Fits a generalized linear model with Normal errors

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NAG[g02gac] NAG[nag_glm_normal] - Fits a generalized linear model with Normal errors

Calling Sequence

g02gac(link, mean, x, sx, y, scale, ex_power, rss, df, b, rank, se, cov, v, tol, max_iter, print_iter, eps, 'n'=n, 'tdx'=tdx, 'm'=m, 'ip'=ip, 'wt'=wt, 'offset'=offset, 'tdv'=tdv, 'outfile'=outfile, 'fail'=fail)

nag_glm_normal(. . .)

Parameters

link - String;

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

An exponent link is used.

An identity link is used. The user is advised not to use nag_glm_normal (g02gac) with an identity link as g02dac (nag_regsn_mult_linear) provides a more efficient way of fitting such a model.

A log link is used.

A square root link is used.

A reciprocal link is used.

Constraint: "Nag_Expo", "Nag_Iden", "Nag_Log", "Nag_Sqrt" or "Nag_Reci". .

mean - String;

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

A mean term, (intercept), will be included in the model.

The model will pass through the origin, zero point.

Constraint: "Nag_MeanInclude" or "Nag_MeanZero". .

x - Matrix(1..n, 1..tdx, datatype=float[8], order=C_order);

On entry: must contain the th observation for the th independent variable, for ; .

sx - Vector(1..m, datatype=integer[kernelopts('wordsize')/8]);

On entry: indicates which independent variables are to be included in the model. If , then the variable contained in the th column of x is included in the regression model.

Constraints:

, for ;

if , then exactly values of sx must be ;

if , then exactly ip values of sx must be .

y - Vector(1..n, datatype=float[8]);

On entry: observations on the dependent variable, , for .

scale - assignable;

Note: On exit the variable scale will have a value of type float.

On entry: indicates the scale argument for the model, . If , then the scale argument is estimated using the residual mean square.

On exit: if on input , then scale contains the estimated value of the scale argument, . If on input , then scale is unchanged on exit.

Constraint: . .

ex_power - float;

On entry: if then ex_power must contain the power of the exponential.

If , ex_power is not referenced.

Constraint: If , . .

rss - assignable;

Note: On exit the variable rss will have a value of type float.

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

df - assignable;

Note: On exit the variable df will have a value of type float.

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

b - Vector(1..ip, datatype=float[8]);

On exit: , contains the estimates of the arguments of the generalized linear model, .

If , then will contain the estimate of the mean argument and will contain the coefficient of the variable contained in column of x, where is the th positive value in the array sx.

If , then will contain the coefficient of the variable contained in column of x, where is the th positive value in the array sx.

rank - assignable;

Note: On exit the variable rank will have a value of type integer.

On exit: the rank of the independent variables.

If the model is of full rank, then .

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

se - Vector(1..ip, datatype=float[8]);

On exit: the standard errors of the linear arguments.

contains the standard error of the argument estimate in , for .

cov - Vector(1.., datatype=float[8]);

Note: the dimension, dim, of the array cov must be at least .

On exit: the elements of cov contain the upper triangular part of the variance-covariance matrix of the ip argument estimates given in b. They are stored packed by column, i.e., the covariance between the argument estimate given in and the argument estimate given in , , is stored in , for and .

v - Matrix(1..n, 1..tdv, datatype=float[8], order=C_order);

On exit: auxiliary information on the fitted model.

, contains the linear predictor value, , for .

, contains the fitted value, , for .

, is only included for consistency with other functions. , for .

, contains the working weight, , for .

, contains the standardized residual, , for .

, contains the leverage, , for .

, for , contains the results of the decomposition or the singular value decomposition.

If the model is not of full rank, i.e., , then the first ip rows of columns to contain the matrix.

tol - float;

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 (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.

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

Constraint: . .

max_iter - integer;

On entry: the maximum number of iterations for the iterative weighted least-squares. If , then a default value of 10 is used.

Constraint: . .

print_iter - integer;

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:

There is no printing.

The following items are printed every print_iter iterations:

the deviance,

the current estimates, and

if the weighted least-squares equations are singular then this is indicated.

eps - float;

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.

If machine precision, then the function will use machine precisioninstead.

Constraint: . .

'n'=n - integer; (optional)

Default value: the first dimension of the arrays x, y, wt, offset, v.

On entry: the number of observations, .

Constraint: . .

'tdx'=tdx - integer; (optional)

On entry: the second dimension of the array x as declared in the function from which nag_glm_normal (g02gac) is called.

Constraint: . .

'm'=m - integer; (optional)

Default value: the first dimension of the array sx and the second dimension of the array sxthe array x.

On entry: the total number of independent variables.

Constraint: . .

'ip'=ip - integer; (optional)

Default value: the first dimension of the arrays b, se and the second dimension of the arrays b, sethe array v.

On entry: the number of independent variables in the model, including the mean or intercept if present.

Constraint: . .

'wt'=wt - Vector(1..n, datatype=float[8]); (optional)

On entry: if weighted estimates are required then wt must contain the weights to be used with the model, . Otherwise wt must be supplied as the null pointer, (double *)0.

If , then the 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 null pointer, then the effective number of observations is .

Constraint: null pointer or , for . .

