g02 Chapter Contents
g02 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_regsn_mult_linear_newyvar (g02dgc)

## 1  Purpose

nag_regsn_mult_linear_newyvar (g02dgc) calculates the estimates of the arguments of a general linear regression model for a new dependent variable after a call to nag_regsn_mult_linear (g02dac).

## 2  Specification

 #include #include
 void nag_regsn_mult_linear_newyvar (Integer n, const double wt[], double *rss, Integer ip, Integer rank, double cov[], double q[], Integer tdq, Nag_Boolean svd, const double p[], const double y[], double b[], double se[], double res[], const double com_ar[], NagError *fail)

## 3  Description

nag_regsn_mult_linear_newyvar (g02dgc) uses the results given by nag_regsn_mult_linear (g02dac) to fit the same set of independent variables to a new dependent variable.
nag_regsn_mult_linear (g02dac) computes a $QR$ decomposition of the matrix of $p$ independent variables and also, if the model is not of full rank, a singular value decomposition (SVD). These results can be used to compute estimates of the arguments for a general linear model with a new dependent variable. The $QR$ decomposition leads to the formation of an upper triangular $p$ by $p$ matrix $R$ and an $n$ by $n$ orthogonal matrix $Q$. In addition the vector $c={Q}^{\mathrm{T}}y$ (or ${Q}^{\mathrm{T}}{W}^{1/2}y$) is computed. For a new dependent variable, ${y}_{\mathrm{new}}$, nag_regsn_mult_linear_newyvar (g02dgc) computes a new value of $c={Q}^{\mathrm{T}}{y}_{\mathrm{new}}$ or ${Q}^{\mathrm{T}}{W}^{1/2}{y}_{\mathrm{new}}$.
If $R$ is of full rank, then the least squares parameter estimates, $\stackrel{^}{\beta }$, are the solution to: $R\stackrel{^}{\beta }={c}_{1}$, where ${c}_{1}$ is the first $p$ elements of $c$.
If $R$ is not of full rank, then nag_regsn_mult_linear (g02dac) will have computed the SVD of $R$,
 $R = Q * D 0 0 0 PT$
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
 $β ^ = P 1 D -1 Q * 1 T c 1$
${P}_{1}$ being the first $k$ columns of $P$, i.e., $P=\left({P}_{1}{P}_{0}\right)$ and ${Q}_{{*}_{1}}$ being the first $k$ columns of ${Q}_{*}$. Details of the SVD are made available by nag_regsn_mult_linear (g02dac) in the form of the matrix ${P}^{*}$:
 $P * = D -1 P1T P0T .$
The matrix ${Q}_{*}$ is made available through the com_ar argument of nag_regsn_mult_linear (g02dac).
In addition to parameter estimates, the new residuals are computed and the variance-covariance matrix of the parameter estimates are found by scaling the variance-covariance matrix for the original regression.

