nag_glm_normal (Nag_Link link, Nag_IncludeMean mean, Integer n, const double x[], Integer tdx, Integer m, const Integer sx[], Integer ip, const double y[], const double wt[], const double offset[], double *scale, double ex_power, double *rss, double *df, double b[], Integer *rank, double se[], double cov[], double v[], Integer tdv, double tol, Integer max_iter, Integer print_iter, const char *outfile, double eps, NagError *fail)

3 Description

A generalized linear model with Normal errors consists of the following elements:

(a)

a set of

n

observations,

y_{i}

, from a Normal distribution with probability density function:

\frac{1}{\sqrt{2 π} σ} \exp (- \frac{{(y - μ)}^{2}}{2 σ^{2}}),

where

μ

is the mean and

σ^{2}

is the variance.

(b)

X

, a set of

p

independent variables for each observation,

x_{1}, x_{2}, \dots, x_{p}

(c)

a linear model:

η = \sum β_{j} x_{j} .

(d)

a link between the linear predictor,

η

, and the mean of the distribution,

μ

, i.e.,

η = g (μ)

. The possible link functions are:

(i)	exponent link: $η = μ^{a}$ , for a constant $a$ ,
(ii)	identity link: $η = μ$ ,
(iii)	log link: $η = \log μ$ ,
(iv)	square root link: $η = \sqrt{μ}$ ,
(v)	reciprocal link: $η = \frac{1}{μ}$ .

(e)

a measure of fit, the residual sum of squares

= \sum {(y_{i} - {\hat{μ}}_{i})}^{2}

The linear arguments are estimated by iterative weighted least squares. An adjusted dependent variable,

z

, is formed:

z = η + (y - μ) \frac{d η}{d μ}

and a working weight,

w

w = {(\frac{d η}{d μ})}^{2} .

At each iteration an approximation to the estimate of

β

\hat{β}

, is found by the weighted least squares regression of

z

X

with weights

w

nag_glm_normal (g02gac) finds a

Q R

decomposition of

w^{\frac{1}{2}} X

, i.e.,

$w^{\frac{1}{2}} X = Q R$ where $R$ is a $p$ by $p$ triangular matrix and $Q$ is a $n$ by $p$ column orthogonal matrix.

R

is of full rank, then

\hat{β}

is the solution to:

R \hat{β} = Q^{T} w^{\frac{1}{2}} z

R

is not of full rank a solution is obtained by means of a singular value decomposition (SVD) of

R

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

where D is a

k

k

diagonal matrix with nonzero diagonal elements,

k

being the rank of

R

and

w^{\frac{1}{2}} X

This gives the solution

\hat{β} = P_{1} D^{- 1} (\begin{matrix} Q_{*} & 0 \\ 0 & I \end{matrix}) Q^{T} w^{\frac{1}{2}} z

P_{1}

being the first

k

columns of

P

, i.e.,

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

The iterations are continued until there is only a small change in the residual sum of squares.

The initial values for the algorithm are obtained by taking

\hat{η} = g (y)

The fit of the model can be assessed by examining and testing the residual sum of squares, in particular comparing the difference in residual sums of squares between nested models, i.e., when one model is a sub-model of the other.

Let

{R S S}_{f}

be the residual sum of squares for the full model with degrees of freedom

ν_{f}

and let

{R S S}_{s}

be the residual sum of squares for the sub-model with degrees of freedom

ν_{s}

then:

F = \frac{({R S S}_{s} - {R S S}_{f}) / (ν_{s} - ν_{f})}{{R S S}_{f} / ν_{f}},

has, approximately, a

F

-distribution with

(ν_{s} - ν_{f})

ν_{f}

degrees of freedom.

The parameter estimates,

\hat{β}

, are asymptotically Normally distributed with variance-covariance matrix:

$C = R^{- 1} R^{- 1^{T}}$ in the full rank case, otherwise
$C = P_{1} D^{- 2} P_{1}^{T}$

The residuals and influence statistics can also be examined.

The estimated linear predictor

\hat{η} = X \hat{β}

, can be written as

{H w}^{\frac{1}{2}} z

for an

n

n

matrix

H

. The

i

th diagonal elements of

H

h_{i}

, give a measure of the influence of the

i

th values of the independent variables on the fitted regression model. These are sometimes known as leverages.

