g02gb Method
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g02gb fits a generalized linear model with binomial errors.

Syntax

C#
public static void g02gb(
	string link,
	string mean,
	string offset,
	string weight,
	int n,
	double[,] x,
	int m,
	int[] isx,
	int ip,
	double[] y,
	double[] t,
	double[] wt,
	out double dev,
	out int idf,
	double[] b,
	out int irank,
	double[] se,
	double[] cov,
	double[,] v,
	double tol,
	int maxit,
	int iprint,
	double eps,
	out int ifail
)
Visual Basic (Declaration)
Public Shared Sub g02gb ( _
	link As String, _
	mean As String, _
	offset As String, _
	weight As String, _
	n As Integer, _
	x As Double(,), _
	m As Integer, _
	isx As Integer(), _
	ip As Integer, _
	y As Double(), _
	t As Double(), _
	wt As Double(), _
	<OutAttribute> ByRef dev As Double, _
	<OutAttribute> ByRef idf As Integer, _
	b As Double(), _
	<OutAttribute> ByRef irank As Integer, _
	se As Double(), _
	cov As Double(), _
	v As Double(,), _
	tol As Double, _
	maxit As Integer, _
	iprint As Integer, _
	eps As Double, _
	<OutAttribute> ByRef ifail As Integer _
)
Visual C++
public:
static void g02gb(
	String^ link, 
	String^ mean, 
	String^ offset, 
	String^ weight, 
	int n, 
	array<double,2>^ x, 
	int m, 
	array<int>^ isx, 
	int ip, 
	array<double>^ y, 
	array<double>^ t, 
	array<double>^ wt, 
	[OutAttribute] double% dev, 
	[OutAttribute] int% idf, 
	array<double>^ b, 
	[OutAttribute] int% irank, 
	array<double>^ se, 
	array<double>^ cov, 
	array<double,2>^ v, 
	double tol, 
	int maxit, 
	int iprint, 
	double eps, 
	[OutAttribute] int% ifail
)
F#
static member g02gb : 
        link:string * 
        mean:string * 
        offset:string * 
        weight:string * 
        n:int * 
        x:float[,] * 
        m:int * 
        isx:int[] * 
        ip:int * 
        y:float[] * 
        t:float[] * 
        wt:float[] * 
        dev:float byref * 
        idf:int byref * 
        b:float[] * 
        irank:int byref * 
        se:float[] * 
        cov:float[] * 
        v:float[,] * 
        tol:float * 
        maxit:int * 
        iprint:int * 
        eps:float * 
        ifail:int byref -> unit 

