NAG FL Interface
g02daf (linregm_​fit)

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1 Purpose

g02daf performs a general multiple linear regression when the independent variables may be linearly dependent. Parameter estimates, standard errors, residuals and influence statistics are computed. g02daf may be used to perform a weighted regression.

2 Specification

Fortran Interface
Subroutine g02daf ( mean, weight, n, x, ldx, m, isx, ip, y, wt, rss, idf, b, se, cov, res, h, q, ldq, svd, irank, p, tol, wk, ifail)
Integer, Intent (In) :: n, ldx, m, isx(m), ip, ldq
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: idf, irank
Real (Kind=nag_wp), Intent (In) :: x(ldx,m), y(n), wt(*), tol
Real (Kind=nag_wp), Intent (Inout) :: q(ldq,ip+1)
Real (Kind=nag_wp), Intent (Out) :: rss, b(ip), se(ip), cov(ip*(ip+1)/2), res(n), h(n), p(2*ip+ip*ip), wk(max(2,5*(ip-1)+ip*ip))
Logical, Intent (Out) :: svd
Character (1), Intent (In) :: mean, weight
C Header Interface
#include <nag.h>
void  g02daf_ (const char *mean, const char *weight, const Integer *n, const double x[], const Integer *ldx, const Integer *m, const Integer isx[], const Integer *ip, const double y[], const double wt[], double *rss, Integer *idf, double b[], double se[], double cov[], double res[], double h[], double q[], const Integer *ldq, logical *svd, Integer *irank, double p[], const double *tol, double wk[], Integer *ifail, const Charlen length_mean, const Charlen length_weight)
The routine may be called by the names g02daf or nagf_correg_linregm_fit.

3 Description

The general linear regression model is defined by
y=Xβ+ε,  
where
If V=I, the identity matrix, then least squares estimation is used. If VI, then for a given weight matrix WV-1, weighted least squares estimation is used.
The least squares estimates β^ of the parameters β minimize (y-Xβ)T (y-Xβ) while the weighted least squares estimates minimize (y-Xβ)T W(y-Xβ) .
g02daf finds a QR decomposition of X (or W1/2X in weighted case), i.e.,
X=QR*(or  W1/2X=QR*),  
where R*=( R 0 ) and R is a p×p upper triangular matrix and Q is an n×n orthogonal matrix. If R is of full rank, then β^ is the solution to
Rβ^=c1,  
where c=QTy (or QTW1/2y) and c1 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,
R=Q* ( D 0 0 0 ) PT,  
where D is a k×k diagonal matrix with nonzero diagonal elements, k being the rank of R, and Q* and P are p×p orthogonal matrices. This gives the solution
β^=P1D-1Q*1Tc1,  
P1 being the first k columns of P, i.e., P=(P1P0), and Q*1 being the first k columns of Q*.
Details of the SVD, are made available, in the form of the matrix P*:
P*=( D-1 P1T P0T ) .  
This will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the parameters. These solutions can be obtained by using g02dkf after using g02daf. Only certain linear combinations of the parameters will have unique estimates; these are known as estimable functions.
The fit of the model can be examined by considering the residuals, ri=yi-y^, where y^=Xβ^ are the fitted values. The fitted values can be written as Hy for an n×n matrix H. The ith diagonal elements of H, hi, give a measure of the influence of the ith values of the independent variables on the fitted regression model. The values hi are sometimes known as leverages. Both ri and hi are provided by g02daf.
The output of g02daf also includes β^, the residual sum of squares and associated degrees of freedom, (n-k), the standard errors of the parameter estimates and the variance-covariance matrix of the parameter estimates.
In many linear regression models the first term is taken as a mean term or an intercept, i.e., Xi,1=1, for i=1,2,,n. This is provided as an option. Also only some of the possible independent variables are required to be included in a model, a facility to select variables to be included in the model is provided.
Details of the QR decomposition and, if used, the SVD, are made available. These allow the regression to be updated by adding or deleting an observation using g02dcf, adding or deleting a variable using g02def and g02dff or estimating and testing an estimable function using g02dnf. For the same matrix of independent variables, a new set of parameter estimates can be quickly calculated from a new vector of dependent variables using g02dgf. The details of the factorizations held in q, p and wk are only for use by this suite of routines and cannot be used by other routines that use such factorizations, e.g., f08agf since these will expect a different storage scheme for the input factorization.

