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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_glm_constrain (g02gk)

## Purpose

nag_correg_glm_constrain (g02gk) calculates the estimates of the parameters of a generalized linear model for given constraints from the singular value decomposition results.

## Syntax

[b, se, cov, ifail] = g02gk(v, c, b, s, 'ip', ip, 'iconst', iconst)
[b, se, cov, ifail] = nag_correg_glm_constrain(v, c, b, s, 'ip', ip, 'iconst', iconst)

## Description

nag_correg_glm_constrain (g02gk) computes the estimates given a set of linear constraints for a generalized linear model which is not of full rank. It is intended for use after a call to nag_correg_glm_normal (g02ga), nag_correg_glm_binomial (g02gb), nag_correg_glm_poisson (g02gc) or nag_correg_glm_gamma (g02gd).
In the case of a model not of full rank the functions use a singular value decomposition to find the parameter estimates, β̂svd${\stackrel{^}{\beta }}_{\text{svd}}$, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix P*${P}^{*}$:
P* =
 ( D − 1 P1T ) P0T
$P*= D-1 P1T P0T$
as described by nag_correg_glm_normal (g02ga), nag_correg_glm_binomial (g02gb), nag_correg_glm_poisson (g02gc) and nag_correg_glm_gamma (g02gd). Alternative solutions can be formed by imposing constraints on the parameters. If there are p$p$ parameters and the rank of the model is k$k$ then nc = pk${n}_{\mathrm{c}}=p-k$ constraints will have to be imposed to obtain a unique solution.
Let C$C$ be a p$p$ by nc${n}_{\mathrm{c}}$ matrix of constraints, such that
 CTβ = 0, $CTβ=0,$
then the new parameter estimates β̂c${\stackrel{^}{\beta }}_{\mathrm{c}}$ are given by:
 β̂c = Aβ̂svd = (I − P0(CTP0) − 1)β̂svd,   where ​I​ is the identity matrix,
$β^c =Aβ^svd =(I-P0(CTP0)-1)β^svd, where ​I​ is the identity matrix,$
and the variance-covariance matrix is given by
 AP1D − 2 P1T AT $AP1D-2 P1T AT$
provided (CTP0)1${\left({C}^{\mathrm{T}}{P}_{0}\right)}^{-1}$ exists.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall
Searle S R (1971) Linear Models Wiley

## Parameters

### Compulsory Input Parameters

1:     v(ldv,ip + 7${\mathbf{ip}}+7$) – double array
ldv, the first dimension of the array, must satisfy the constraint ldvip$\mathit{ldv}\ge {\mathbf{ip}}$..
2:     c(ldc,iconst) – double array
ldc, the first dimension of the array, must satisfy the constraint ldcip$\mathit{ldc}\ge {\mathbf{ip}}$.
Contains the iconst constraints stored by column, i.e., the i$i$th constraint is stored in the i$i$th column of c.
3:     b(ip) – double array
ip, the dimension of the array, must satisfy the constraint ip1${\mathbf{ip}}\ge 1$.
The parameter estimates computed by using the singular value decomposition, β̂svd${\stackrel{^}{\beta }}_{\text{svd}}$.
4:     s – double scalar
The estimate of the scale parameter.
For results from nag_correg_glm_normal (g02ga) and nag_correg_glm_gamma (g02gd) then s is the scale parameter for the model.
For results from nag_correg_glm_binomial (g02gb) and nag_correg_glm_poisson (g02gc) then s should be set to 1.0$1.0$.
Constraint: s > 0.0${\mathbf{s}}>0.0$.

