Chapter 3: Correlation and Covariance > partial_covariances

partial_covariances

     

Computes partial covariances or partial correlations from the covariance or correlation matrix.

Synopsis

#include <imsls.h>

float *imsls_f_partial_covariances (int n_independent, int n_dependent, float x, ..., 0)

The type double function is imsls_d_partial_covariances.

Required Argument

int n_independent   (Input)
Number of “independent” variables to be used in the partial covariances/correlations. The partial covariances/correlations are the covariances/correlations between the dependent variables after removing the linear effect of the independent variables.

int n_dependent   (Input)
Number of variables for which partial covariances/correlations are desired (the number of “dependent” variables).

float x   (Input)
The n × n covariance or correlation matrix, where n = n_independent + n_dependent. The rows/columns must be ordered such that the first n_independent rows/columns contain the independent variables, and the last n_dependent row/columns contain the dependent variables. Matrix x must always be square symmetric.

Return Value

Matrix of size n_dependent by n_dependent containing the partial covariances (the default) or partial correlations (use keyword IMSLS_PARTIAL_CORR).

Synopsis with Optional Arguments

#include <imsls.h>

float *imsls_f_partial_covariances (int n_independent, int n_dependent, float x[],
IMSLS_X_COL_DIM, int x_col_dim,
IMSLS_X_INDICES, int indices[],
IMSLS_PARTIAL_COV, or
IMSLS_PARTIAL_CORR,
IMSLS_TEST, int df, int *df_out, float **p_values,
IMSLS_TEST_USER, int df, int *df_out, float p_values[],
IMSLS_RETURN_USER, float c[],
0)

Optional Arguments

IMSLS_X_COL_DIM, int x_col_dim   (Input)
Row/Column dimension of x.
Default: x_col_dim = n_independent + n_dependent.

IMSLS_X_INDICES, int indices[]   (Input)
An array of length x_col_dim containing values indicating the status of the variable as in the following table:

 

indices[i]

Variable is...

−1

not used in analysis

0

dependent variable

1

independent variable

 

            By default, the first n_independent elements of indices are equal to 1, and the last n_dependent elements are equal to 0.

IMSLS_PARTIAL_COV, or

IMSLS_PARTIAL_CORR,
By default, and if IMSLS_PARTIAL_COV is specified, partial covariances are calcu­lated. Partial correlations are calculated if IMSLS_PARTIAL_CORR is specified.

IMSLS_TEST, int df, int *df_out, float **p_values  
(Input, Output, Output)
Argument df is an input integer indicating the number of degrees of freedom associ­ated with the input matrix x. If the number of degrees of freedom in x varies from element to element, then a conservative choice for df is the minimum degrees of freedom for all elements in x.

            Argument df_out contains the number of degrees of freedom in the test that the partial covariances/correlations are zero. This value will usually be df − n_independent, but will be greater than this value if the independent variables are computationally lin­early related.

            Argument p_values is the address of a pointer to an internally allocated array of size n_dependent by n_dependent containing the p-values for testing the null hypothe­sis that the associated partial covariance/correlation is zero. It is assumed that the observations from which x was computed follows a multivariate normal distribution and that each element in x has df degrees of freedom.

IMSLS_TEST_USER, int df, int *df_out, float p_values[]  
(Input, Output, Output)
Storage for array p_values is provided by the user. See IMSLS_TEST above.

IMSLS_RETURN_USER, float c[]   (Output)
If specified, c returns the partial covariances/correlations. Storage for array c is pro­vided by the user.

Description

Function imsls_f_partial_covariances computed partial covariances or partial correlations from an input covariance or correlation matrix. If the “independent” variables (the linear “effect” of the independent variables is removed in computing the partial covariances/correlations) are linearly related to one another, imsls_f_partial_covariances 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 df_out.

Given a covariance or correlation matrix Σ partitioned as

function imsls_f_partial_covariances computed 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 Σ 11 is singular, then as many variables as required are deleted from Σ 11 (and Σ 12) in order to eliminate the linear dependencies. The computations then proceed as above.

The p-value for a partial covariance tests the null hypothesis H0: σ ij|1 = 0, where σij|1 is the (i, j) element in matrix Σ22|1. The p-value for a partial correlation tests the null hypothesis H0: ρij|1 = 0, where ρij|1 is the (i, j) element in matrix P22|1. The p-values are returned in p_values. If the degrees of freedom for x, df, is not known, the resulting p-values may be useful for comparison, but they should not by used as an approximation to the actual probabilities.