The fitted values are given by

\hat{μ} = g^{- 1} (\hat{η})

nag_glm_normal (g02gac) also computes the residuals,

r

r_{i} = y_{i} - {\hat{μ}}_{i}

An option allows prior weights,

ω_{i}

to be used, this gives a model with:

σ_{i}^{2} = \frac{σ^{2}}{ω_{i}} .

In many linear regression models the first term is taken as a mean term or an intercept, i.e.,

x_{i, 1} = 1

, for

i = 1, 2, \dots, n

; this is provided as an option.

Often only some of the possible independent variables are included in a model, the facility to select variables to be included in the model is provided.

If part of the linear predictor can be represented by a variable with a known coefficient, then this can be included in the model by using an offset,

o

η = o + \sum β_{j} x_{j} .

If the model is not of full rank the solution given will be only one of the possible solutions. Other estimates be may be obtained by applying constraints to the arguments. These solutions can be obtained by using nag_glm_tran_model (g02gkc) after using nag_glm_normal (g02gac). Only certain linear combinations of the arguments will have unique estimates; these are known as estimable functions and can be estimated and tested using nag_glm_est_func (g02gnc).

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}) .

4 References

Cook R D and Weisberg S (1982) Residuals and Influence in Regression Chapman and Hall

McCullagh P and Nelder J A (1983) Generalized Linear Models Chapman and Hall

5 Arguments

1: $link$ – Nag_LinkInput

On entry: indicates which link function is to be used.

$link = Nag_Expo$: An exponent link is used.
$link = Nag_Iden$: An identity link is used. You are advised not to use nag_glm_normal (g02gac) with an identity link as nag_regsn_mult_linear (g02dac) provides a more efficient way of fitting such a model.
$link = Nag_Log$: A log link is used.
$link = Nag_Sqrt$: A square root link is used.
$link = Nag_Reci$: A reciprocal link is used.

Constraint:

link = Nag_Expo

Nag_Iden

Nag_Log

Nag_Sqrt

Nag_Reci

2: $mean$ – Nag_IncludeMeanInput

On entry: indicates if a mean term is to be included.

$mean = Nag_MeanInclude$: A mean term, (intercept), will be included in the model.
$mean = Nag_MeanZero$: The model will pass through the origin, zero point.

Constraint:

mean = Nag_MeanInclude

Nag_MeanZero

3: $n$ – IntegerInput

On entry: the number of observations,

n

Constraint:

n \geq 2

4: $x [n \times tdx]$ – const doubleInput

On entry:

x [(i - 1) \times tdx + j - 1]

must contain the

i

th observation for the

j

th independent variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

5: $tdx$ – IntegerInput

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq m

6: $m$ – IntegerInput

On entry: the total number of independent variables.

Constraint:

m \geq 1

7: $sx [m]$ – const IntegerInput

On entry: indicates which independent variables are to be included in the model. If

sx [j - 1] > 0

, then the variable contained in the

j

th column of x is included in the regression model.

Constraints:

$sx [j - 1] \geq 0$ , for $j = 1, 2, \dots, m$ ;
if $mean = Nag_MeanInclude$ , then exactly $ip - 1$ values of sx must be $> 0$ ;
if $mean = Nag_MeanZero$ , then exactly ip values of sx must be $> 0$ .

8: $ip$ – IntegerInput

On entry: the number

p

of independent variables in the model, including the mean or intercept if present.

Constraint:

ip > 0

9: $y [n]$ – const doubleInput

On entry: observations on the dependent variable,

y_{i}

, for

i = 1, 2, \dots, n

10: $wt [n]$ – const doubleInput

On entry: if weighted estimates are required, then wt must contain the weights to be used. Otherwise wt need not be defined and may be set to NULL.

wt [i - 1] = 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 positive weights.

If wt is NULL, then the effective number of observations is

n

Constraint:

wt is NULL

wt [i - 1] \geq 0.0

, for

i = 1, 2, \dots, n

11: $offset [n]$ – const doubleInput

On entry: if an offset is required then offset must contain the values of the offset

o

. Otherwise offset must be supplied as NULL.

12: $scale$ – double *Input/Output

On entry: indicates the scale argument for the model,

σ^{2}

. If

scale = 0.0

, then the scale argument is estimated using the residual mean square.