### Optional Input Parameters

1:     ip – int64int32nag_int scalar
Default: The dimension of the array b and the first dimension of the arrays c, v. (An error is raised if these dimensions are not equal.)
p$p$, the number of terms in the linear model.
Constraint: ip1${\mathbf{ip}}\ge 1$.
2:     iconst – int64int32nag_int scalar
Default: The second dimension of the array c.
The number of constraints to be imposed on the parameters, nc${n}_{\mathrm{c}}$.
Constraint: 0 < iconst < ip$0<{\mathbf{iconst}}<{\mathbf{ip}}$.

ldv ldc wk

### Output Parameters

1:     b(ip) – double array
The parameter estimates of the parameters with the constraints imposed, β̂c${\stackrel{^}{\beta }}_{\mathrm{c}}$.
2:     se(ip) – double array
The standard error of the parameter estimates in b.
3:     cov(ip × (ip + 1) / 2${\mathbf{ip}}×\left({\mathbf{ip}}+1\right)/2$) – 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 b(i)${\mathbf{b}}\left(i\right)$ and the parameter estimate given in b(j)${\mathbf{b}}\left(j\right)$, ji$j\ge i$, is stored in cov((j × (j1) / 2 + i))${\mathbf{cov}}\left(\left(j×\left(j-1\right)/2+i\right)\right)$.
4:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, ip < 1${\mathbf{ip}}<1$. or ${\mathbf{iconst}}\ge {\mathbf{ip}}$, or iconst ≤ 0${\mathbf{iconst}}\le 0$, or ldv < ip$\mathit{ldv}<{\mathbf{ip}}$, or ldc < ip$\mathit{ldc}<{\mathbf{ip}}$, or s ≤ 0.0${\mathbf{s}}\le 0.0$.
ifail = 2${\mathbf{ifail}}=2$
c does not give a model of full rank.

## Accuracy

It should be noted that due to rounding errors a parameter that should be zero when the constraints have been imposed may be returned as a value of order machine precision.

## Further Comments

nag_correg_glm_constrain (g02gk) is intended for use in situations in which dummy (01$0–1$) variables have been used such as in the analysis of designed experiments when you do not wish to change the parameters of the model to give a full rank model. The function is not intended for situations in which the relationships between the independent variables are only approximate.