Parameters

mean
Type: System..::.String
On entry: indicates if a mean term is to be included.
mean="M"
A mean term, intercept, will be included in the model.
mean="Z"
The model will pass through the origin, zero-point.
Constraint: mean="M" or "Z".
offset
Type: System..::.String
On entry: indicates if an offset is required.
offset="Y"
An offset is required and the offsets must be supplied in the seventh column of v.
offset="N"
No offset is required.
Constraint: offset="N" or "Y".
weight
Type: System..::.String
On entry: indicates if prior weights are to be used.
weight="U"
No prior weights are used.
weight="W"
Prior weights are used and weights must be supplied in wt.
Constraint: weight="U" or "W".
n
Type: System..::.Int32
On entry: n, the number of observations.
Constraint: n2.
x
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldx, m]
Note: ldx must satisfy the constraint: ldxn
On entry: x[i-1,j-1] must contain the ith observation for the jth independent variable, for i=1,2,,n and j=1,2,,m.
m
Type: System..::.Int32
On entry: m, the total number of independent variables.
Constraint: m1.
isx
Type: array< System..::.Int32 >[]()[]
An array of size [m]
On entry: indicates which independent variables are to be included in the model.
isx[j-1]>0
The variable contained in the jth column of x is included in the regression model.
Constraints:
  • isx[j]0, for j=0,1,,m-1;
  • if mean="M", exactly ip-1 values of isx must be >0;
  • if mean="Z", exactly ip values of isx must be >0.
ip
Type: System..::.Int32
On entry: the number of independent variables in the model, including the mean or intercept if present.
Constraint: ip>0.
y
Type: array< System..::.Double >[]()[]
An array of size [n]
On entry: the observations on the dependent variable, yi, for i=1,2,,n.
Constraint: 0.0y[i]t[i], for i=0,1,,n-1.
t
Type: array< System..::.Double >[]()[]
An array of size [n]
On entry: t, the binomial denominator.
Constraint: t[i]0.0, for i=0,1,,n-1.
wt
Type: array< System..::.Double >[]()[]
An array of size [dim1]
Note: the dimension of the array wt must be at least n if weight="W", and at least 1 otherwise.
On entry: if weight="W", wt must contain the weights to be used in the weighted regression. If wt[i-1]=0.0, the ith observation is not included in the model, in which case the effective number of observations is the number of observations with nonzero weights.
If weight="U", wt is not referenced and the effective number of observations is n.
Constraint: if weight="W", wt[i]0.0 for i=0,1,,n-1.
dev
Type: System..::.Double %
On exit: the deviance for the fitted model.
idf
Type: System..::.Int32 %
On exit: the degrees of freedom associated with the deviance for the fitted model.
b
Type: array< System..::.Double >[]()[]
An array of size [ip]
On exit: the estimates of the parameters of the generalized linear model, β^.
If mean="M", the first element of b will contain the estimate of the mean parameter and b[i] will contain the coefficient of the variable contained in column j of x, where isx[j-1] is the ith positive value in the array isx.
If mean="Z", b[i-1] will contain the coefficient of the variable contained in column j of x, where isx[j-1] is the ith positive value in the array isx.
irank
Type: System..::.Int32 %
On exit: the rank of the independent variables.
If the model is of full rank, irank=ip.
If the model is not of full rank, irank is an estimate of the rank of the independent variables. irank is calculated as the number of singular values greater that eps×(largest singular value).
It is possible for the SVD to be carried out but for irank to be returned as ip.
se
Type: array< System..::.Double >[]()[]
An array of size [ip]
On exit: the standard errors of the linear parameters.
se[i-1] contains the standard error of the parameter estimate in b[i-1], for i=1,2,,ip.
cov
Type: array< System..::.Double >[]()[]
An array of size [ip×ip+1/2]
On exit: the upper triangular part of the variance-covariance matrix of the ip parameter estimates given in b. They are stored in packed form by column, i.e., the covariance between the parameter estimate given in b[i-1] and the parameter estimate given in b[j-1], ji, is stored in cov[j×j-1/2+i-1].
v
Type: array< System..::.Double ,2>[,](,)[,]
An array of size [ldv, ip+7]
Note: ldv must satisfy the constraint: ldvn
On entry: if offset="N", v need not be set.
If offset="Y", v[i-1,6], for i=1,2,,n must contain the offset values oi. All other values need not be set.
On exit: auxiliary information on the fitted model.
v[i-1,0] contains the linear predictor value, ηi, for i=1,2,,n.
v[i-1,1] contains the fitted value, μ^i, for i=1,2,,n.
v[i-1,2] contains the variance standardization, 1τi , for i=1,2,,n.
v[i-1,3] contains the square root of the working weight, wi12, for i=1,2,,n.
v[i-1,4] contains the deviance residual, ri, for i=1,2,,n.
v[i-1,5] contains the leverage, hi, for i=1,2,,n.
v[i-1,6] contains the offset, oi, for i=1,2,,n. If offset="N", all values will be zero.
v[i-1,j-1] for j=8,,ip+7, contains the results of the QR decomposition or the singular value decomposition.
If the model is not of full rank, i.e., irank<ip, the first ip rows of columns 8 to ip+7 contain the P* matrix.
tol
Type: System..::.Double
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 iterations is less than tol×1.0+Current Deviance. This is approximately an absolute precision if the deviance is small and a relative precision if the deviance is large.
If 0.0tol<machine precision, the method will use 10×machine precision instead.
Constraint: tol0.0.
maxit
Type: System..::.Int32
On entry: the maximum number of iterations for the iterative weighted least-squares.
If maxit=0, a default value of 10 is used.
Constraint: maxit0.
iprint
Type: System..::.Int32
On entry: indicates if the printing of information on the iterations is required.
iprint0
There is no printing.
iprint>0
The following is printed every iprint iterations:
the deviance,
the current estimates,
and if the weighted least-squares equations are singular, then this is indicated.
eps
Type: System..::.Double
On entry: the value of eps is used to decide if the independent variables are of full rank and, if not, what is the rank of the independent variables. The smaller the value of eps the stricter the criterion for selecting the singular value decomposition.
If 0.0eps<machine precision, the method will use machine precision instead.
Constraint: eps0.0.
ifail
Type: System..::.Int32 %
On exit: ifail=0 unless the method detects an error (see [Error Indicators and Warnings]).

