NAG Library Routine Document
G02DDF
1 Purpose
G02DDF calculates the regression arguments for a general linear regression model. It is intended to be called after
G02DCF,
G02DEF or
G02DFF.
2 Specification
SUBROUTINE G02DDF ( 
N, IP, Q, LDQ, RSS, IDF, B, SE, COV, SVD, IRANK, P, TOL, WK, IFAIL) 
INTEGER 
N, IP, LDQ, IDF, IRANK, IFAIL 
REAL (KIND=nag_wp) 
Q(LDQ,IP+1), RSS, B(IP), SE(IP), COV(IP*(IP+1)/2), P(IP*IP+2*IP), TOL, WK(IP*IP+(IP1)*5) 
LOGICAL 
SVD 

3 Description
A general linear regression model fitted by
G02DAF may be adjusted by adding or deleting an observation using
G02DCF, adding a new independent variable using
G02DEF or deleting an existing independent variable using
G02DFF. Alternatively a model may be constructed by a forward selection procedure using
G02EEF. These routines compute the vector
$c$ and the upper triangular matrix
$R$. G02DDF takes these basic results and computes the regression coefficients,
$\hat{\beta}$, their standard errors and their variancecovariance matrix.
If
$R$ is of full rank, then
$\hat{\beta}$ is the solution to
where
${c}_{1}$ is the first
$p$ elements of
$c$.
If
$R$ is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of
$R$,
where
$D$ is a
$k$ by
$k$ diagonal matrix with nonzero diagonal elements,
$k$ being the rank of
$R$, and
${Q}_{*}$ and
$P$ are
$p$ by
$p$ orthogonal matrices. This gives the solution
${P}_{1}$ being the first
$k$ columns of
$P$, i.e.,
$P=\left({P}_{1}{P}_{0}\right)$, and
${Q}_{{*}_{1}}$ being the first
$k$ columns of
${Q}_{*}$.
Details of the SVD are made available in the form of the matrix
${P}^{*}$:
This will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the arguments. These solutions can be obtained by calling
G02DKF after calling G02DDF. Only certain linear combinations of the arguments will have unique estimates; these are known as estimable functions. These can be estimated using
G02DNF.
The residual sum of squares required to calculate the standard errors and the variancecovariance matrix can either be input or can be calculated if additional information on $c$ for the whole sample is provided.
4 References
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25
Searle S R (1971) Linear Models Wiley
5 Arguments
 1: $\mathrm{N}$ – INTEGERInput

On entry: the number of observations.
Constraint:
${\mathbf{N}}\ge 1$.
 2: $\mathrm{IP}$ – INTEGERInput

On entry: $p$, the number of terms in the regression model.
Constraint:
${\mathbf{IP}}\ge 1$.
 3: $\mathrm{Q}\left({\mathbf{LDQ}},{\mathbf{IP}}+1\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: must be the array
Q as output by
G02DCF,
G02DEF,
G02DFF or
G02EEF. If on entry
${\mathbf{RSS}}\le 0.0$ then all
N elements of
$c$ are needed. This is provided by routines
G02DEF,
G02DFF or
G02EEF.
 4: $\mathrm{LDQ}$ – INTEGERInput

On entry: the first dimension of the array
Q as declared in the (sub)program from which G02DDF is called.
Constraints:
 if ${\mathbf{RSS}}\le 0.0$, ${\mathbf{LDQ}}\ge {\mathbf{N}}$;
 otherwise ${\mathbf{LDQ}}\ge {\mathbf{IP}}$.

On entry: either the residual sum of squares or a value less than or equal to $0.0$ to indicate that the residual sum of squares is to be calculated by the routine.
On exit: if
${\mathbf{RSS}}\le 0.0$ on entry, then on exit
RSS will contain the residual sum of squares as calculated by G02DDF.
If
RSS was positive on entry, it will be unchanged.
 6: $\mathrm{IDF}$ – INTEGEROutput

