NAG Library Routine Document
G02CHF
1 Purpose
G02CHF performs a multiple linear regression with no constant on a set of variables whose sums of squares and crossproducts about zero and correlationlike coefficients are given.
2 Specification
SUBROUTINE G02CHF ( 
N, K1, K, SSPZ, LDSSPZ, RZ, LDRZ, RESULT, COEF, LDCOEF, RZNV, LDRZNV, CZ, LDCZ, WKZ, LDWKZ, IFAIL) 
INTEGER 
N, K1, K, LDSSPZ, LDRZ, LDCOEF, LDRZNV, LDCZ, LDWKZ, IFAIL 
REAL (KIND=nag_wp) 
SSPZ(LDSSPZ,K1), RZ(LDRZ,K1), RESULT(13), COEF(LDCOEF,3), RZNV(LDRZNV,K), CZ(LDCZ,K), WKZ(LDWKZ,K) 

3 Description
G02CHF fits a curve of the form
to the data points
such that
The routine calculates the regression coefficients,
${b}_{1},{b}_{2},\dots ,{b}_{k}$, (and various other statistical quantities) by minimizing
The actual data values
$\left({x}_{1i},{x}_{2i},\dots ,{x}_{ki},{y}_{i}\right)$ are not provided as input to the routine. Instead, input to the routine consists of:
(i) 
The number of cases, $n$, on which the regression is based. 
(ii) 
The total number of variables, dependent and independent, in the regression, $\left(k+1\right)$. 
(iii) 
The number of independent variables in the regression, $k$. 
(iv) 
The $\left(k+1\right)$ by $\left(k+1\right)$ matrix $\left[{\stackrel{~}{S}}_{ij}\right]$ of sums of squares and crossproducts about zero of all the variables in the regression; the terms involving the dependent variable, $y$, appear in the $\left(k+1\right)$th row and column. 
(v) 
The $\left(k+1\right)$ by $\left(k+1\right)$ matrix $\left[{\stackrel{~}{R}}_{ij}\right]$ of correlationlike coefficients for all the variables in the regression; the correlations involving the dependent variable, $y$, appear in the $\left(k+1\right)$th row and column. 
The quantities calculated are:
(a) 
The inverse of the $k$ by $k$ partition of the matrix of correlationlike coefficients, $\left[{\stackrel{~}{R}}_{ij}\right]$, involving only the independent variables. The inverse is obtained using an accurate method which assumes that this submatrix is positive definite (see Section 9). 
(b) 
The modified matrix, $C=\left[{c}_{ij}\right]$, where
where ${\stackrel{~}{r}}^{ij}$ is the $\left(i,j\right)$th element of the inverse matrix of $\left[{\stackrel{~}{R}}_{ij}\right]$ as described in (a) above. Each element of $C$ is thus the corresponding element of the matrix of correlationlike coefficients multiplied by the corresponding element of the inverse of this matrix, divided by the corresponding element of the matrix of sums of squares and crossproducts about zero. 
(c) 
The regression coefficients:
where ${\stackrel{~}{S}}_{j\left(k+1\right)}$ is the sum of crossproducts about zero for the independent variable ${x}_{j}$ and the dependent variable $y$. 
(d) 
The sum of squares attributable to the regression, $SSR$, the sum of squares of deviations about the regression, $SSD$, and the total sum of squares, $SST$:
 $SST={\stackrel{~}{S}}_{\left(k+1\right)\left(k+1\right)}$, the sum of squares about zero for the dependent variable, $y$;
 $SSR={\displaystyle \sum _{j=1}^{k}}{b}_{j}{\stackrel{~}{S}}_{j\left(k+1\right)}\text{; \hspace{1em}}SSD=SSTSSR$.

(e) 
The degrees of freedom attributable to the regression, $DFR$, the degrees of freedom of deviations about the regression, $DFD$, and the total degrees of freedom, $DFT$:

(f) 
The mean square attributable to the regression, $MSR$, and the mean square of deviations about the regression, $MSD$:

(g) 
The $F$ value for the analysis of variance:

(h) 
The standard error estimate:

(i) 
The coefficient of multiple correlation, $R$, the coefficient of multiple determination, ${R}^{2}$, and the coefficient of multiple determination corrected for the degrees of freedom, ${\stackrel{}{R}}^{2}$:

(j) 
The standard error of the regression coefficients:

(k) 
The $t$ values for the regression coefficients:

4 References
Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the number of cases used in calculating the sums of squares and crossproducts and correlationlike coefficients.
 2: $\mathrm{K1}$ – INTEGERInput

On entry: the total number of variables, independent and dependent $\left(k+1\right)$, in the regression.
Constraint:
$2\le {\mathbf{K1}}\le {\mathbf{N}}$.
 3: $\mathrm{K}$ – INTEGERInput