## Example

```function nag_correg_glm_constrain_example
v = [4.890297476493481, 132.9931304975522, 0.08671323636775569, 11.53226476012202, ...
0.6875039713237874, 0.6035396163882434, 0, 0.0190106784478845, 0.008405335397184037, ...
0.008587821997748737, 0.002017521052951722, 0.006018698150011753, ...
0.002569439508216107, 0.005710268118949224, 0.003195800397035001, 0.001516472273672414;
4.150630280659896, 63.47399412403458, 0.1255168658332746, 7.96705680436851, ...
0.4385677120269487, 0.5137644803624514, 0, -0.0002104052850739208, -0.03362644601906961, ...
0.03352834001769896, -0.0001122992837032618, -8.410523978672018e-05, ...
-1.834202533241677e-05, -7.350180559025446e-05, -2.50373181392074e-05, -9.418896225321663e-06;
4.847173050021614, 127.3797841010689, 0.08860326907783028, 11.28626528578293, ...
-1.207211262022778, 0.5962906923855547, 0, -0.000593266263451917, -0.0003445349204235381, ...
-0.000333175581116977, 8.444423808860036e-05, 0.04164067761137802, ...
-0.000834602710139113, -0.03974432490123243, -0.001281738894244829, -0.0003732773692135418;
4.347583499449088, 77.29146221158076, 0.1137455037136346, 8.791556302019613, ...
0.193629026736895, 0.5316079856446033, 0, -0.005604448606553034, -0.00473076223435367, ...
-0.004660408745794929, 0.003786722373595579, 0.03079195529372083, ...
-0.01961182893501804, 0.03434336417204345, -0.04455401443460466, -0.006573924702694605;
3.660007365784039, 38.86162911664574, 0.160412976990412, 6.23390961729842, ...
0.02218333268101244, 0.4819807366090336, 0, -0.01246501079735563, 0.02562719062269757, ...
0.02538801446619446, -0.06348021588624767, -0.003268009116713865, ...
0.01218311975746848, -0.003443738617825787, -0.01974708083814387, 0.001810698017859427;
4.906081344202832, 135.1089303020151, 0.08603159451608589, 11.62363670724507, ...
-0.3553126814370362, 0.6083327470988832, 0, -0.007494797532342946, 0.006087596749368498, ...
0.006034528387724169, -0.01961692266943561, 0.009789429893167587, ...
-0.06230084141169696, 0.01026746382279108, 0.04102802603588968, -0.006278875872494353;
4.166414148369245, 64.48380766743846, 0.1245301935277972, 8.030181048235367, ...
0.1880789664600944, 0.5196429754104227, 0, -0.007856295692343652, -0.001520639263699578, ...
-0.001513684000176248, -0.004821972428467818, 0.01450100243772308, ...
0.03053649776467425, 0.01481400539841174, 0.02215701246445583, -0.08986481375760852;
4.862956917730965, 129.4062806673598, 0.08790676991659417, 11.37568814038781, ...
1.174924303092439, 0.6011714612100958, 0, 0.4644148349507297, -0.508180337140321, ...
-0.508180337140321, -0.508180337140321, 0.04376550218959127, ...
0.04376550218959111, 0.04376550218959127, 0.04376550218959124, 0.04376550218959125;
4.363367367158438, 78.52109911103648, 0.1128513646061788, 8.861213185057478, ...
-0.7464706890225739, 0.5372707561039687, 0, -0.3635174659273034, -0.05120849696428324, ...
-0.05120849696428324, -0.05120849696428316, 0.4147259628915867, ...
0.4147259628915865, 0.4147259628915868, 0.4147259628915868, 0.4147259628915868];
c = [0, 0;
1, 0;
1, 0;
1, 0;
0, 1;
0, 1;
0, 1;
0, 1;
0, 1];
b = [2.597657842414576;
1.261948923584132;
1.277732791293482;
0.05797612753696259;
1.030690710494773;
0.2910235146611871;
0.9875662840229057;
0.4879767334503795;
-0.1995994002146691];
s = 1;
[bOut, se, covar, ifail] = nag_correg_glm_constrain(v, c, b, s)
```
```

bOut =

3.9831
0.3961
0.4118
-0.8079
0.5112
-0.2285
0.4680
-0.0316
-0.7191

se =

0.0396
0.0458
0.0457
0.0622
0.0562
0.0727
0.0569
0.0675
0.0887

covar =

0.0016
-0.0006
0.0021
-0.0006
-0.0002
0.0021
0.0012
-0.0019
-0.0019
0.0039
-0.0006
0.0000
-0.0000
0.0000
0.0032
0.0002
-0.0000
0.0000
-0.0000
-0.0008
0.0053
-0.0005
0.0000
-0.0000
-0.0000
-0.0001
-0.0008
0.0032
-0.0001
0.0000
-0.0000
-0.0000
-0.0006
-0.0013
-0.0006
0.0046
0.0010
-0.0000
0.0000
0.0000
-0.0017
-0.0024
-0.0017
-0.0021
0.0079

ifail =

0