Examples

Example 1

The following example computes partial covariances, scaled from a nine-variable correlation matrix originally given by Emmett (1949). The first three rows and columns contain the independent variables and the final six rows and columns contain the dependent variables.

 

#include <imsls.h>

#include <math.h>

int main()

{

    float *pcov;

    float x[9][9] = {

        6.300, 3.050, 1.933, 3.365, 1.317, 2.293, 2.586, 1.242, 4.363,

        3.050, 5.400, 2.170, 3.346, 1.473, 2.303, 2.274, 0.750, 4.077,

        1.933, 2.170, 3.800, 1.970, 0.798, 1.062, 1.576, 0.487, 2.673,

        3.365, 3.346, 1.970, 8.100, 2.983, 4.828, 2.255, 0.925, 3.910,

        1.317, 1.473, 0.798, 2.983, 2.300, 2.209, 1.039, 0.258, 1.687,

        2.293, 2.303, 1.062, 4.828, 2.209, 4.600, 1.427, 0.768, 2.754,

        2.586, 2.274, 1.576, 2.255, 1.039, 1.427, 3.200, 0.785, 3.309,

        1.242, 0.750, 0.487, 0.925, 0.258, 0.768, 0.785, 1.300, 1.458,

        4.363, 4.077, 2.673, 3.910, 1.687, 2.754, 3.309, 1.458, 7.400};

    pcov = imsls_f_partial_covariances(3, 6, x, 0);

    imsls_f_write_matrix("Partial Covariances", 6, 6, pcov, 0);

    imsls_free(pcov);

    return;

}

Output

                           Partial Covariances

            1           2           3           4           5           6

1       0.000       0.000       0.000       0.000       0.000       0.000

2       0.000       0.000       0.000       0.000       0.000       0.000

3       0.000       0.000       0.000       0.000       0.000       0.000

4       0.000       0.000       0.000       5.495       1.895       3.084

5       0.000       0.000       0.000       1.895       1.841       1.476

6       0.000       0.000       0.000       3.084       1.476       3.403

Example 2

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 numbers, not variable numbers. The corresponding variable numbers would be 2, 4, 5, 6, 7 and 8, respectively.

 

#include <imsls.h>

 

int main()

{

    float *pcorr, *pval;

    int   df;

    float x[9][9] = {

        1.0, 0.523, 0.395, 0.471, 0.346, 0.426, 0.576, 0.434, 0.639,

        0.523, 1.0, 0.479, 0.506, 0.418, 0.462, 0.547, 0.283, 0.645,

        0.395, 0.479, 1.0, .355, 0.27, 0.254, 0.452,  0.219, 0.504,

        0.471, 0.506, 0.355, 1.0, 0.691, 0.791, 0.443, 0.285, 0.505,

        0.346, 0.418, 0.27, 0.691, 1.0, 0.679,  0.383, 0.149, 0.409,

        0.426, 0.462, 0.254, 0.791, 0.679, 1.0, 0.372, 0.314, 0.472,

        0.576, 0.547, 0.452, 0.443, 0.383, 0.372, 1.0, 0.385, 0.68,

        0.434, 0.283, 0.219, 0.285, 0.149, 0.314, 0.385, 1.0, 0.47,

        0.639, 0.645, 0.504, 0.505, 0.409, 0.472, 0.68, 0.47, 1.0};

    int indices[9] = {1, 0, 1, 0, 0, 0, 0, 0, 1};

    pcorr = imsls_f_partial_covariances(3, 6, &x[0][0],

                                        IMSLS_PARTIAL_CORR,

                                        IMSLS_X_INDICES, indices,

                                        IMSLS_TEST, 30, &df, &pval,

                                        0);


    printf ("The degrees of freedom are %d\n\n", df);

    imsls_f_write_matrix("Partial Correlations", 6, 6, pcorr, 0);

    imsls_f_write_matrix("P-Values", 6, 6, pval, 0);

    imsls_free(pcorr);

    imsls_free(pval);

    return;

}

Output

The degrees of freedom are 27

                          Partial Correlations

            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-Values

            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

 

Warning Errors

IMSLS_NO_HYP_TESTS                              The input matrix “x” has # degrees of freedom, and the rank of the dependent variables is #. There are not enough degrees of freedom for hypothesis testing. The elements of “p_values” are set to NaN (not a number).

Fatal Errors

IMSLS_INVALID_MATRIX_1                     The input matrix “x” is incorrectly specified. A computed correlation is greater than 1 for variables # and #.

IMSLS_INVALID_PARTIAL                       A computed partial correlation for variables # and # is greater than 1. The input matrix “x” is not positive semi-definite.

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