NAG Library Function Document
nag_regsn_mult_linear_est_func (g02dnc)
1 Purpose
nag_regsn_mult_linear_est_func (g02dnc) gives the estimate of an estimable function along with its standard error.
2 Specification
#include <nag.h> |
#include <nagg02.h> |
void |
nag_regsn_mult_linear_est_func (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) |
|
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
nag_regsn_mult_linear (g02dac) or
nag_regsn_mult_linear_upd_model (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,
, and their variance-covariance matrix. Given the upper triangular matrix
obtained from the
decomposition of the independent variables the SVD gives:
where
is a
by
diagonal matrix with nonzero diagonal elements,
being the rank of
, and
and
are
by
orthogonal matrices. This leads to a solution:
being the first
columns of
, i.e.,
,
being the first
columns of
and
being the first
elements of
.
Details of the SVD are made available, in the form of the matrix
:
as given by
nag_regsn_mult_linear (g02dac) and
nag_regsn_mult_linear_upd_model (g02ddc).
A linear function of the arguments, , can be tested to see if it is estimable by computing . If is zero, then the function is estimable, if not, the function is not estimable. In practice is tested against some small quantity .
Given that
is estimable it can be estimated by
and its standard error calculated from the variance-covariance matrix of
,
, as
Also a
-statistic:
can be computed. The
-statistic will have a Student's
-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 – IntegerInput
On entry: the number of terms in the linear model, .
Constraint:
.
- 2:
rank – IntegerInput
-
On entry: the rank of the independent variables, .
Constraint:
.
- 3:
b[ip] – const doubleInput
-
On entry: the
ip values of the estimates of the arguments of the model,
.
- 4:
cov[] – const doubleInput
-
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
and the parameter estimate given in
,
, is stored in
, for
and
.
- 5:
p[] – const doubleInput
-
- 6:
f[ip] – const doubleInput
-
On entry: the linear function to be estimated, .
- 7:
est – Nag_Boolean *Output
-
On exit:
est indicates if the function was estimable.
- The function is estimable.
- The function is not estimable and stat, sestat and t are not set.
- 8:
stat – double *Output
-
On exit: if
,
stat contains the estimate of the function,
.
- 9:
sestat – double *Output
-
On exit: if
,
sestat contains the standard error of the estimate of the function,
.
- 10:
t – double *Output
-
On exit: if
,
t contains the
-statistic for the test of the function being equal to zero.
- 11:
tol – doubleInput
-
On entry:
tol is the tolerance value used in the check for estimability,
. If
,
is used instead.
- 12:
fail – NagError *Input/Output
-
The NAG error argument (see
Section 3.6 in the Essential Introduction).
6 Error Indicators and Warnings
- NE_2_INT_ARG_GT
-
On entry, while . These arguments must satisfy .
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
- NE_INT_ARG_LT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_RANK_EQ_IP
-
On entry,
. In this case, the boolean variable
est is returned as Nag_TRUE and all statistics are calculated.
- NE_STDES_ZERO
-
se
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
Not applicable.
The value of estimable functions is independent of the solution chosen from the many possible solutions. While nag_regsn_mult_linear_est_func (g02dnc) may be used to estimate functions of the arguments of the model as computed by
nag_regsn_mult_linear_tran_model (g02dkc),
, these must be expressed in terms of the original arguments,
. 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
nag_regsn_mult_linear (g02dac). The number of functions to be tested is read in, then the linear functions themselves are read in and tested with nag_regsn_mult_linear_est_func (g02dnc). The results of nag_regsn_mult_linear_est_func (g02dnc) are printed.
10.1 Program Text
Program Text (g02dnce.c)
10.2 Program Data
Program Data (g02dnce.d)
10.3 Program Results
Program Results (g02dnce.r)