g02jef fits a multi-level linear mixed effects regression model using maximum likelihood (ML). Prior to calling g02jef the initialization routine g02jcf must be called.
The routine may be called by the names g02jef or nagf_correg_mixeff_hier_ml.
3Description
g02jef fits a model of the form:
where
is a vector of observations on the dependent variable,
is a known design matrix for the fixed independent variables,
is a vector of length of unknown fixed effects,
is a known design matrix for the random independent variables,
is a vector of length of unknown random effects,
and
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
where , 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 . Rather than working directly with , g02jef uses an iterative process to estimate . Due to the iterative nature of the estimation a set of initial values, , for is required. g02jef allows these initial values either to be supplied by you or calculated from the data using the minimum variance quadratic unbiased estimators (MIVQUE0) suggested by Rao (1972).
g02jef fits the model by maximizing the log-likelihood function:
where
Once the final estimates for have been obtained, the value of is given by
Case weights, , can be incorporated into the model by replacing and with and respectively, for a diagonal weight matrix .
The log-likelihood, , is calculated using the sweep algorithm detailed in Wolfinger et al. (1994).
4References
Goodnight J H (1979) A tutorial on the SWEEP operator The American Statistician33(3) 149–158
Harville D A (1977) Maximum likelihood approaches to variance component estimation and to related problems JASA72 320–340
Rao C R (1972) Estimation of variance and covariance components in a linear model J. Am. Stat. Assoc.67 112–115
Stroup W W (1989) Predictable functions and prediction space in the mixed model procedure Applications of Mixed Models in Agriculture and Related DisciplinesSouthern Cooperative Series Bulletin No. 343 39–48
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
5Arguments
Note: prior to calling g02jef the initialization routine g02jcf must be called, therefore, this documention should be read in conjunction with the document for g02jcf. In particular some argument names and conventions described in that document are also relevant here, but their definition has not been repeated. Specifically, rndm,
weight, n, nff, nrf, nlsv, levels, fixed,
dat, licomm and lrcomm should be interpreted identically in both routines.
1: – IntegerInput
On entry: the sum of the number of random parameters and the random intercept flags specified in the call to g02jcf.
Constraint:
.
2: – Integer arrayInput
On entry: a vector of flags indicating the mapping between the random variables specified in rndm and the variance components, . See Section 9 for more details.
Constraint:
, for .
3: – IntegerInput
On entry: , the number of variance components being estimated (excluding the overall variance, ).
Constraint:
.
4: – Real (Kind=nag_wp) arrayInput/Output
On entry: holds the initial values of the variance components, , with
the initial value for , for .
If , the remaining elements of gamma are ignored and the initial values for the variance components are estimated from the data using MIVQUE0.
On exit: , for , holds the final estimate of and holds the final estimate for .
Constraint:
or , for .
5: – IntegerOutput
On exit: effective number of observations. If no weights were supplied to g02jcf or all supplied weights were nonzero, .
6: – IntegerOutput
On exit: the rank of the design matrix, , for the fixed effects.
7: – IntegerOutput
On exit: number of variance components not estimated to be zero. If none of the variance components are estimated to be zero, then .
8: – Real (Kind=nag_wp)Output
On exit: where is the log of the maximum likelihood calculated at , the estimated variance components returned in gamma.
9: – IntegerInput
On entry: the dimension of the arrays b and se as declared in the (sub)program from which g02jef is called.
Constraint:
.
10: – Integer arrayOutput
Note: the second dimension of the array id
must be at least
(see g02jcf).
On exit: an array describing the parameter estimates returned in b. The first columns of id describe the parameter estimates for the random effects and the last nff columns the parameter estimates for the fixed effects.
The example program for this routine includes a demonstration of decoding the parameter estimates given in b using information from id.
For fixed effects:
for
if contains the parameter estimate for the intercept, then
if contains the parameter estimate for the th level of the th fixed variable, that is the vector of values held in the th column of dat when then
if the th variable is continuous or binary, that is , ;
any remaining rows of the th column of id are set to .
For random effects:
let
denote the number of random variables in the th random statement, that is ;
denote the th random variable from the th random statement, that is the vector of values held in the th column of dat when ;
denote the number of subject variables in the th random statement, that is ;
denote the th subject variable from the th random statement, that is the vector of values held in the th column of dat when ;
denote the number of levels for , that is ;
then
for , if contains the parameter estimate for the th level of when
, for and , i.e., is a valid value for the th subject variable, then
if the parameter being estimated is for the intercept, then ;
if the th variable is continuous, or binary, that is , then ;
the remaining rows of the th column of id are set to .
In some situations, certain combinations of variables are never observed. In such circumstances all elements of the th row of id are set to .
11: – IntegerInput
On entry: the first dimension of the array id as declared in the (sub)program from which g02jef is called.
Constraint:
, i.e., maximum number of subject variables (see g02jcf).
12: – Real (Kind=nag_wp) arrayOutput
On exit: the parameter estimates, with the first elements of containing the parameter estimates for the random effects, , and the remaining nff elements containing the parameter estimates for the fixed effects, . The order of these estimates are described by the id argument.
