The routine may be called by the names g02byf or nagf_correg_corrmat_partial.
3Description
Partial correlation can be used to explore the association between pairs of random variables in the presence of other variables. For three variables, ${y}_{1}$, ${y}_{2}$ and ${x}_{3}$, the partial correlation coefficient between ${y}_{1}$ and ${y}_{2}$ given ${x}_{3}$ is computed as:
where ${r}_{ij}$ is the product-moment correlation coefficient between variables with subscripts $i$ and $j$. The partial correlation coefficient is a measure of the linear association between ${y}_{1}$ and ${y}_{2}$ having eliminated the effect due to both ${y}_{1}$ and ${y}_{2}$ being linearly associated with ${x}_{3}$. That is, it is a measure of association between ${y}_{1}$ and ${y}_{2}$ conditional upon fixed values of ${x}_{3}$. Like the full correlation coefficients the partial correlation coefficient takes a value in the range ($\mathrm{-1},1$) with the value $0$ indicating no association.
In general, let a set of variables be partitioned into two groups $Y$ and $X$ with ${n}_{y}$ variables in $Y$ and ${n}_{x}$ variables in $X$ and let the variance-covariance matrix of all ${n}_{y}+{n}_{x}$ variables be partitioned into,
To test the hypothesis that a partial correlation is zero under the assumption that the data has an approximately Normal distribution a test similar to the test for the full correlation coefficient can be used. If $r$ is the computed partial correlation coefficient then the appropriate $t$ statistic is
$$r\sqrt{\frac{n-{n}_{x}-2}{1-{r}^{2}}}\text{,}$$
which has approximately a Student's $t$-distribution with $n-{n}_{x}-2$ degrees of freedom, where $n$ is the number of observations from which the full correlation coefficients were computed.
4References
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press
Morrison D F (1967) Multivariate Statistical Methods McGraw–Hill
Osborn J F (1979) Statistical Exercises in Medical Research Blackwell
Snedecor G W and Cochran W G (1967) Statistical Methods Iowa State University Press
5Arguments
1: $\mathbf{m}$ – IntegerInput
On entry: the number of variables in the variance-covariance/correlation matrix given in r.
Constraint:
${\mathbf{m}}\ge 3$.
2: $\mathbf{ny}$ – IntegerInput
On entry: the number of $Y$ variables, ${n}_{y}$, for which partial correlation coefficients are to be computed.
Constraint:
${\mathbf{ny}}\ge 2$.
3: $\mathbf{nx}$ – IntegerInput
On entry: the number of $X$ variables, ${n}_{x}$, which are to be considered as fixed.
5: $\mathbf{r}({\mathbf{ldr}},{\mathbf{m}})$ – Real (Kind=nag_wp) arrayInput
On entry: the variance-covariance or correlation matrix for the m variables as given by g02bxf. Only the upper triangle need be given.
Note: the matrix must be a full rank variance-covariance or correlation matrix and so be positive definite. This condition is not directly checked by the routine.
6: $\mathbf{ldr}$ – IntegerInput
On entry: the first dimension of the array r as declared in the (sub)program from which g02byf is called.
Constraint:
${\mathbf{ldr}}\ge {\mathbf{m}}$.
7: $\mathbf{p}({\mathbf{ldp}},{\mathbf{ny}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the strict upper triangle of p contains the strict upper triangular part of the ${n}_{y}\times {n}_{y}$ partial correlation matrix. The lower triangle contains the lower triangle of the ${n}_{y}\times {n}_{y}$ partial variance-covariance matrix if the matrix given in r is a variance-covariance matrix. If the matrix given in r is a partial correlation matrix then the variance-covariance matrix is for standardized variables.
8: $\mathbf{ldp}$ – IntegerInput
On entry: the first dimension of the array p as declared in the (sub)program from which g02byf is called.
Constraint:
${\mathbf{ldp}}\ge {\mathbf{ny}}$.
9: $\mathbf{wk}\left({\mathbf{ny}}\times {\mathbf{nx}}+{\mathbf{nx}}\times ({\mathbf{nx}}+1)/2\right)$ – Real (Kind=nag_wp) arrayWorkspace
10: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{ldp}}=\u27e8\mathit{\text{value}}\u27e9$${\mathbf{ny}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldp}}\ge {\mathbf{ny}}$.
On entry, ${\mathbf{ldr}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ldr}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{m}}>2$.
On entry, ${\mathbf{nx}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nx}}\ge 1$.
On entry, ${\mathbf{ny}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{nx}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ny}}+{\mathbf{nx}}\le {\mathbf{m}}$.
On entry, ${\mathbf{ny}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{ny}}>1$.
${\mathbf{ifail}}=2$
On entry, $\u27e8\mathit{\text{value}}\u27e9$ values of isz are greater than zero and ${\mathbf{nx}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: exactly nx values of isz must be greater than zero.
On entry, $\u27e8\mathit{\text{value}}\u27e9$ values of isz are less than zero and ${\mathbf{ny}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: exactly ny values of isz must be less than zero.
${\mathbf{ifail}}=3$
On entry, the correlation matrix of the independent variables is singular.
Try removing some of the $X$ variables by setting the appropriate element of ${\mathbf{isz}}=0$.
On entry, the square root of the correlation matrix of the independent variables is singular.
Try removing some of the $X$ variables by setting the appropriate element of ${\mathbf{isz}}=0$.
${\mathbf{ifail}}=4$
On entry, a diagonal element of the partial covariance matrix is zero.
Constraint: r must be positive definite.
On entry, a diagonal element of the partial covariance matrix is zero and an element of the partial correlation matrix is greater than $1$.
Constraint: r must be positive definite.
On entry, an element of the partial correlation matrix is greater than $1$.
Constraint: r must be positive definite.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
g02byf computes the partial variance-covariance matrix, ${\Sigma}_{y\mid x}$, by computing the Cholesky factorization of ${\Sigma}_{xx}$. If ${\Sigma}_{xx}$ is not of full rank the computation will fail. For a statement on the accuracy of the Cholesky factorization see f07gdf.
8Parallelism and Performance
g02byf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02byf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
Models that represent the linear associations given by partial correlations can be fitted using the multiple regression routine g02daf.
10Example
Data, given by Osborn (1979), on the number of deaths, smoke ($\mathrm{mg}/{\mathrm{m}}^{3}$) and sulphur dioxide (parts/million) during an intense period of fog is input. The correlations are computed using g02bxf and the partial correlation between deaths and smoke given sulphur dioxide is computed using g02byf. Both correlation matrices are printed using the routine x04caf.