4 References

Cook R D and Weisberg S (1982) Residuals and Influence in Regression Chapman and Hall
Draper N R and Smith H (1985) Applied Regression Analysis (2nd Edition) Wiley
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
McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall
Searle S R (1971) Linear Models Wiley

5 Arguments

1: mean Character(1) Input
On entry: indicates if a mean term is to be included.
mean='M'
A mean term, intercept, will be included in the model.
mean='Z'
The model will pass through the origin, zero-point.
Constraint: mean='M' or 'Z'.
2: weight Character(1) Input
On entry: indicates if weights are to be used.
weight='U'
Least squares estimation is used.
weight='W'
Weighted least squares is used and weights must be supplied in array wt.
Constraint: weight='U' or 'W'.
3: n Integer Input
On entry: n, the number of observations.
Constraint: n2.
4: x(ldx,m) Real (Kind=nag_wp) array Input
On entry: x(i,j) must contain the ith observation for the jth independent variable, for i=1,2,,n and j=1,2,,m.
5: ldx Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02daf is called.
Constraint: ldxn.
6: m Integer Input
On entry: m, the total number of independent variables in the dataset.
Constraint: m1.
7: isx(m) Integer array Input
On entry: indicates which independent variables are to be included in the model.
isx(j)>0
The variable contained in the jth column of x is included in the regression model.
Constraints:
  • isx(j)0, for j=1,2,,m;
  • if mean='M', exactly ip-1 values of isx must be >0;
  • if mean='Z', exactly ip values of isx must be >0.
8: ip Integer Input
On entry: the number of independent variables in the model, including the mean or intercept if present.
Constraints:
  • if mean='M', 1ipm+1;
  • if mean='Z', 1ipm;
  • otherwise 1ipn.
9: y(n) Real (Kind=nag_wp) array Input
On entry: y, the observations on the dependent variable.
10: wt(*) Real (Kind=nag_wp) array Input
Note: the dimension of the array wt must be at least n if weight='W'.
On entry: if weight='W' wt must contain the weights to be used with the model.
If wt(i)=0.0, the ith observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights. The values of res and h will be set to zero for observations with zero weights.
If weight='U', wt is not referenced and the effective number of observations is n.
Constraint: if weight='W', wt(i)0.0, for i=1,2,,n.
11: rss Real (Kind=nag_wp) Output
On exit: the residual sum of squares for the regression.
12: idf Integer Output
On exit: the degrees of freedom associated with the residual sum of squares.
13: b(ip) Real (Kind=nag_wp) array Output
On exit: b(i), i=1,2,,ip contains the least squares estimates of the parameters of the regression model, β^.
If mean='M', b(1) will contain the estimate of the mean parameter and b(i+1) will contain the coefficient of the variable contained in column j of x, where isx(j) is the ith positive value in the array isx.
If mean='Z', b(i) will contain the coefficient of the variable contained in column j of x, where isx(j) is the ith positive value in the array isx.
14: se(ip) Real (Kind=nag_wp) array Output
On exit: se(i), i=1,2,,ip contains the standard errors of the ip parameter estimates given in b.
15: cov(ip×(ip+1)/2) Real (Kind=nag_wp) array Output
On exit: the first ip×(ip+1)/2 elements of cov contain the upper triangular part of the variance-covariance matrix of the ip parameter estimates given in b. They are stored packed by column, i.e., the covariance between the parameter estimate given in b(i) and the parameter estimate given in b(j), ji, is stored in cov(j×(j-1)/2+i).
16: res(n) Real (Kind=nag_wp) array Output
On exit: the (weighted) residuals, ri, for i=1,2,,n.
17: h(n) Real (Kind=nag_wp) array Output
On exit: the diagonal elements of H, hi, for i=1,2,,n.
18: q(ldq,ip+1) Real (Kind=nag_wp) array Output
On exit: the results of the QR decomposition:
  • the first column of q contains c;
  • the upper triangular part of columns 2 to ip+1 contain the R matrix;
  • the strictly lower triangular part of columns 2 to ip+1 contain details of the Q matrix.