On exit: if on input

scale = 0.0

, then scale contains the estimated value of the scale argument,

{\hat{σ}}^{2}

. If on input

scale \neq 0.0

, then scale is unchanged on exit.

Constraint:

scale \geq 0.0

13: $ex_power$ – doubleInput

On entry: if

link = Nag_Expo

then ex_power must contain the power

a

of the exponential.

link \neq Nag_Expo

, ex_power is not referenced.

Constraint: If

link = Nag_Expo

ex_power \neq 0.0

14: $rss$ – double *Output

On exit: the residual sum of squares for the fitted model.

15: $df$ – double *Output

On exit: the degrees of freedom associated with the residual sum of squares for the fitted model.

16: $b [ip]$ – doubleOutput

On exit:

b [i - 1]

i = 1, 2, \dots, ip

contains the estimates of the arguments of the generalized linear model,

\hat{β}

mean = Nag_MeanInclude

, then

b [0]

will contain the estimate of the mean argument and

b [i]

will contain the coefficient of the variable contained in column

j

of x, where

sx [j - 1]

is the

i

th positive value in the array sx.

mean = Nag_MeanZero

, then

b [i - 1]

will contain the coefficient of the variable contained in column

j

of x, where

sx [j - 1]

is the

i

th positive value in the array sx.

17: $rank$ – Integer *Output

On exit: the rank of the independent variables.

If the model is of full rank, then

rank = ip

If the model is not of full rank, then rank is an estimate of the rank of the independent variables. rank is calculated as the number of singular values greater than

eps \times

(largest singular value). It is possible for the SVD to be carried out but rank to be returned as ip.

18: $se [ip]$ – doubleOutput

On exit: the standard errors of the linear arguments.

se [i - 1]

contains the standard error of the parameter estimate in

b [i - 1]

, for

i = 1, 2, \dots, ip

19: $cov [ip \times (ip + 1) / 2]$ – doubleOutput

On exit: the

ip \times (ip + 1) / 2

elements of cov contain 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

20: $v [n \times tdv]$ – doubleOutput

On exit: auxiliary information on the fitted model.

v [(i - 1) \times tdv]

, contains the linear predictor value,

η_{i}

, for

i = 1, 2, \dots, n

v [(i - 1) \times tdv + 1]

, contains the fitted value,

{\hat{μ}}_{i}

, for

i = 1, 2, \dots, n

v [(i - 1) \times tdv + 2]

, is only included for consistency with other functions.

v [(i - 1) \times tdv + 2] = 1.0

, for

i = 1, 2, \dots, n

v [(i - 1) \times tdv + 3]

, contains the working weight,

w_{i}

, for

i = 1, 2, \dots, n

v [(i - 1) \times tdv + 4]

, contains the standardized residual,

r_{i}

, for

i = 1, 2, \dots, n

v [(i - 1) \times tdv + 5]

, contains the leverage,

h_{i}

, for

i = 1, 2, \dots, n

v [(i - 1) \times tdv + j - 1]

, for

j = 7, 8, \dots, ip + 6

, contains the results of the

Q R

decomposition or the singular value decomposition.

If the model is not of full rank, i.e.,

rank < ip

, then the first ip rows of columns

7

ip + 6

contain the

P^{*}

matrix.

21: $tdv$ – IntegerInput

On entry: the stride separating matrix column elements in the array v.

Constraint:

tdv \geq ip + 6

22: $tol$ – doubleInput

On entry: indicates the accuracy required for the fit of the model.

The iterative weighted least squares procedure is deemed to have converged if the absolute change in deviance between interactions is less than

tol \times

(1.0+current residual sum of squares). This is approximately an absolute precision if the residual sum of squares is small and a relative precision if the residual sum of squares is large.

0.0 \leq tol <

machine precision, then the function will use

10 \times

machine precision.

Constraint:

tol \geq 0.0

23: $max_iter$ – IntegerInput

On entry: the maximum number of iterations for the iterative weighted least squares. If

max_iter = 0

, then a default value of 10 is used.

Constraint:

max_iter \geq 0

24: $print_iter$ – IntegerInput

On entry: indicates if the printing of information on the iterations is required and the rate at which printing is produced. The following values are available:

$print_iter \leq 0$

There is no printing.

