NAG Library Function Document
nag_regsn_mult_linear_est_func (g02dnc) gives the estimate of an estimable function along with its standard error.
||nag_regsn_mult_linear_est_func (Integer ip,
const double b,
const double cov,
const double p,
const double f,
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:
diagonal matrix with nonzero diagonal elements,
being the rank of
orthogonal matrices. This leads to a solution:
being the first
being the first
being the first
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 .
is estimable it can be estimated by
and its standard error calculated from the variance-covariance matrix of
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.
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
ip – IntegerInput
On entry: the number of terms in the linear model, .
rank – IntegerInput
On entry: the rank of the independent variables, .
b[ip] – const doubleInput
: the ip
values of the estimates of the arguments of the model,
cov – const doubleInput
: 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
p – const doubleInput
f[ip] – const doubleInput
On entry: the linear function to be estimated, .
est – Nag_Boolean *Output
indicates if the function was estimable.
- The function is estimable.
- The function is not estimable and stat, sestat and t are not set.
stat – double *Output
contains the estimate of the function,
sestat – double *Output
contains the standard error of the estimate of the function,
t – double *Output
-statistic for the test of the function being equal to zero.
tol – doubleInput
is the tolerance value used in the check for estimability,
is used instead.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
On entry, while . These arguments must satisfy .
Dynamic memory allocation failed.
On entry, .
On entry, .
. In this case, the boolean variable est
is returned as Nag_TRUE and all statistics are calculated.
probably due to rounding error or due to incorrectly specified inputs cov
The computations are believed to be stable.
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.
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.
9.1 Program Text
Program Text (g02dnce.c)
9.2 Program Data
Program Data (g02dnce.d)
9.3 Program Results
Program Results (g02dnce.r)