'offset'=offset - Vector(1..n, datatype=float[8]); (optional)

On entry: if an offset is required then offset must contain the values of the offset . Otherwise offset must be supplied as the null pointer, (double *)0.

'tdv'=tdv - integer; (optional)

On entry: the second dimension of the array v as declared in the function from which nag_glm_normal (g02gac) is called.

Constraint: . .

'outfile'=outfile - character; (optional)

On entry: The name of a file to which intermediate or diagnostic output should be appended. If a value is not provided for this parameter then the behaviour of this routine is platform dependent. Usually all output will be suppressed, however on some platforms output will be produced and will be displayed in the Maple session.

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_glm_normal (g02gac) fits a generalized linear model with normal errors.

Description

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

a. a set of observations, , from a Normal distribution with probability density function:

1/(sqrt(2Pi)sigma)exp(-,((y,-,mu)^2)/(2sigma^2))

where is the mean and is the variance.

b. , a set of independent variables for each observation, .

c. a linear model:

a link between the linear predictor, , and the mean of the distribution, , i.e., . The possible link functions are:

exponent link: , for a constant ,

identity link: ,

log link: ,

square root link: ,

reciprocal link: .

e. a measure of fit, the residual sum of squares

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

and a working weight, ,

w = (d*eta*(1/(d*mu)))^2

At each iteration an approximation to the estimate of , , is found by the weighted least-squares regression of on with weights .

nag_glm_normal (g02gac) finds a decomposition of , i.e.,

where is a by triangular matrix and is a by column orthogonal matrix.

If is of full rank, then is the solution to:

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

R=Q[*]([[D,0],[0,0]])P^T

where D is a by diagonal matrix with non-zero diagonal elements, being the rank of and .

This gives the solution

beta=P[1]D^(-1)([[Q[*],0],[0,I]])Q^Tw^(1/2)z

being the first columns of , i.e., .

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

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 be the residual sum of squares for the full model with degrees of freedom and let be the residual sum of squares for the sub-model with degrees of freedom then:

F=(((RSS)[s],-,(RSS)[f])/(nu[s],-,nu[f]))/((RSS)[f]/nu[f])

has, approximately, a -distribution with , degrees of freedom.

The argument estimates, , are asymptotically Normally distributed with variance-covariance matrix:

in the full rank case, otherwise

The residuals and influence statistics can also be examined.

The estimated linear predictor , can be written as for an by matrix . The th diagonal elements of , , give a measure of the influence of the th values of the independent variables on the fitted regression model. These are sometimes known as leverages.

The fitted values are given by .

nag_glm_normal (g02gac) also computes the residuals, :

An option allows prior weights, to be used, this gives a model with:

sigma[i]^2 = sigma^2/omega[i]

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

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 g02gkc (nag_glm_tran_model) 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 g02gnc (nag_glm_est_func).

Details of the SVD, are made available, in the form of the matrix :

P^(*)=([[D^(-1)P[1]^T],[P[0]^T]])

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument link had an illegal value.

"NE_INT_ARG_LT"

On entry, n must not be less than 2: .

"NE_IP_GT_OBSERV"

Argument ip is greater than the effective number of observations.

"NE_IP_INCOMP_SX"

Argument ip is incompatible with arguments mean and sx.

"NE_LSQ_ITER_NOT_CONV"

The iterative weighted least-squares has failed to converge in 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 for appending.

"NE_NOT_CLOSE_FILE"

Cannot close file .

"NE_RANK_CHANGED"

The rank of the model has changed during the weighted least-squares iterations. The estimate for returned may be reasonable, but the user should check how the deviance has changed during iterations.

"NE_REAL_ARG_LT"

On entry, scale must not be less than 0.0: .

"NE_REAL_ENUM_ARG_CONS"

On entry, , . These arguments must satisfy and .

"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 "Nag_Expo", "Nag_Log" or "Nag_Reci". This may occur if there are small values of 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.

	Accuracy
	The accuracy is determined by tol as described in Section [Parameters]. As the residual sum of squares is a function of the accuracy of the 's will depend on the link used and may be of the order .

	Further Comments
	None.

Examples

link := "Nag_Reci":
mean := "Nag_MeanInclude":
n := 5:
tdx := 1:
m := 1:
ip := 2:
scale := 0:
ex_power := 0:
tdv := 8:
tol := 5e-05:
max_iter := 10:
print_iter := 0:
eps := 1e-06:
x := Matrix([[1], [2], [3], [4], [5]], datatype=float[8], order='C_order'):
sx := Vector([1], datatype=integer[kernelopts('wordsize')/8]):
y := Vector([25, 10, 6, 4, 3], datatype=float[8]):
wt := Vector([], datatype=float[8]):
offset := Vector([], datatype=float[8]):
b := Vector(2, datatype=float[8]):
se := Vector(2, datatype=float[8]):
cov := Vector(3, datatype=float[8]):
v := Matrix(5, 8, datatype=float[8], order='C_order'):
NAG:-g02gac(link, mean, x, sx, y, scale, ex_power, rss, df, b, rank, se, cov, v, tol, max_iter, print_iter, eps, 'n' = n, 'tdx' = tdx, 'm' = m, 'ip' = ip, 'wt' = wt, 'offset' = offset, 'tdv' = tdv):

scale;

rss;

df;

rank;

se;

cov;

	See Also
	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 g02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.