$print_iter > 0$

The following items are printed every print_iter iterations:

(i)	the deviance,
(ii)	the current estimates, and
(iii)	if the weighted least squares equations are singular then this is indicated.

25: $outfile$ – const char *Input

On entry: a null terminated character string giving the name of the file to which results should be printed. If outfile is NULL or an empty string then the stdout stream is used. Note that the file will be opened in the append mode.

26: $eps$ – doubleInput

On entry: the value of eps is used to decide if the independent variables are of full rank and, if not, what the rank of the independent variables is. The smaller the value of eps the stricter the criterion for selecting the singular value decomposition.

0.0 \leq eps <

machine precision, then the function will use machine precision instead.

Constraint:

eps \geq 0.0

27: $fail$ – NagError *Input/Output

The NAG error argument (see Section 2.7 in How to Use the NAG Library and its Documentation).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT: On entry, $tdv = 〈value〉$ while $ip = 〈value〉$ . These arguments must satisfy $tdv \geq ip + 6$ .

On entry, $tdx = 〈value〉$ while $m = 〈value〉$ . These arguments must satisfy $tdx \geq m$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument link had an illegal value.

On entry, argument mean had an illegal value.
NE_INT_ARG_LT: On entry, $ip = 〈value〉$ .
Constraint: $ip \geq 1$ .

On entry, $m = 〈value〉$ .
Constraint: $m \geq 1$ .

On entry, max_iter must not be less than 0: $max_iter = 〈value〉$ .

On entry, $n = 〈value〉$ .
Constraint: $n \geq 2$ .

On entry, $sx [〈value〉]$ must not be less than 0: $sx [〈value〉] = 〈value〉$ .
NE_IP_GT_OBSERV: Parameter ip is greater than the effective number of observations.
NE_IP_INCOMP_SX: Parameter ip is incompatible with arguments mean and sx.
NE_LSQ_ITER_NOT_CONV: The iterative weighted least squares has failed to converge in $max_iter = 〈value〉$ iterations. The value of max_iter could be increased but it may be advantageous to examine the convergence using the print_iter option. This may indicate that the convergence is slow because the solution is at a boundary in which case it may be better to reformulate the model.
NE_NOT_APPEND_FILE: Cannot open file $〈string〉$ for appending.
NE_NOT_CLOSE_FILE: Cannot close file $〈string〉$ .
NE_RANK_CHANGED: The rank of the model has changed during the weighted least squares iterations. The estimate for $β$ returned may be reasonable, but you should check how the deviance has changed during iterations.
NE_REAL_ARG_LT: On entry, eps must not be less than 0.0: $eps = 〈value〉$ .

On entry, scale must not be less than 0.0: $scale = 〈value〉$ .

On entry, tol must not be less than 0.0: $tol = 〈value〉$ .

On entry, $wt [〈value〉]$ must not be less than 0.0: $wt [〈value〉] = 〈value〉$ .
NE_REAL_ENUM_ARG_CONS: On entry, $ex_power = 0.0$ , $link = Nag_Expo$ . These arguments must satisfy $link = Nag_Expo$ and $ex_power \neq 0.0$ .
NE_SVD_NOT_CONV: The singular value decomposition has failed to converge.
NE_VALUE_AT_BOUNDARY_A: A fitted value is at a boundary. This will only occur with $link = Nag_Expo$ , $Nag_Log$ or $Nag_Reci$ . This may occur if there are small values of $y$ and the model is not suitable for the data. The model should be reformulated with, perhaps, some observations dropped.
NE_ZERO_DOF_ERROR: The degrees of freedom for error are 0. A saturated model has been fitted.

7 Accuracy

The accuracy is determined by tol as described in Section 5. As the residual sum of squares is a function of

μ^{2}

the accuracy of the

\hat{β}

's will depend on the link used and may be of the order

\sqrt{tol}

8 Parallelism and Performance

nag_glm_normal (g02gac) is not threaded in any implementation.

9 Further Comments

None.

10 Example

The model:

y = \frac{1}{β_{1} + β_{2} x} + ε

for a sample of five observations.

NAG Library Function Documentnag_glm_normal (g02gac)

▸▿ 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 Library Function Document

nag_glm_normal (g02gac)