On entry: the number of independent variables $k$ in the regression.
Constraint:
${\mathbf{K}}={\mathbf{K1}}1$.
 4: $\mathrm{SSPZ}\left({\mathbf{LDSSPZ}},{\mathbf{K1}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: ${\mathbf{SSPZ}}\left(\mathit{i},\mathit{j}\right)$ must be set to ${\stackrel{~}{S}}_{\mathit{i}\mathit{j}}$, the sum of crossproducts about zero for the $\mathit{i}$th and $\mathit{j}$th variables, for $\mathit{i}=1,2,\dots ,k+1$ and $\mathit{j}=1,2,\dots ,k+1$; terms involving the dependent variable appear in row $k+1$ and column $k+1$.
 5: $\mathrm{LDSSPZ}$ – INTEGERInput

On entry: the first dimension of the array
SSPZ as declared in the (sub)program from which G02CHF is called.
Constraint:
${\mathbf{LDSSPZ}}\ge {\mathbf{K1}}$.
 6: $\mathrm{RZ}\left({\mathbf{LDRZ}},{\mathbf{K1}}\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: ${\mathbf{RZ}}\left(\mathit{i},\mathit{j}\right)$ must be set to ${\stackrel{~}{R}}_{\mathit{i}\mathit{j}}$, the correlationlike coefficient for the $\mathit{i}$th and $\mathit{j}$th variables, for $\mathit{i}=1,2,\dots ,k+1$ and $\mathit{j}=1,2,\dots ,k+1$; coefficients involving the dependent variable appear in row $k+1$ and column $k+1$.
 7: $\mathrm{LDRZ}$ – INTEGERInput

On entry: the first dimension of the array
RZ as declared in the (sub)program from which G02CHF is called.
Constraint:
${\mathbf{LDRZ}}\ge {\mathbf{K1}}$.
 8: $\mathrm{RESULT}\left(13\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the following information:
${\mathbf{RESULT}}\left(1\right)$  $SSR$, the sum of squares attributable to the regression; 
${\mathbf{RESULT}}\left(2\right)$  $DFR$, the degrees of freedom attributable to the regression; 
${\mathbf{RESULT}}\left(3\right)$  $MSR$, the mean square attributable to the regression; 
${\mathbf{RESULT}}\left(4\right)$  $F$, the $F$ value for the analysis of variance; 
${\mathbf{RESULT}}\left(5\right)$  $SSD$, the sum of squares of deviations about the regression; 
${\mathbf{RESULT}}\left(6\right)$  $DFD$, the degrees of freedom of deviations about the regression; 
${\mathbf{RESULT}}\left(7\right)$  $MSD$, the mean square of deviations about the regression; 
${\mathbf{RESULT}}\left(8\right)$  $SST$, the total sum of squares; 
${\mathbf{RESULT}}\left(9\right)$  $DFT$, the total degrees of freedom; 
${\mathbf{RESULT}}\left(10\right)$  $s$, the standard error estimate; 
${\mathbf{RESULT}}\left(11\right)$  $R$, the coefficient of multiple correlation; 
${\mathbf{RESULT}}\left(12\right)$  ${R}^{2}$, the coefficient of multiple determination; 
${\mathbf{RESULT}}\left(13\right)$  ${\stackrel{}{R}}^{2}$, the coefficient of multiple determination corrected for the degrees of freedom. 
 9: $\mathrm{COEF}\left({\mathbf{LDCOEF}},3\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: for
$i=1,2,\dots ,k$, the following information:
 ${\mathbf{COEF}}\left(i,1\right)$
 ${b}_{i}$, the regression coefficient for the $i$th variable.
 ${\mathbf{COEF}}\left(i,2\right)$
 $se\left({b}_{i}\right)$, the standard error of the regression coefficient for the $i$th variable.
 ${\mathbf{COEF}}\left(i,3\right)$
 $t\left({b}_{i}\right)$, the $t$ value of the regression coefficient for the $i$th variable.
 10: $\mathrm{LDCOEF}$ – INTEGERInput

On entry: the first dimension of the array
COEF as declared in the (sub)program from which G02CHF is called.
Constraint:
${\mathbf{LDCOEF}}\ge {\mathbf{K}}$.
 11: $\mathrm{RZNV}\left({\mathbf{LDRZNV}},{\mathbf{K}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the inverse of the matrix of correlationlike coefficients for the independent variables; that is, the inverse of the matrix consisting of the first
$k$ rows and columns of
RZ.
 12: $\mathrm{LDRZNV}$ – INTEGERInput

On entry: the first dimension of the array
RZNV as declared in the (sub)program from which G02CHF is called.
Constraint:
${\mathbf{LDRZNV}}\ge {\mathbf{K}}$.
 13: $\mathrm{CZ}\left({\mathbf{LDCZ}},{\mathbf{K}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the modified inverse matrix,
$C$, where
 14: $\mathrm{LDCZ}$ – INTEGERInput

