g02gn gives the estimate of an estimable function along with its standard error from the results from fitting a generalized linear model.
Public Shared Sub g02gn ( _ ip As Integer, _ irank As Integer, _ b As Double(), _ cov As Double(), _ v As Double(,), _ f As Double(), _ <OutAttribute> ByRef est As Boolean, _ <OutAttribute> ByRef stat As Double, _ <OutAttribute> ByRef sestat As Double, _ <OutAttribute> ByRef z As Double, _ tol As Double, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g02gn( int ip, int irank, array<double>^ b, array<double>^ cov, array<double,2>^ v, array<double>^ f, [OutAttribute] bool% est, [OutAttribute] double% stat, [OutAttribute] double% sestat, [OutAttribute] double% z, double tol, [OutAttribute] int% ifail )
- Type: System..::..Int32On entry: , the number of terms in the linear model.Constraint: .
- Type: System..::..Int32On entry: , the rank of the dependent variables.Constraint: .
- Type: array<System..::..Double>()An array of size [ip]On entry: the ip values of the estimates of the parameters of the model, .
- Type: array<System..::..Double>()An array of size 
- Type: array<System..::..Double,2>[,](,)[,][,]An array of size [dim1, ]Note: dim1 must satisfy the constraint:
- Type: array<System..::..Double>()An array of size [ip]On entry: , the linear function to be estimated.
- Type: System..::..Boolean%
- Type: System..::..Double%On exit: if , sestat contains the standard error of the estimate of the function, .
- Type: System..::..Double%On exit: if , z contains the statistic for the test of the function being equal to zero.
- Type: System..::..DoubleOn entry: the tolerance value used in the check for estimability, .If then , where is the machine precision, is used instead.
g02gn computes the estimates of an estimable function for a generalized linear model which is not of full rank. It is intended for use after a call to g02ga, g02gb, g02gc or g02gd. An estimable function is a linear combination of the parameters such that it has a unique estimate. For a full rank model all linear combinations of parameters are estimable.
In the case of a model not of full rank the methods use a singular value decomposition (SVD) to find the parameter estimates, , and their variance-covariance matrix. Given the upper triangular matrix obtained from the decomposition of the independent variables the SVD gives
where is a by diagonal matrix with nonzero diagonal elements, being the rank of , and and are by orthogonal matrices. This leads to a solution:
being the first columns of , i.e., ; being the first columns of , and being the first elements of .
Details of the SVD are made available in the form of the matrix :
as described by g02ga, g02gb, g02gc and g02gd.
A linear function of the parameters, , can be tested to see if it is estimable by computing . If is zero, then the function is estimable, if not; the function is not estimable. In practice is tested against some small quantity .
Given that is estimable it can be estimated by and its standard error calculated from the variance-covariance matrix of , , as
Also a statistic
can be computed. The distribution of will be approximately Normal.
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
Note: g02gn 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 (LDV) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
On entry, , or , or , On entry, . In this case est is returned as true and all statistics are calculated.
The computations are believed to be stable.
The value of estimable functions is independent of the solution chosen from the many possible solutions. While g02gn may be used to estimate functions of the parameters of the model as computed by g02gk, , these must be expressed in terms of the original parameters, . The relation between the two sets of parameters may not be straightforward.