13: – Real (Kind=nag_wp) arrayOutput
On exit: the standard errors of the parameter estimates given in b.
14: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array czz
must be at least
(see g02jcf).
On exit: if , then czz holds the lower triangular portion of the matrix , where and are the estimates of and respectively. If , then czz holds this matrix in compressed form, with the first nrf columns holding the part of the matrix corresponding to the first level of the overall subject variable, the next nrf columns the part corresponding to the second level of the overall subject variable etc.
15: – IntegerInput
On entry: the first dimension of the array czz as declared in the (sub)program from which g02jef is called.
Constraint:
.
16: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array cxx
must be at least
.
On exit: cxx holds the lower triangular portion of the matrix , where is the estimated value of .
17: – IntegerInput
On entry: the first dimension of the array cxx as declared in the (sub)program from which g02jef is called.
Constraint:
.
18: – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array cxz
must be at least
(see g02jcf).
On exit: if , then cxz holds the matrix , where and are the estimates of and respectively. If , then cxz holds this matrix in compressed form, with the first nrf columns holding the part of the matrix corresponding to the first level of the overall subject variable, the next nrf columns the part corresponding to the second level of the overall subject variable etc.
19: – IntegerInput
On entry: the first dimension of the array cxz as declared in the (sub)program from which g02jef is called.
Constraint:
.
20: – Real (Kind=nag_wp) arrayCommunication Array
On entry: communication array initialized by a call to g02jcf.
21: – Integer arrayCommunication Array
On entry: communication array initialized by a call to g02jcf.
22: – Integer arrayInput
On entry: optional parameters passed to the optimization routine.
By default g02jef fits the specified model using a modified Newton optimization algorithm as implemented in e04lbf. In some cases, where the calculation of the derivatives is computationally expensive it may be more efficient to use a sequential QP algorithm. The sequential QP algorithm as implemented in e04uca can be chosen by setting . If or , then the modified Newton algorithm will be used.
Different optional parameters are available depending on the optimization routine used. In all cases, using a value of will cause the default value to be used. In addition only the first liopt values of iopt are used, so for example, if only the first element of iopt needs changing and default values for all other optional parameters are sufficient liopt can be set to .
The following table lists the association between elements of iopt and arguments in the optimizer when the modified Newton algorithm is being used.
If requested, monitoring information is displayed in a similar format to that given by the modified Newton optimizer.
The following table lists the association between elements of iopt and optional parameters in the optimizer when the sequential QP algorithm is being used.
On entry: optional parameters passed to the optimization routine.
Different optional parameters are available depending on the optimization routine used. In all cases, using a value of will cause the default value to be used. In addition only the first lropt values of ropt are used, so for example, if only the first element of ropt needs changing and default values for all other optional parameters are sufficient lropt can be set to .
The following table lists the association between elements of ropt and arguments in the optimizer when the modified Newton algorithm is being used.
The following table lists the association between elements of ropt and optional parameters in the optimizer when the sequential QP algorithm is being used.
On entry: ifail must be set to , or to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value or is recommended. If message printing is undesirable, then the value is recommended. Otherwise, the value is recommended. When the value or is used it is essential to test the value of ifail on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: or .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, icomm has not been initialized correctly.
On entry, at least one value of i, for , does not appear in vpr.
Optimal solution found, but requested accuracy not achieved.
Too many major iterations.
Current point cannot be improved upon.
At least one negative estimate for gamma was obtained. All negative estimates have been set to zero.
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL 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 FL Interface for further information.
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
g02jef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02jef 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
The argument vpr gives the mapping between the random variables and the variance components. In most cases , for . However, in some cases it might be necessary to associate more than one random variable with a single variance component, for example, when the columns of dat hold dummy variables.
Consider a dataset with three variables:
where the first column corresponds to a categorical variable with three levels, the next to a categorical variable with two levels and the last column to a continuous variable. So in a call to g02jcf
also assume a model with no fixed effects, no random intercept, no nesting and all three variables being included as random effects, then
Each of the three columns in dat, therefore, correspond to a single variable and hence there are three variance components, one for each random variable included in the model, so
This is the recommended way of supplying the data to g02jef, however it is possible to reformat the above dataset by replacing each of the categorical variables with a series of dummy variables, one for each level. The dataset then becomes
where each column only has one level
Again a model with no fixed effects, no random intercept, no nesting and all variables being included as random effects is required, so
With the data entered in this manner, the first three columns of dat correspond to a single variable (the first column of the original dataset) as do the next two columns (the second column of the original dataset). Therefore, vpr must reflect this
In most situations it is more efficient to supply the data to g02jcf in terms of categorical variables rather than transform them into dummy variables.
9.1Internal Changes
Internal changes have been made to this routine as follows:
At Mark 27: The algorithm underlying this routine was altered to improve efficiency. The design matrices are now reordered internally before performing the analysis. It is therefore no longer necessary to construct rndm in the previously recommended manner.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.
10Example
This example fits a random effects model with three levels of nesting to a simulated dataset with observations and variables.