On entry: the first dimension of the array
CZ as declared in the (sub)program from which G02CHF is called.
Constraint:
${\mathbf{LDCZ}}\ge {\mathbf{K}}$.
 15: $\mathrm{WKZ}\left({\mathbf{LDWKZ}},{\mathbf{K}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
 16: $\mathrm{LDWKZ}$ – INTEGERInput

On entry: the first dimension of the array
WKZ as declared in the (sub)program from which G02CHF is called.
Constraint:
${\mathbf{LDWKZ}}\ge {\mathbf{K}}$.
 17: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, 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,  ${\mathbf{K1}}<2$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{K1}}\ne \left({\mathbf{K}}+1\right)$. 
 ${\mathbf{IFAIL}}=3$

On entry,  ${\mathbf{N}}<{\mathbf{K1}}$. 
 ${\mathbf{IFAIL}}=4$

On entry,  ${\mathbf{LDSSPZ}}<{\mathbf{K1}}$, 
or  ${\mathbf{LDRZ}}<{\mathbf{K1}}$, 
or  ${\mathbf{LDCOEF}}<{\mathbf{K}}$, 
or  ${\mathbf{LDRZNV}}<{\mathbf{K}}$, 
or  ${\mathbf{LDCZ}}<{\mathbf{K}}$, 
or  ${\mathbf{LDWKZ}}<{\mathbf{K}}$. 
 ${\mathbf{IFAIL}}=5$

This indicates that the
$k$ by
$k$ partition of the matrix held in
RZ, which is to be inverted, is not positive definite.
 ${\mathbf{IFAIL}}=6$

This indicates that the refinement following the actual inversion fails, indicating that the
$k$ by
$k$ partition of the matrix held in
RZ, which is to be inverted, is illconditioned. The use of
G02DAF, which employs a different numerical technique, may avoid the difficulty.
 ${\mathbf{IFAIL}}=7$

Unexpected error in
F04ABF.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
The accuracy of any regression routine is almost entirely dependent on the accuracy of the matrix inversion method used. In G02CHF, it is the matrix of correlationlike coefficients rather than that of the sums of squares and crossproducts about zero that is inverted; this means that all terms in the matrix for inversion are of a similar order, and reduces the scope for computational error. For details on absolute accuracy, the relevant section of the document describing the inversion routine used,
F04ABF, should be consulted.
G02DAF uses a different method, based on
F04AMF, and that routine may well prove more reliable numerically. It does not handle missing values, nor does it provide the same output as this routine.
If, in calculating
$F$ or any of the
$t\left({b}_{i}\right)$
(see
Section 3), the numbers involved are such that the result would be outside the range of numbers which can be stored by the machine, then the answer is set to the largest quantity which can be stored as a real variable, by means of a call to
X02ALF.
8 Parallelism and Performance
G02CHF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G02CHF 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 implementationspecific information.
The time taken by G02CHF depends on $k$.
This routine assumes that the matrix of correlationlike coefficients for the independent variables in the regression is positive definite; it fails if this is not the case.
This correlation matrix will in fact be positive definite whenever the correlationlike matrix and the sums of squares and crossproducts (about zero) matrix have been formed either without regard to missing values, or by eliminating
completely any cases involving missing values for any variable. If, however, these matrices are formed by eliminating cases with missing values from only those calculations involving the variables for which the values are missing, no such statement can be made, and the correlationlike matrix may or may not be positive definite. You should be aware of the possible dangers of using correlation matrices formed in this way (see the
G02 Chapter Introduction), but if they nevertheless wish to carry out regressions using such matrices, this routine is capable of handling the inversion of such matrices, provided they are positive definite.
If a matrix is positive definite, its subsequent reorganisation by either of
G02CEF or
G02CFF will not affect this property and the new matrix can safely be used in this routine. Thus correlation matrices produced by any of
G02BDF,
G02BEF,
G02BKF or
G02BLF, even if subsequently modified by either
G02CEF or
G02CFF, can be handled by this routine.
It should be noted that the routine requires the dependent variable to be the last of the
$k+1$ variables whose statistics are provided as input to the routine. If this variable is not correctly positioned in the original data, the means, standard deviations, sums of squares and crossproducts about zero, and correlationlike coefficients can be manipulated by using
G02CEF or
G02CFF to reorder the variables as necessary.
10 Example
This example reads in the sums of squares and crossproducts about zero, and correlationlike coefficients for three variables. A multiple linear regression with no constant is then performed with the third and final variable as the dependent variable. Finally the results are printed.
10.1 Program Text
Program Text (g02chfe.f90)
10.2 Program Data
Program Data (g02chfe.d)
10.3 Program Results
Program Results (g02chfe.r)