Description

A generalized linear model with binomial errors consists of the following elements:
(a) a set of n observations, yi, from a binomial distribution:
t y πy1-πt-y.
(b) X, a set of p independent variables for each observation, x1,x2,,xp.
(c) a linear model:
η=βjxj.
(d) a link between the linear predictor, η, and the mean of the distribution, μ=πt, the link function, η=gμ. The possible link functions are:
(i) logistic link: η=logμ t-μ ,
(ii) probit link: η=Φ-1 μt ,
(iii) complementary log-log link: log-log1-μt .
(e) a measure of fit, the deviance:
i=1ndevyi,μ^i=i=1n2 yilogyiμ^i+ti-yilog ti-yi ti-μ^i .
The linear parameters are estimated by iterative weighted least-squares. An adjusted dependent variable, z, is formed:
z=η+y-μdη dμ
and a working weight, w,
w= τdη dμ 2,   where ​τ=tμt-μ .
At each iteration an approximation to the estimate of β, β^, is found by the weighted least-squares regression of z on X with weights w.
g02gb finds a QR decomposition of w1/2X, i.e., w1/2X=QR where R is a p by p triangular matrix and Q is an n by p column orthogonal matrix.
If R is of full rank, then β^ is the solution to
Rβ^=QTw1/2z.
If R is not of full rank a solution is obtained by means of a singular value decomposition (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 w1/2X.
This gives the solution
β^=P1D-1 Q* 0 0 I QTw1/2z,
P1 being the first k columns of P, i.e., P=P1P0.
The iterations are continued until there is only a small change in the deviance.
The initial values for the algorithm are obtained by taking
η^=gy.
The fit of the model can be assessed by examining and testing the deviance, in particular by comparing the difference in deviance between nested models, i.e., when one model is a sub-model of the other. The difference in deviance between two nested models has, asymptotically, a χ2-distribution with degrees of freedom given by the difference in the degrees of freedom associated with the two deviances.
The parameters estimates, β^, are asymptotically Normally distributed with variance-covariance matrix
  • C = R-1 R-1T  in the full rank case, otherwise
  • C=P1D-2 P1T .
The residuals and influence statistics can also be examined.
The estimated linear predictor η^=Xβ^, can be written as Hw1/2z for an n by n matrix H. The ith diagonal elements of H, hi, give a measure of the influence of the ith values of the independent variables on the fitted regression model. These are sometimes known as leverages.
The fitted values are given by μ^=g-1η^.
g02gb also computes the deviance residuals, r:
ri=signyi-μ^idevyi,μ^i.
An option allows the use of prior weights in the model.
In many linear regression models the first term is taken as a mean term or an intercept, i.e., xi,1=1, for i=1,2,,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 variables with a known coefficient then this can be included in the model by using an offset, o:
η=o+βjxj.
If the model is not of full rank the solution given will be only one of the possible solutions. Other estimates may be obtained by applying constraints to the parameters. These solutions can be obtained by using g02gk after using g02gb. Only certain linear combinations of the parameters will have unique estimates, these are known as estimable functions and can be estimated and tested using g02gn.
Details of the SVD are made available in the form of the matrix P*:
P*= D-1 P1T P0T .

References

Error Indicators and Warnings

Note: g02gb may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (ldx, ldv, wk) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
ifail=1
On entry,n<2,
orm<1,
orip<1,
orlink"G", "P" or "C".
ormean"M" or "Z".
orweight"U" or "W".
oroffset"N" or "Y".
ormaxit<0,
ortol<0.0,
oreps<0.0.
ifail=2
On entry,weight="W" and a value of wt<0.0.
ifail=3
On entry,a value of isx<0,
orthe value of ip is incompatible with the values of mean and isx,
orip is greater than the effective number of observations.
ifail=4
On entry,t[i-1]<0 for some i=1,2,,n.
ifail=5
On entry,y[i-1]<0.0,
ory[i-1]>t[i-1] for some i=1,2,,n.
ifail=6
A fitted value is at the boundary, i.e., 0.0 or 1.0. This may occur if there are y values of 0.0 or t and the model is too complex for the data. The model should be reformulated with, perhaps, some observations dropped.
ifail=7
The singular value decomposition has failed to converge. This is an unlikely error exit.
ifail=8
The iterative weighted least-squares has failed to converge in maxit (or default 10) iterations. The value of maxit could be increased but it may be advantageous to examine the convergence using the iprint 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.
ifail=9
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.
ifail=10
The degrees of freedom for error are 0. A saturated model has been fitted.
ifail=-4000
ifail=-8000
ifail=-6000

Accuracy

The accuracy will depend on the value of tol as described in [Parameters]. As the deviance is a function of logμ the accuracy of the β^ will be only a function of tol, so tol should be set smaller than the required accuracy for β^.

Further Comments

Example

See Also