The function may be called by the names: g02jfc or nag_correg_lmm_init.
g02jfc must be called prior to fitting a linear mixed effects regression model via g02jhc.
The model is of the form:
is a vector of observations on the dependent variable,
is an design matrix of fixed independent variables,
is a vector of unknown fixed effects,
is an design matrix of random independent variables,
is a vector of length of unknown random effects,
is a vector of length of unknown random errors.
Both and are assumed to have a Gaussian distribution with expectation zero and variance/covariance matrix defined by
, is the identity matrix and is a diagonal matrix. It is assumed that the random variables, , can be subdivided into groups with each group being identically distributed with expectation zero and variance . The diagonal elements of matrix , therefore, take one of the values , depending on which group the associated random variable belongs to.
The model, therefore, contains three sets of unknowns: the fixed effects , the random effects and a vector of variance components , where
Case weights can be incorporated into the model by replacing and with and respectively where is a diagonal weight matrix.
The design matrices, and , are constructed from an data matrix, , a description of the fixed independent variables, , and a description of the random independent variables, . See Section 11 for further details.
Rao C R (1972) Estimation of variance and covariance components in a linear model J. Am. Stat. Assoc.67 112–115
Wolfinger R, Tobias R and Sall J (1994) Computing Gaussian likelihoods and their derivatives for general linear mixed models SIAM Sci. Statist. Comput.15 1294–1310
1: – void **Input/Output
On entry: must be set to NULL or, alternatively, an existing G22 handle may be supplied in which case g02jfc will destroy the supplied G22 handle as if g22zac had been called.
On exit: holds a G22 handle to the internal data structure containing a description of the model. You must not change the G22 handle other than through the functions in Chapters G02 or G22.
2: – void *Input
On entry: a G22 handle to the internal data structure containing a description of the data matrix, , as returned in hddesc by g22ybc.
3: – void *Input
On entry: a G22 handle to the internal data structure containing a description of the fixed part of the model as returned in hform by g22yac.
If hfixed is NULL then the model is assumed to not have a fixed part.
4: – IntegerInput
On entry: the number of elements used to describe the random part of the model.
5: – void *Input
On entry: a series of G22 handles to internal data structures containing a description of the random part of the model as returned in hform by g22yac. If , hrndm is not referenced and may be NULL.
6: – IntegerInput
On entry: , the number of observations in the dataset, .
, where is the value supplied in nobs when hddesc was created.
7: – const doubleInput
On entry: , the vector of observations on the dependent variable.
for at least one .
8: – const doubleInput
On entry: optionally, the diagonal elements of the weight matrix .
If , the th observation is not included in the model and the effective number of observations is the number of observations with nonzero weights.
If weights are not provided then wt must be set to NULL, and the effective number of observations is .
if , , for
9: – const doubleInput
Note: the th element of the matrix is stored in .
On entry: the data matrix, . By default,
, the th value for the th variable, for and , should be supplied in .
If the optional parameter , described in g22ybc, is set to , should be supplied in .
If either , or , for a variable used in the model, is NaN (Not A Number) then that value is treated as missing and the whole observation is excluded from the analysis.
10: – IntegerInput
On entry: the stride separating matrix row elements in the array dat.
if the optional parameter , described in g22ybc, is set to , ;
On entry, hlmm is not NULL or a recognised G22 handle.
On entry, .
On entry, and . Constraint: , where is the value supplied in nobs when hddesc was created.
On entry, no observations due to zero weights or missing values.
On entry, . Constraint: .
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
On entry, column of the data matrix, , is not consistent with information supplied in hddesc, .
On entry, and . Constraint: .
No model has been specified.
On entry, and . Constraint: and . The minimum array sizes for licomm and lrcomm are held in the first two elements of icomm repectively.
Column of the data matrix, , required rounding more than expected when being treated as a categorical variable, .
All output is returned using the rounded value(s).
The fixed part of the model contains categorical variables, but no intercept or main effects terms have been requested.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g02jfc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
This example fits a random effects model with three random submodels and two fixed effects to a simulated dataset with observations and variables. The model is fit using maximum likelihood (ML). Standard labels for the parameter estimates and variance components are obtained from g22ydc. See g02jhc for an example of how to construct custom labels.
The fixed effects design matrix, , is constructed from the data matrix and , as encoded in hfixed. Details of the construction are described in Section 3 in g22yac and Section 3 in g22ycc.
It is possible to store the cross-product matrix, in a block diagonal form if contains an overall subject effect, . In this context is defined as a main effect or interaction term that is contained in all other terms. For example, if simplifies to , then . If it is advantageous to do so, g02jfc will make use of this block diagonal structure and fnlsv will be set to the number of levels in , otherwise .
11.2Random Effects Design Matrix,
The random effects design matrix, , is constructed from the data matrix and which is made up of nrndm submodels, , where is encoded in . Each submodel is made up of two parts, the random effects and a subject term. The random effects are specified as described in Section 3 in g22yac and the subject term is specified via the g22yac optional parameter . The design matrix is constructed as described in Section 3 in g22ycc using a model constructed from the nrndm submodels. As an example, if there were submodels:
-1+V07+V08+V09 / SUBJECT = V13
-1+V05+V06 / SUBJECT = V11.V12
V03+V04 / SUBJECT = V10.V11.V12
then would be constructed as if g22ycc was called using the model
It should be noted that unless specified otherwise (by the inclusion of -1) a submodel will contain an intercept. This results in a term corresponding to the subject term being included in the combined model (V10.V11.V12 in this instance).
Each term in the expanded model corresponds to a variance component, so in this case, .
When constructing all contrast information specified when the submodels are constructed in calls to g22yac is ignored and dummy variables are used throughout.
It is possible to store the cross-product matrix, in a block diagonal form if contains an overall subject effect, . In this context is defined as a main effect or interaction term that is contained in all other subject terms. For example, if the random effects model is constructed from submodels with subject terms , and , then and rnlsv will be set to the number of levels in , otherwise .
As well as the optional parameters common to all G22 handles described in g22zmcandg22znc, a number of additional optional parameters can be specified for a G22 handle holding the description of a linear mixed model, as returned by g02jfc in hlmm.
Each writeable optional parameter has an associated default value; to set any of them to a non-default value, use g22zmc. The value of any optional parameter can be queried using g22znc.
Most of the optional parameters described in this section are related to the behaviour g02jhc when fitting the model. These descriptions should, therefore, be read in conjunction with the documentation for that function.
The remainder of this section can be skipped if you wish to use the default values for all optional parameters.
The following is a list of the optional parameters available. A full description of each optional parameter is provided in Section 12.1.