## 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:     nIntegerInput
On entry: the number of observations, $n$.
Constraint: ${\mathbf{n}}\ge 2$.
2:     wt[n]const doubleInput
On entry: if weighted estimates are required then wt must contain the weights to be used in the weighted regression. Otherwise wt need not be defined and may be set to the null pointer NULL, i.e., (double *)0.
If ${\mathbf{wt}}\left[i\right]=0.0$, then the $i$th observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights. The values of res and h will be set to zero for observations with zero weights. If wt is NULL, then the effective number of observations is $n$.
Constraint:  or ${\mathbf{wt}}\left[\mathit{i}\right]\ge 0.0$, for $\mathit{i}=0,1,\dots ,n-1$.
On entry: the residual sum of squares for the original dependent variable.
On exit: the residual sum of squares for the new dependent variable.
4:     ipIntegerInput
On entry: the number $p$ of independent variables in the model (including the mean if fitted).
Constraint: $1\le {\mathbf{ip}}\le {\mathbf{n}}$.
5:     rankIntegerInput
On entry: the rank of the independent variables, as given by nag_regsn_mult_linear (g02dac).
Constraint: ${\mathbf{rank}}>0$ and if ${\mathbf{svd}}=\mathrm{Nag_FALSE}$, ${\mathbf{rank}}={\mathbf{ip}}$ otherwise ${\mathbf{rank}}\le {\mathbf{ip}}$.
6:     cov[${\mathbf{ip}}×\left({\mathbf{ip}}+1\right)/2$]doubleInput/Output
On entry: the covariance matrix of the parameter estimates as given by nag_regsn_mult_linear (g02dac).
On exit: 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 ${\mathbf{b}}\left[i\right]$ and the parameter estimate given in ${\mathbf{b}}\left[j\right]$, $j\ge i$, is stored in ${\mathbf{cov}}\left[j\left(j+1\right)/2+i\right]$ for $i=0,1,\dots ,{\mathbf{ip}}-1$ and $j=i,i+1,\dots ,{\mathbf{ip}}-1$.
7:     q[${\mathbf{n}}×{\mathbf{tdq}}$]doubleInput/Output
Note: the $\left(i,j\right)$th element of the matrix $Q$ is stored in ${\mathbf{q}}\left[\left(i-1\right)×{\mathbf{tdq}}+j-1\right]$.
On entry: the results of the $QR$ decomposition as returned by nag_regsn_mult_linear (g02dac).
On exit: the first column of q contains the new values of $c$, the remainder of q will be unchanged.
8:     tdqIntegerInput
On entry: the stride separating matrix column elements in the array q.
Constraint: ${\mathbf{tdq}}\ge {\mathbf{ip}}+1$.
9:     svdNag_BooleanInput
On entry: indicates if a singular value decomposition was used by nag_regsn_mult_linear (g02dac).
${\mathbf{svd}}=\mathrm{Nag_TRUE}$
A singular value decomposition was used by nag_regsn_mult_linear (g02dac).
${\mathbf{svd}}=\mathrm{Nag_FALSE}$
A singular value decomposition was not used by nag_regsn_mult_linear (g02dac).
10:   p[$2×{\mathbf{ip}}+{\mathbf{ip}}×{\mathbf{ip}}$]const doubleInput
On entry: details of the $QR$ decomposition and SVD, if used, as returned in array p by nag_regsn_mult_linear (g02dac).
If ${\mathbf{svd}}=\mathrm{Nag_FALSE}$, only the first ip elements of p are used, these will contain details of the Householder vector in the $QR$ decomposition (Sections 2.2.1 and 3.3.6 in the f08 Chapter Introduction).
If ${\mathbf{svd}}=\mathrm{Nag_TRUE}$, the first ip elements of p will contain details of the Householder vector in the $QR$ decomposition (Sections 2.2.1 and 3.3.6 in the f08 Chapter Introduction) and the next ip elements of p contain singular values. The following ip by ip elements contain the matrix ${P}^{*}$ stored by rows.
11:   y[n]const doubleInput
On entry: the new dependent variable ${y}_{\mathrm{new}}$.
12:   b[ip]doubleOutput
On exit: ${\mathbf{b}}\left[i\right]$, $i=0,1,\dots ,{\mathbf{ip}}-1$ contain the least squares estimates of the arguments of the regression model, $\stackrel{^}{\beta }$.
13:   se[ip]doubleOutput
On exit: ${\mathbf{se}}\left[i\right]$, $i=0,1,\dots ,{\mathbf{ip}}-1$ contain the standard errors of the ip parameter estimates given in b.
14:   res[n]doubleOutput
On exit: the residuals for the new regression model.
15:   com_ar[$5×\left({\mathbf{ip}}-1\right)×{\mathbf{ip}}$]const doubleInput
On entry: if ${\mathbf{svd}}=\mathrm{Nag_TRUE}$, com_ar must be unaltered from the previous call to nag_regsn_mult_linear (g02dac).
16:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_2_INT_ARG_LT
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$ while ${\mathbf{ip}}=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{n}}\ge {\mathbf{ip}}$.
On entry, ${\mathbf{tdq}}=〈\mathit{\text{value}}〉$ while ${\mathbf{ip}}+1=〈\mathit{\text{value}}〉$. These arguments must satisfy ${\mathbf{tdq}}\ge {\mathbf{ip}}+1$.
NE_INT_ARG_LE
On entry, ${\mathbf{rank}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rank}}>0$.
NE_INT_ARG_LT
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
NE_REAL_ARG_LE
On entry, rss must not be less than or equal to 0.0: ${\mathbf{rss}}=〈\mathit{\text{value}}〉$.
NE_REAL_ARG_LT
On entry, ${\mathbf{wt}}\left[〈\mathit{\text{value}}〉\right]$ must not be less than 0.0: ${\mathbf{wt}}\left[〈\mathit{\text{value}}〉\right]=〈\mathit{\text{value}}〉$.
NE_SVD_RANK_GT_IP
On entry, the Boolean variable, svd, is Nag_TRUE and rank must not be greater than ip: rank = $〈\mathit{\text{value}}〉$, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
NE_SVD_RANK_NE_IP
On entry, the Boolean variable, svd, is Nag_FALSE and rank must be equal to ip: ${\mathbf{rank}}=〈\mathit{\text{value}}〉$, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.

## 7  Accuracy

The same accuracy as nag_regsn_mult_linear (g02dac) is obtained.

The values of the leverages, ${h}_{i}$, are unaltered by a change in the dependent variable so a call to nag_regsn_std_resid_influence (g02fac) can be made using the value of h from nag_regsn_mult_linear (g02dac).

## 9  Example

A dataset consisting of 12 observations with four independent variables and two dependent variables is read in. A model with all four independent variables is fitted to the first dependent variable by nag_regsn_mult_linear (g02dac) and the results printed. The model is then fitted to the second dependent variable by nag_regsn_mult_linear_newyvar (g02dgc) and those results printed.

### 9.1  Program Text

Program Text (g02dgce.c)

### 9.2  Program Data

Program Data (g02dgce.d)

### 9.3  Program Results

Program Results (g02dgce.r)