FRESI
Computes communalities and the standardized factor residual correlation matrix.
Required Arguments
COV — NVAR by NVAR matrix containing the variance‑covariance or correlation matrix. (Input)
Only the upper triangular part of COV is referenced.
A — NVAR by NF orthogonal factor‑loading matrix. (Input)
Y — Vector of length NVAR containing the communalities. (Output)
RESID — NVAR by NVAR matrix containing the normalized residual variance‑covariance or correlation matrix. (Output)
Optional Arguments
NVAR — Number of variables. (Input)
Default: NVAR = size (COV,1).
LDCOV — Leading dimension of COV exactly as specified in the dimension statement in the calling program. (Input)
Default: LDCOV = size (COV,1).
NF — Number of factors. (Input)
Default: NF = size (A,2).
LDA — Leading dimension of A exactly as specified in the dimension statement in the calling program. (Input)
Default: LDA = size (A,1).
LDRESI — Leading dimension of RESID exactly as specified in the dimension statement in the calling program. (Input)
Default: LDRESI = size (RESID,1).
FORTRAN 90 Interface
Generic: CALL FRESI (COV, A, Y, RESID [, …])
Specific: The specific interface names are S_FRESI and D_FRESI.
FORTRAN 77 Interface
Single: CALL FRESI (NVAR, COV, LDCOV, NF, A, LDA, Y, RESID, LDRESI)
Double: The double precision name is DFRESI.
Description
Routine FRESI computes the communalities and a standardized residual covariance/correlation matrix for input covariance/correlation matrix COV. The user must also input the orthogonal (unrotated) factor loadings, A, obtained from the matrix COV. Let ai denote the i‑th row of matrix A. Then, the communalities are given as
where yi is the i‑th communality. The residual covariance/correlation matrix is given by
where sij denotes an element of the covariance/correlation matrix and R = (rij) denotes the residual matrix. Standardization is performed by dividing the rij by
where ui= sii ‑yi is the unique error variance for the i‑th variable. If ui is zero (or slightly less than zero due to roundoff error), ui = 1.0 is assumed and division by zero is avoided.
Example
The following example computes the residual correlation matrix with communalities in a 9‑factor problem. The resulting residual correlations do not seem to exhibit any pattern.
USE FRESI_INT
USE WRRRN_INT
IMPLICIT NONE
INTEGER LDA, LDCOV, LDRESI, NF, NVAR
PARAMETER (LDA=9, LDCOV=9, LDRESI=9, NF=3, NVAR=9)
!
REAL A(9,3), COV(9,9), RESID(9,9), Y(9)
!
DATA COV/1.000, 0.523, 0.395, 0.471, 0.346, 0.426, 0.576, 0.434, &
0.639, 0.523, 1.000, 0.479, 0.506, 0.418, 0.462, 0.547, &
0.283, 0.645, 0.395, 0.479, 1.000, 0.355, 0.270, 0.254, &
0.452, 0.219, 0.504, 0.471, 0.506, 0.355, 1.000, 0.691, &
0.791, 0.443, 0.285, 0.505, 0.346, 0.418, 0.270, 0.691, &
1.000, 0.679, 0.383, 0.149, 0.409, 0.426, 0.462, 0.254, &
0.791, 0.679, 1.000, 0.372, 0.314, 0.472, 0.576, 0.547, &
0.452, 0.443, 0.383, 0.372, 1.000, 0.385, 0.680, 0.434, &
0.283, 0.219, 0.285, 0.149, 0.314, 0.385, 1.000, 0.470, &
0.639, 0.645, 0.504, 0.505, 0.409, 0.472, 0.680, 0.470, &
1.000/
!
DATA A/.6642, .6888, .4926, .8372, .7050, .8187, .6615, .4579, &
.7657, -.3209, -.2471, -.3022, .2924, .3148, .3767, -.3960, &
-.2955, -.4274, .0735, -.1933, -.2224, -.0354, -.1528, .1045, &
-.0778, .4914, -.0117/
!
CALL FRESI (COV, A, Y, RESID)
!
CALL WRRRN ('Communalities', Y, 1, NVAR, 1)
CALL WRRRN ('Residuals', RESID)
END
Output
Communalities
1 2 3 4 5 6 7 8
0.5495 0.5729 0.3834 0.7877 0.6195 0.8231 0.6005 0.5385
9
0.7691
Residuals
1 2 3 4 5 6 7 8 9
1 1.000 0.001 -0.024 0.037 -0.024 -0.016 0.036 -0.002 -0.018
2 0.001 1.000 0.043 -0.017 -0.048 0.041 -0.052 -0.023 0.031
3 -0.024 0.043 1.000 0.064 -0.033 -0.037 -0.022 0.025 -0.013
4 0.037 -0.017 0.064 1.000 0.012 -0.004 0.008 0.017 -0.052
5 -0.024 -0.048 -0.033 0.012 1.000 -0.003 0.075 -0.014 0.007
6 -0.016 0.041 -0.037 -0.004 -0.003 1.000 -0.046 -0.003 0.036
7 0.036 -0.052 -0.022 0.008 0.075 -0.046 1.000 0.008 0.011
8 -0.002 -0.023 0.025 0.017 -0.014 -0.003 0.008 1.000 -0.004
9 -0.018 0.031 -0.013 -0.052 0.007 0.036 0.011 -0.004 1.000
