Chapter 9: Covariance Structures and Factor Analysis

FRESI

Computes communalities and the standardized factor residual correlation matrix.

Required Arguments

COVNVAR by NVAR matrix containing the variance-covariance or correlation matrix.   (Input)
Only the upper triangular part of COV is referenced.

ANVAR by NF orthogonal factor-loading matrix.   (Input)

Y — Vector of length NVAR containing the communalities.   (Output)

RESIDNVAR 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



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