19: ldq Integer Input
On entry: the first dimension of the array q as declared in the (sub)program from which g02daf is called.
Constraint: ldqn.
20: svd Logical Output
On exit: if a singular value decomposition has been performed then svd will be .TRUE., otherwise svd will be .FALSE..
21: irank Integer Output
On exit: the rank of the independent variables.
If svd=.FALSE., irank=ip.
If svd=.TRUE., irank is an estimate of the rank of the independent variables.
irank is calculated as the number of singular values greater that tol× (largest singular value). It is possible for the SVD to be carried out but irank to be returned as ip.
22: p(2×ip+ip×ip) Real (Kind=nag_wp) array Output
On exit: details of the QR decomposition and SVD if used.
If svd=.FALSE., only the first ip elements of p are used these will contain the zeta values for the QR decomposition (see f08aef for details).
If svd=.TRUE., the first ip elements of p will contain the zeta values for the QR decomposition (see f08aef for details) and the next ip elements of p contain singular values. The following ip by ip elements contain the matrix P* stored by columns.
23: 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 tol=0.0, the singular value decomposition will never be used; this may cause run time errors or inaccurate results if the independent variables are not of full rank.
Suggested value: tol=0.000001.
Constraint: tol0.0.
24: wk(max(2,5×(ip-1)+ip×ip)) Real (Kind=nag_wp) array Output
On exit: if on exit svd=.TRUE., wk contains information which is needed by g02dgf; otherwise wk is used as workspace.
25: ifail Integer Input/Output
On entry: ifail must be set to 0, −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 −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 −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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry 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:
ifail=1
On entry, ip=value.
Constraint: ip1.
On entry, ip=value and n=value.
Constraint: ipn.
On entry, ldq=value and n=value.
Constraint: ldqn.
On entry, ldx=value and n=value.
Constraint: ldxn.
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n2.
On entry, tol=value.
Constraint: tol0.0.
ifail=2
On entry, mean=value.
Constraint: mean='M' or 'Z'.
On entry, weight=value.
Constraint: weight='U' or 'W'.
ifail=3
On entry, wt(value)<0.0.
Constraint: wt(i)0.0, for i=1,2,,n.
ifail=4
On entry, ip=value.
Constraint: ip must be compatible with the number of nonzero elements in isx.
On entry, isx(value)<0.
Constraint: isx(i)0.0, for i=1,2,,m.
ifail=5
The degrees of freedom for the residuals are zero, i.e., the designated number of arguments is equal to the effective number of observations. In this case the parameter estimates will be returned along with the diagonal elements of H, but neither standard errors nor the variance-covariance matrix will be calculated.
ifail=6
SVD solution failed to converge.
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.
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The accuracy of g02daf is closely related to the accuracy of f02wuf and f08aef. These routine documents should be consulted.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
g02daf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02daf 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.

9 Further Comments

Standardized residuals and further measures of influence can be computed using g02faf. g02faf requires, in particular, the results stored in res and h.

10 Example

Data from an experiment with four treatments and three observations per treatment are read in. The treatments are represented by dummy (0-1) variables. An unweighted model is fitted with a mean included in the model. g02buf is then called to calculate the total sums of squares and the coefficient of determination (R2), adjusted R2 and Akaike's information criteria (AIC) are calculated.

10.1 Program Text

Program Text (g02dafe.f90)

10.2 Program Data

Program Data (g02dafe.d)

10.3 Program Results

Program Results (g02dafe.r)