NAG Library Routine Document
G02DKF calculates the estimates of the parameters of a general linear regression model for given constraints from the singular value decomposition results.
|SUBROUTINE G02DKF (
||IP, ICONST, P, C, LDC, B, RSS, IDF, SE, COV, WK, IFAIL)
||IP, ICONST, LDC, IDF, IFAIL
||P(IP*IP+2*IP), C(LDC,ICONST), B(IP), RSS, SE(IP), COV(IP*(IP+1)/2), WK(2*IP*IP+IP*ICONST+2*ICONST*ICONST+4*ICONST)
G02DKF computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to G02DAF
In the case of a model not of full rank the routines use a singular value decomposition (SVD) to find the parameter estimates,
, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix
as described by G02DAF
Alternative solutions can be formed by imposing constraints on the parameters. If there are parameters and the rank of the model is , then constraints will have to be imposed to obtain a unique solution.
matrix of constraints, such that
then the new parameter estimates
are given by
is the identity matrix, and the variance-covariance matrix is given by
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
- 1: IP – INTEGERInput
On entry: , the number of terms in the linear model.
- 2: ICONST – INTEGERInput
On entry: the number of constraints to be imposed on the parameters, .
- 3: P() – REAL (KIND=nag_wp) arrayInput
: as returned by G02DAF
- 4: C(LDC,ICONST) – REAL (KIND=nag_wp) arrayInput
: the ICONST
constraints stored by column, i.e., the
th constraint is stored in the
th column of C
- 5: LDC – INTEGERInput
: the first dimension of the array C
as declared in the (sub)program from which G02DKF is called.
- 6: B(IP) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the parameter estimates computed by using the singular value decomposition, .
On exit: the parameter estimates of the parameters with the constraints imposed, .
: the residual sum of squares as returned by G02DAF
- 8: IDF – INTEGERInput
: the degrees of freedom associated with the residual sum of squares as returned by G02DAF
- 9: SE(IP) – REAL (KIND=nag_wp) arrayOutput
: the standard error of the parameter estimates in B
- 10: COV() – REAL (KIND=nag_wp) arrayOutput
: 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
and the parameter estimate given in
, is stored in
- 11: WK() – REAL (KIND=nag_wp) arrayWorkspace
Note that a simple upper bound for the size of the workspace is .
- 12: IFAIL – INTEGERInput/Output
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
does not give a model of full rank.
It should be noted that due to rounding errors a parameter that should be zero when the constraints have been imposed may be returned as a value of order machine precision.
G02DKF is intended for use in situations in which dummy () variables have been used such as in the analysis of designed experiments when you do not wish to change the parameters of the model to give a full rank model. The routine is not intended for situations in which the relationships between the independent variables are only approximate.
Data from an experiment with four treatments and three observations per treatment are read in. A model, including the mean term, is fitted by G02DAF
and the results printed. The constraint that the sum of treatment effect is zero is then read in and the parameter estimates with this constraint imposed are computed by G02DKF and printed.
9.1 Program Text
Program Text (g02dkfe.f90)
9.2 Program Data
Program Data (g02dkfe.d)
9.3 Program Results
Program Results (g02dkfe.r)