290197 9.8359098 9.9396095 ##CW 191.579500 162.465450 365.20785 455.11445 12.9964281 12.2573068 12.3865353 ##lFL 5.470820 4.639428 10.42902 12.99643 0.3711311 0.3500245 0.3537148 ##lRW 5.159688 4.375578 9.83591 12.25731 0.3500245 0.3301182 0.3335986 ##lCL 5.214087 4.421710 9.93961 12.38654 0.3537148 0.3335986 0.3371158 ##lCW 5.584150 4.735535 10.64506 13.26565 0.3788193 0.3572754 0.3610421 ## lCW ##FL 5.5841501 ##RW 4.7355352 ##CL 10.6450610 ##CW 13.2656543 ##lFL 0.3788193 ##lRW 0.3572754 ##lCL 0.3610421 ##lCW 0.3866667 ##$r ##[1] 1 ##$call ##glhHmat.formula(formula = cbind(FL, RW, CL, CW, lFL, lRW, lCL, ## lCW) ~ sp * sex, C = c(0, 0, 0, 1), data = crabs) ## 3) Create the T and H matrices associated with an analysis ## of the effect of the sp factor after controlling for sex C <- matrix(0.,2,4) C[1,3] = C[2,4] = 1. C ## [,1] [,2] [,3] [,4] ## [1,] 0 0 1 0 ## [2,] 0 0 0 1 Hmat3 <- glhHmat(cbind(FL,RW,CL,CW,lFL,lRW,lCL,lCW) ~ sp*sex,C,crabs) Hmat3 ##$mat ## FL RW CL CW lFL lRW lCL ##FL 1964.8964 1375.92420 4221.6722 4765.1928 131.977728 113.906076 138.315643 ##RW 1375.9242 1186.41150 2922.6779 3354.5236 93.560559 96.961292 97.428477 ##CL 4221.6722 2922.67790 9246.8527 10401.3878 285.023931 243.479136 303.358489 ##CW 4765.1928 3354.52360 10401.3878 11755.2667 322.144623 279.160241 341.776779 ##lFL 131.9777 93.56056 285.0239 322.1446 9.088336 7.905989 9.556135 ##lRW 113.9061 96.96129 243.4791 279.1602 7.905989 8.094783 8.273439 ##lCL 138.3156 97.42848 303.3585 341.7768 9.556135 8.273439 10.183194 ##lCW 137.6258 98.38041 300.6960 340.1509 9.503886 8.338091 10.097091 ## lCW ##FL 137.625801 ##RW 98.380414 ##CL 300.696018 ##CW 340.150874 ##lFL 9.503886 ##lRW 8.338091 ##lCL 10.097091 ##lCW 10.050426 ##$H ## FL RW CL CW lFL lRW ##FL 85.205200 45.784800 176.247600 209.231400 5.7967443 3.45859277 ##RW 45.784800 170.046900 18.769500 74.965800 3.0238356 12.80782993 ##CL 176.247600 18.769500 404.216100 452.356800 12.0381364 1.43745463 ##CW 209.231400 74.965800 452.356800 523.442500 14.2580360 5.67260253 ##lFL 5.796744 3.023836 12.038136 14.258036 0.3944254 0.22844463 ##lRW 3.458593 12.807830 1.437455 5.672603 0.2284446 0.96467943 ##lCL 5.865986 1.190274 13.158093 14.909948 0.4003070 0.09041999 ##lCW 6.004088 2.653921 12.718332 14.891177 0.4088329 0.20062548 ## lCL lCW ##FL 5.86598627 6.0040883 ##RW 1.19027431 2.6539211 ##CL 13.15809339 12.7183319 ##CW 14.90994753 14.8911765 ##lFL 0.40030704 0.4088329 ##lRW 0.09041999 0.2006255 ##lCL 0.43030750 0.4210740 ##lCW 0.42107404 0.4253378 ##$r ##[1] 2 ##$call ##glhHmat.formula(formula = cbind(FL, RW, CL, CW, lFL, lRW, lCL, ## lCW) ~ sp * sex, C = C, data = crabs)
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