On exit: the degrees of freedom associated with the residual sum of squares.
 7: $\mathrm{B}\left({\mathbf{IP}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the estimates of the $p$ parameters, $\hat{\beta}$.
 8: $\mathrm{SE}\left({\mathbf{IP}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the standard errors of the
$p$ parameters given in
B.
 9: $\mathrm{COV}\left({\mathbf{IP}}\times \left({\mathbf{IP}}+1\right)/2\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the upper triangular part of the variancecovariance matrix of the
$p$ parameter estimates given in
B. They are stored packed by column, i.e., the covariance between the parameter estimate given in
${\mathbf{B}}\left(i\right)$ and the parameter estimate given in
${\mathbf{B}}\left(j\right)$,
$j\ge i$, is stored in
${\mathbf{COV}}\left(j\times \left(j1\right)/2+i\right)$.
 10: $\mathrm{SVD}$ – LOGICALOutput

On exit: if a singular value decomposition has been performed, ${\mathbf{SVD}}=\mathrm{.TRUE.}$, otherwise ${\mathbf{SVD}}=\mathrm{.FALSE.}$.
 11: $\mathrm{IRANK}$ – INTEGEROutput

On exit: the rank of the independent variables.
If ${\mathbf{SVD}}=\mathrm{.FALSE.}$, ${\mathbf{IRANK}}={\mathbf{IP}}$.
If
${\mathbf{SVD}}=\mathrm{.TRUE.}$,
IRANK is an estimate of the rank of the independent variables.
IRANK is calculated as the number of singular values greater than
${\mathbf{TOL}}\times \text{}$ (largest singular value). It is possible for the SVD to be carried out but
IRANK to be returned as
IP.
 12: $\mathrm{P}\left({\mathbf{IP}}\times {\mathbf{IP}}+2\times {\mathbf{IP}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: contains details of the singular value decomposition if used.
If
${\mathbf{SVD}}=\mathrm{.FALSE.}$,
P is not referenced.
If
${\mathbf{SVD}}=\mathrm{.TRUE.}$, the first
IP elements of
P will not be referenced, the next
IP values contain the singular values. The following
${\mathbf{IP}}\times {\mathbf{IP}}$ values contain the matrix
${P}^{*}$ stored by columns.
 13: $\mathrm{TOL}$ – REAL (KIND=nag_wp)Input

On entry: the value of
TOL is used to decide if the independent variables are of full rank and, if not, what is the rank of the independent variables. The smaller the value of
TOL the stricter the criterion for selecting the singular value decomposition. If
${\mathbf{TOL}}=0.0$, the singular value decomposition will never be used, this may cause run time errors or inaccuracies if the independent variables are not of full rank.
Suggested value:
${\mathbf{TOL}}=0.000001$.
Constraint:
${\mathbf{TOL}}\ge 0.0$.
 14: $\mathrm{WK}\left({\mathbf{IP}}\times {\mathbf{IP}}+\left({\mathbf{IP}}1\right)\times 5\right)$ – REAL (KIND=nag_wp) arrayWorkspace

 15: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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
${\mathbf{IFAIL}}=0$ or
$1$, 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,  ${\mathbf{N}}<1$, 
or  ${\mathbf{IP}}<1$, 
or  ${\mathbf{LDQ}}<{\mathbf{IP}}$, 
or  ${\mathbf{LDQ}}<{\mathbf{N}}$, 
or  ${\mathbf{TOL}}<0.0$. 
 ${\mathbf{IFAIL}}=2$

The degrees of freedom for error are less than or equal to $0$. In this case the estimates of $\beta $ are returned but not the standard errors or covariances.
 ${\mathbf{IFAIL}}=3$

The singular value decomposition, if used, has failed to converge, see
F02WUF. This is an unlikely error exit.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7 Accuracy
The accuracy of the results will depend on the accuracy of the input $R$ matrix, which may lose accuracy if a large number of observations or variables have been dropped.
8 Parallelism and Performance
G02DDF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
G02DDF 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.
None.
10 Example
A dataset consisting of
$12$ observations and four independent variables is input and a regression model fitted by calls to
G02DEF. The arguments are then calculated by G02DDF and the results printed.
10.1 Program Text
Program Text (g02ddfe.f90)
10.2 Program Data
Program Data (g02ddfe.d)
10.3 Program Results
Program Results (g02ddfe.r)