```
```function g02gk_example
v = [4.890297476493481, 132.9931304975522, 0.08671323636775569, 11.53226476012202, ...
0.6875039713237874, 0.6035396163882434, 0, 0.0190106784478845, 0.008405335397184037, ...
0.008587821997748737, 0.002017521052951722, 0.006018698150011753, ...
0.002569439508216107, 0.005710268118949224, 0.003195800397035001, 0.001516472273672414;
4.150630280659896, 63.47399412403458, 0.1255168658332746, 7.96705680436851, ...
0.4385677120269487, 0.5137644803624514, 0, -0.0002104052850739208, -0.03362644601906961, ...
0.03352834001769896, -0.0001122992837032618, -8.410523978672018e-05, ...
-1.834202533241677e-05, -7.350180559025446e-05, -2.50373181392074e-05, -9.418896225321663e-06;
4.847173050021614, 127.3797841010689, 0.08860326907783028, 11.28626528578293, ...
-1.207211262022778, 0.5962906923855547, 0, -0.000593266263451917, -0.0003445349204235381, ...
-0.000333175581116977, 8.444423808860036e-05, 0.04164067761137802, ...
-0.000834602710139113, -0.03974432490123243, -0.001281738894244829, -0.0003732773692135418;
4.347583499449088, 77.29146221158076, 0.1137455037136346, 8.791556302019613, ...
0.193629026736895, 0.5316079856446033, 0, -0.005604448606553034, -0.00473076223435367, ...
-0.004660408745794929, 0.003786722373595579, 0.03079195529372083, ...
-0.01961182893501804, 0.03434336417204345, -0.04455401443460466, -0.006573924702694605;
3.660007365784039, 38.86162911664574, 0.160412976990412, 6.23390961729842, ...
0.02218333268101244, 0.4819807366090336, 0, -0.01246501079735563, 0.02562719062269757, ...
0.02538801446619446, -0.06348021588624767, -0.003268009116713865, ...
0.01218311975746848, -0.003443738617825787, -0.01974708083814387, 0.001810698017859427;
4.906081344202832, 135.1089303020151, 0.08603159451608589, 11.62363670724507, ...
-0.3553126814370362, 0.6083327470988832, 0, -0.007494797532342946, 0.006087596749368498, ...
0.006034528387724169, -0.01961692266943561, 0.009789429893167587, ...
-0.06230084141169696, 0.01026746382279108, 0.04102802603588968, -0.006278875872494353;
4.166414148369245, 64.48380766743846, 0.1245301935277972, 8.030181048235367, ...
0.1880789664600944, 0.5196429754104227, 0, -0.007856295692343652, -0.001520639263699578, ...
-0.001513684000176248, -0.004821972428467818, 0.01450100243772308, ...
0.03053649776467425, 0.01481400539841174, 0.02215701246445583, -0.08986481375760852;
4.862956917730965, 129.4062806673598, 0.08790676991659417, 11.37568814038781, ...
1.174924303092439, 0.6011714612100958, 0, 0.4644148349507297, -0.508180337140321, ...
-0.508180337140321, -0.508180337140321, 0.04376550218959127, ...
0.04376550218959111, 0.04376550218959127, 0.04376550218959124, 0.04376550218959125;
4.363367367158438, 78.52109911103648, 0.1128513646061788, 8.861213185057478, ...
-0.7464706890225739, 0.5372707561039687, 0, -0.3635174659273034, -0.05120849696428324, ...
-0.05120849696428324, -0.05120849696428316, 0.4147259628915867, ...
0.4147259628915865, 0.4147259628915868, 0.4147259628915868, 0.4147259628915868];
c = [0, 0;
1, 0;
1, 0;
1, 0;
0, 1;
0, 1;
0, 1;
0, 1;
0, 1];
b = [2.597657842414576;
1.261948923584132;
1.277732791293482;
0.05797612753696259;
1.030690710494773;
0.2910235146611871;
0.9875662840229057;
0.4879767334503795;
-0.1995994002146691];
s = 1;
[bOut, se, covar, ifail] = g02gk(v, c, b, s)
```
```

bOut =

3.9831
0.3961
0.4118
-0.8079
0.5112
-0.2285
0.4680
-0.0316
-0.7191

se =

0.0396
0.0458
0.0457
0.0622
0.0562
0.0727
0.0569
0.0675
0.0887

covar =

0.0016
-0.0006
0.0021
-0.0006
-0.0002
0.0021
0.0012
-0.0019
-0.0019
0.0039
-0.0006
0.0000
-0.0000
0.0000
0.0032
0.0002
-0.0000
0.0000
-0.0000
-0.0008
0.0053
-0.0005
0.0000
-0.0000
-0.0000
-0.0001
-0.0008
0.0032
-0.0001
0.0000
-0.0000
-0.0000
-0.0006
-0.0013
-0.0006
0.0046
0.0010
-0.0000
0.0000
0.0000
-0.0017
-0.0024
-0.0017
-0.0021
0.0079

ifail =

0

```

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Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2013