PCORR
Computes partial correlations or covariances from the covariance or correlation matrix.
Required Arguments
COR — NVAR by NVAR correlation or covariance matrix. (Input)
NIND — Number of “independent” variables to to be used in the partial correlations. (Input)
If NIND is ‑1, the independent variables are taken to be the NVAR ‑ NDEP variables not in INDDEP. If NIND is zero, no independent variables are used, and p-values for the input dependent (see INDDEP) correlations (or covariances) are computed. The partial correlations (covariances) are the correlations (covariances) between the dependent variables after removing the linear effect of the independent variables. NIND and NDEP cannot simultaneously be ‑1.
IND — Vector of length NIND containing the column (or row) numbers in COR of the independent variables. (Input, if NIND > 0; not referenced otherwise)
If NIND is negative or zero, IND is not used and can be dimensioned of length 1 in the calling program.
NDEP — Number of variables for which partial correlations (covariances) are desired (the number of “dependent” variables). (Input)
If NDEP is ‑1, the dependent variables are taken as the NVAR ‑ NIND variables not in IND. NIND and NDEP cannot simultaneously be ‑1.
INDDEP — Vector of length NDEP containing the indices of the dependent variables. (Input, if NDEP > 0; not referenced otherwise)
If NDEP is 1, INDDEP is not used and can be dimensioned of length 1 in the calling program.
PCOR — Matrix of size m by m containing the partial correlations or partial covariances. (Output)
m = NDEP if NDEP > 0, and m = NVAR ‑ NIND otherwise. If NIND = 0, then COR and PCOR can share the same memory location.
Optional Arguments
NVAR — Number of variables in COR. (Input)
Default: NVAR = size (COR,1).
LDCOR — Leading dimension of COR exactly as specified in the dimension statement in the calling program. (Input)
Default: LDCOR = size (COR,1).
NDF — Number of degrees of freedom in COR. (Input)
If the number of degrees of freedom in COR varies from element to element, then a conservative choice for NDF is the minimum degrees of freedom for all elements in COR. If NDF is not known, then NDF ≤ 0 defaults to NDF = 100.
Default: NDF = 0.
ICOR — Partial correlations/covariances option. (Input)
Default: ICOR = 0.
|
ICOR |
Action |
|
1 |
Partial correlations are desired. |
|
0 |
Partial covariances are desired. |
Partial correlations can be computed when either a correlation or a covariance matrix in input in COR. To compute partial covariances, COR must contain a covariance matrix.
LDPCOR — Leading dimension of PCOR exactly as specified in the dimension statement of the calling program. (Input)
Default: LDPCOR = size (PCOR,1).
NDFP — Number of degrees of freedom in the test that the partial correlation (covariance) is zero. (Output)
This will usually be NDF ‑ NIND but will be greater than this value if the variables in IND are computationally linearly related.
PVAL — Matrix of size m by m (see PCOR) containing the p-values for testing the null hypothesis that the associated partial correlation (covariance) is zero. (Output)
The p-values reported in PVAL assume that the observations from which COR was computed follow a multivariate normal distribution and that each element in COR has NDF degrees of freedom.
LDPVAL — Leading dimension of PVAL exactly as specified in the dimension statement in the calling program. (Input)
Default: LDPVAL = size(PVAL, 1).
FORTRAN 90 Interface
Generic: CALL PCORR (COR, NIND, IND, NDEP, INDDEP, PCOR [, …])
Specific: The specific interface names are S_PCORR and D_PCORR.
FORTRAN 77 Interface
Single: CALL PCORR (NVAR, COR, LDCOR, NDF, ICOR, NIND, IND, NDEP, INDDEP, PCOR, LDPCOR, NDFP, PVAL, LDPVAL)
Double: The double precision name is DPCORR.
Description
Routine PCORR computes partial correlations or partial covariances from an input correlation or covariance matrix. If the “independent” variables (the linear “effect” of the independent variables is removed in computing the partial correlations/covariances) are linearly related to one another, PCORR detects the linearity and eliminates one or more of the independent variables from the list of independent variables. The number of variables eliminated, if any, can be determined from argument NDFP.
