void	g02dnc (Integer ip, Integer rank, const double b[], const double cov[], const double p[], const double f[], Nag_Boolean est, double stat, double sestat, double t, double tol, NagError *fail)

The function may be called by the names: g02dnc, nag_correg_linregm_estfunc or nag_regsn_mult_linear_est_func.

3 Description

This function 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 g02dac or g02ddc. 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,

\hat{β}

, and their variance-covariance matrix. Given the upper triangular matrix

R

obtained from the

Q R

decomposition of the independent variables the SVD gives:

R = Q_{*} (\begin{matrix} D & 0 \\ 0 & 0 \end{matrix}) P^{T}

where

D

is a

k \times k

diagonal matrix with nonzero diagonal elements,

k

being the rank of

R

, and

Q_{*}

and

P

are

p \times p

orthogonal matrices. This leads to a solution:

\hat{β} = P_{1} D^{−1} Q_{*_{1}}^{T} c_{1}

P_{1}

being the first

k

columns of

P

, i.e.,

P = (P_{1} P_{0})

Q_{*_{1}}

being the first

k

columns of

Q_{*}

and

c_{1}

being the first

p

elements of

c

Details of the SVD are made available, in the form of the matrix

P^{*}

P^{*} = (\begin{matrix} D^{−1} P_{1}^{T} \\ P_{0}^{T} \end{matrix})

as given by g02dac and g02ddc.

A linear function of the arguments,

F = f^{T} β

, can be tested to see if it is estimable by computing

ζ = P_{0}^{T} 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

f^{T} \hat{β}

and its standard error calculated from the variance-covariance matrix of

\hat{β}

C_{β}

, as

se (F) = \sqrt{f^{T} C_{β} f}

Also a

t

-statistic:

t = \frac{f^{T} \hat{β}}{se (F)},

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.

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: $ip$ – Integer Input

On entry: the number of terms in the linear model,

p

Constraint:

ip \geq 1

2: $rank$ – Integer Input

On entry: the rank of the independent variables,

k

Constraint:

1 \leq rank \leq ip

3: $b [ip]$ – const double Input

On entry: the ip values of the estimates of the arguments of the model,

\hat{β}

4: $cov [ip \times (ip + 1) / 2]$ – const double Input

On entry: 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]

j \geq i

, is stored in

cov [j (j + 1) / 2 + i]

, for

i = 0, 1, \dots, ip - 1

and

j = i, \dots, ip - 1

5: $p [ip \times ip + 2 \times ip]$ – const double Input

On entry: p as returned by g02dac or g02ddc.

6: $f [ip]$ – const double Input

On entry: the linear function to be estimated,

f

7: $est$ – Nag_Boolean * Output

On exit: est indicates if the function was estimable.

$est = Nag_TRUE$: The function is estimable.
$est = Nag_FALSE$: The function is not estimable and stat, sestat and t are not set.

8: $stat$ – double * Output

On exit: if

est = Nag_TRUE

, stat contains the estimate of the function,

f^{T} \hat{β}

9: $sestat$ – double * Output

On exit: if

est = Nag_TRUE

, sestat contains the standard error of the estimate of the function,

se (F)

10: $t$ – double * Output

On exit: if

est = Nag_TRUE

, t contains the

t

-statistic for the test of the function being equal to zero.

11: $tol$ – double Input

On entry: tol is the tolerance value used in the check for estimability,

η

. If

tol \leq 0.0

\sqrt{machine precision}

is used instead.

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_GT: On entry, $ip = ⟨ value ⟩$ while $rank = ⟨ value ⟩$ . These arguments must satisfy $rank \leq ip$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_INT_ARG_LT: On entry, $ip = ⟨ value ⟩$ .
Constraint: $ip \geq 1$ .

On entry, $rank = ⟨ value ⟩$ .
Constraint: $rank \geq 1$ .
NE_RANK_EQ_IP: On entry, $rank = ip$ . In this case, the boolean variable est is returned as Nag_TRUE and all statistics are calculated.
NE_STDES_ZERO: se $(F) = 0.0$ probably due to rounding error or due to incorrectly specified inputs cov and f.

7 Accuracy

The computations are believed to be stable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g02dnc is not threaded in any implementation.

9 Further Comments

The value of estimable functions is independent of the solution chosen from the many possible solutions. While g02dnc may be used to estimate functions of the arguments of the model as computed by g02dkc,

β_{c}

, these must be expressed in terms of the original arguments,

β

. The relation between the two sets of arguments may not be straightforward.

10 Example

Data from an experiment with four treatments and three observations per treatment are read in. A model, with a mean term, is fitted by g02dac. The number of functions to be tested is read in, then the linear functions themselves are read in and tested with g02dnc. The results of g02dnc are printed.

g02dn: FL CL CPP AD PY MB

NAG CL Interfaceg02dnc (linregm_​estfunc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g02dnc (linregm_estfunc)