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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_correg_linregm_estfunc (g02dn)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_correg_linregm_estfunc (g02dn) gives the estimate of an estimable function along with its standard error.


[est, stat, sestat, t, ifail] = g02dn(irank, b, covar, p, f, tol, 'ip', ip)
[est, stat, sestat, t, ifail] = nag_correg_linregm_estfunc(irank, b, covar, p, f, tol, 'ip', ip)


nag_correg_linregm_estfunc (g02dn) computes the estimates of an estimable function for a general linear regression model which is not of full rank. It is intended for use after a call to nag_correg_linregm_fit (g02da) or nag_correg_linregm_update (g02dd). An estimable function is a linear combination of the arguments such that it has a unique estimate. For a full rank model all linear combinations of arguments are estimable.
In the case of a model not of full rank the functions use a singular value decomposition (SVD) to find the parameter estimates, β^, and their variance-covariance matrix. Given the upper triangular matrix R obtained from the QR decomposition of the independent variables the SVD gives
R=Q* D 0 0 0 PT,  
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
P1 being the first k columns of P, i.e., P=P1P0, Q*1 being the first k columns of Q*, and c1 being the first p elements of c.
Details of the SVD are made available in the form of the matrix P*:
P*= D-1 P1T P0T ,  
as given by nag_correg_linregm_fit (g02da) and nag_correg_linregm_update (g02dd).
A linear function of the arguments, F=fTβ, can be tested to see if it is estimable by computing ζ= P0T f. If ζ is zero, then the function is estimable; if not, the function is not estimable. In practice ζ is tested against some small quantity η.
Given that F is estimable it can be estimated by fTβ^ and its standard error calculated from the variance-covariance matrix of β^, Cβ, as
Also a t-statistic,
t=fTβ^ seF ,  
can be computed. The t-statistic will have a Student's t-distribution with degrees of freedom as given by the degrees of freedom for the residual sum of squares for the model.


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


Compulsory Input Parameters

1:     irank int64int32nag_int scalar
k, the rank of the independent variables.
Constraint: 1irankip.
2:     bip – double array
The ip values of the estimates of the arguments of the model, β^.
3:     covarip×ip+1/2 – double array
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 bi and the parameter estimate given in bj, ji, is stored in covarj×j-1/2+i.
4:     pip×ip+2×ip – double array
5:     fip – double array
f, the linear function to be estimated.
6:     tol – double scalar
η, the tolerance value used in the check for estimability.
ε, where ε is the machine precision, is used instead.

Optional Input Parameters

1:     ip int64int32nag_int scalar
Default: the dimension of the arrays b, f. (An error is raised if these dimensions are not equal.)
p, the number of terms in the linear model.
Constraint: ip1.

Output Parameters

1:     est – logical scalar
Indicates if the function was estimable.
The function is estimable.
The function is not estimable and stat, sestat and t are not set.
2:     stat – double scalar
If est=true, stat contains the estimate of the function, fTβ^.
3:     sestat – double scalar
If est=true, sestat contains the standard error of the estimate of the function, seF.
4:     t – double scalar
If est=true, t contains the t-statistic for the test of the function being equal to zero.
5:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Note: nag_correg_linregm_estfunc (g02dn) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

On entry,ip<1,
W  ifail=2
On entry,irank=ip. In this case est is returned as true and all statistics are calculated.
Standard error of statistic =0.0; this may be due to rounding errors if the standard error is very small or due to mis-specified inputs covar and f.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


The computations are believed to be stable.

Further Comments

The value of estimable functions is independent of the solution chosen from the many possible solutions. While nag_correg_linregm_estfunc (g02dn) may be used to estimate functions of the arguments of the model as computed by nag_correg_linregm_constrain (g02dk), βc, these must be expressed in terms of the original arguments, β. The relation between the two sets of arguments may not be straightforward.


Data from an experiment with four treatments and three observations per treatment are read in. A model, with a mean term, is fitted by nag_correg_linregm_fit (g02da). The number of functions to be tested is read in, then the linear functions themselves are read in and tested with nag_correg_linregm_estfunc (g02dn). The results of nag_correg_linregm_estfunc (g02dn) are printed.
function g02dn_example

fprintf('g02dn example results\n\n');

x = [1, 0, 0, 0;
     0, 0, 0, 1;
     0, 1, 0, 0;
     0, 0, 1, 0;
     0, 0, 0, 1;
     0, 1, 0, 0;
     0, 0, 0, 1;
     1, 0, 0, 0;
     0, 0, 1, 0;
     1, 0, 0, 0;
     0, 0, 1, 0;
     0, 1, 0, 0];
y = [33.63;     39.62;     38.18;     41.46;     38.02;     35.83;
     35.99;     36.58;     42.92;     37.80;     40.43;     37.89];

[n,m]  = size(x);
isx    = ones(m,1,'int64');
mean_p = 'M';
ip     = int64(m+1);

% Fit general linear regression model
[rss, idf, b, se, covar, res, h, q, svd, irank, p, wk, ifail] = ...
  g02da(mean_p, x, isx, ip, y);

% Display initial results
fprintf('Estimates from g02da\n\n');
fprintf('Residual sum of squares = %12.4e\n', rss);
fprintf('Degrees of freedom      = %4d\n', idf);
fprintf('\nVariable   Parameter estimate   Standard error\n\n');
ivar = double([1:ip]');
fprintf('%6d%20.4e%20.4e\n',[ivar b se]');

% Estimable functions
f = [1 1  0 0 0;
     0 1 -1 0 0;
     0 1  0 0 0];
nf = size(f,1);
tol = 1e-05;

% Loop over estimable functions
for j = 1:nf
  [est, stat, sestat, t, ifail] = ...
  g02dn(irank, b, covar, p, f(j,:), tol);

  % Display results
   fprintf('\nFunction %2d\n\n', j);
   fprintf('%8.2f', f(j,:));
   if est
     fprintf('\n\nstat = %10.4f, se = %10.4f, t = %10.4f\n', stat, sestat, t);
     fprintf('\n\nFunction not estimable\n');

g02dn example results

Estimates from g02da

Residual sum of squares =   2.2227e+01
Degrees of freedom      =    8

Variable   Parameter estimate   Standard error

     1          3.0557e+01          3.8494e-01
     2          5.4467e+00          8.3896e-01
     3          6.7433e+00          8.3896e-01
     4          1.1047e+01          8.3896e-01
     5          7.3200e+00          8.3896e-01

Function  1

    1.00    1.00    0.00    0.00    0.00

stat =    36.0033, se =     0.9623, t =    37.4119

Function  2

    0.00    1.00   -1.00    0.00    0.00

stat =    -1.2967, se =     1.3610, t =    -0.9528

Function  3

    0.00    1.00    0.00    0.00    0.00

Function not estimable

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