Given a correlation or covariance matrix Σ partitioned as
Routine PCORR computes the partial covariances (of the standardized variables if Σ is a correlation matrix) as
If partial correlations are desired, these are computed as
where “diag” denotes the matrix containing the diagonal of its argument along its diagonal with zeros off the diagonal. If Σ11is singular, then as many variables as required are deleted from Σ11 (and Σ12) in order to eliminate the linear dependency(ies). The computations then proceed as above.
The p-value for a partial correlation (covariance) tests the null hypothesis H0 : ρij∣1 = 0 (H0 : σij∣1 = 0), where ρij∣1(σij∣1) is the (i, j) element in matrix P22∣1 (Σ22∣1). The p-values are returned in PVAL. If NDF is not known, the p-values are computed as if each element in COR had 100 degrees of freedom. When NDF is not known, the resulting p-values may be useful for comparison, but they should not be used as an approximation to the actual probabilities.
Comments
1. Workspace may be explicitly provided, if desired, by use of P2ORR/DP2ORR. The reference is:
CALL P2ORR (NVAR, COR, LDCOR, NDF, ICOR, NIND, IND, NDEP, INDDEP, PCOR, LDPCOR, NDFP, PVAL, LDPVAL, SXY, SXX, LDSXX, IY, IX)
The additional arguments are as follows:
SXY — Work vector of length m * n.
SXX — Work vector of length n2.
LDSXX — The value of n.
IY — Work vector of length NVAR.
IX — Work vector of length NVAR.
2. Informational errors
|
Type |
Code |
Description |
|
4 |
1 |
COR is incorrectly specified for two independent variables. |
|
4 |
2 |
COR is incorrectly specified for an independent variable and a dependent variable. |
|
4 |
3 |
COR is incorrectly specified for two dependent variables. |
|
4 |
4 |
A computed partial correlation is greater than one. |
Example
The following example computes partial correlations from a 9 variable correlation matrix originally given by Emmett (1949). The partial correlations between the remaining variables, after adjusting for variables 1, 3, and 9, are computed. Note in the output that the row and column labels are column numbers, not variable numbers. The corresponding variable numbers would be 2, 4, 5, 6, 7, and 8, respectively.
! SPECIFICATIONS FOR PARAMETERS
USE PCORR_INT
USE UMACH_INT
USE WRRRN_INT
IMPLICIT NONE
INTEGER ICOR, LDCOR, LDP, LDPCOR, NDEP, NDF, NIND, NVAR
PARAMETER (ICOR=1, LDCOR=9, LDP=6, LDPCOR=6, NDEP=-1, NDF=30, &
NIND=3, NVAR=9)
!
INTEGER IND(NIND), INDDEP(1), NDFP, NOUT
REAL COR(LDCOR,NVAR), P(LDP,LDP), PCOR(LDPCOR,LDPCOR)
!
DATA IND/1, 3, 9/
!
DATA COR/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/
!
CALL PCORR (COR, NIND, IND, NDEP, INDDEP, PCOR, NDF=NDF, &
ICOR=ICOR, NDFP=NDFP, PVAL=P)
!
CALL UMACH (2, NOUT)
WRITE (NOUT,*) 'The degrees of freedom are ', NDFP
CALL WRRRN ('PCOR', PCOR)
CALL WRRRN ('P', P)
!
END
Output
The degrees of freedom are 27
PCOR
1 2 3 4 5 6
1 1.000 0.224 0.194 0.211 0.125 -0.061
2 0.224 1.000 0.605 0.720 0.092 0.025
3 0.194 0.605 1.000 0.598 0.123 -0.077
4 0.211 0.720 0.598 1.000 0.035 0.086
5 0.125 0.092 0.123 0.035 1.000 0.062
6 -0.061 0.025 -0.077 0.086 0.062 1.000
P
1 2 3 4 5 6
1 0.0000 0.2525 0.3232 0.2801 0.5249 0.7576
2 0.2525 0.0000 0.0006 0.0000 0.6417 0.9000
3 0.3232 0.0006 0.0000 0.0007 0.5328 0.6982
4 0.2801 0.0000 0.0007 0.0000 0.8602 0.6650
5 0.5249 0.6417 0.5328 0.8602 0.0000 0.7532
6 0.7576 0.9000 0.6982 0.6650 0.7532 0.0000
