g02jfc (void **hlmm, void *hddesc, void *hfixed, Integer nrndm, void * hrndm[], Integer n, const double y[], const double wt[], const double dat[], Integer pddat, Integer sddat, Integer *fnlsv, Integer *nff, Integer *rnlsv, Integer *nrf, Integer *nvpr, double rcomm[], Integer lrcomm, Integer icomm[], Integer licomm, NagError *fail)

The function may be called by the names: g02jfc or nag_correg_lmm_init.

3 Description

g02jfc must be called prior to fitting a linear mixed effects regression model via g02jhc.

The model is of the form:

y = X β + Z ν + ε

where	$y$ is a vector of $n$ observations on the dependent variable,
	$X$ is an $n \times p$ design matrix of fixed independent variables,
	$β$ is a vector of $p$ unknown fixed effects,
	$Z$ is an $n \times q$ design matrix of random independent variables,
	$ν$ is a vector of length $q$ of unknown random effects,
	$ε$ is a vector of length $n$ of unknown random errors.

Both

ν

and

ε

are assumed to have a Gaussian distribution with expectation zero and variance/covariance matrix defined by

Var [\begin{matrix} ν \\ ε \end{matrix}] = [\begin{matrix} G & 0 \\ 0 & R \end{matrix}]

where

R = σ_{R}^{2} I

I

is the

n \times n

identity matrix and

G

is a diagonal matrix. It is assumed that the random variables,

Z

, can be subdivided into

g \leq q

groups with each group being identically distributed with expectation zero and variance

σ_{i}^{2}

. The diagonal elements of matrix

G

, therefore, take one of the values

{σ_{i}^{2} : i = 1, 2, \dots, g}

, depending on which group the associated random variable belongs to.

The model, therefore, contains three sets of unknowns: the fixed effects

β

, the random effects

ν

and a vector of

g + 1

variance components

γ

, where

γ = {σ_{1}^{2}, σ_{2}^{2}, \dots, σ_{g - 1}^{2}, σ_{g}^{2}, σ_{R}^{2}}

Case weights can be incorporated into the model by replacing

X

and

Z

with

W_{c}^{1 / 2} X

and

W_{c}^{1 / 2} Z

respectively where

W_{c}

is a diagonal weight matrix.

The design matrices,

X

and

Z

, are constructed from an

n \times m_{d}

data matrix,

D

, a description of the fixed independent variables,

M_{f}

, and a description of the random independent variables,

M_{r}

. See Section 11 for further details.

4 References

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

5 Arguments

1: $hlmm$ – 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: $hddesc$ – void * Input

On entry: a G22 handle to the internal data structure containing a description of the data matrix,

D

, as returned in hddesc by g22ybc.

3: $hfixed$ – void * Input

On entry: a G22 handle to the internal data structure containing a description of the fixed part of the model

M_{f}

as returned in hform by g22yac.

If hfixed is NULL then the model is assumed to not have a fixed part.

4: $nrndm$ – Integer Input

On entry: the number of elements used to describe the random part of the model.

Constraint:

nrndm \geq 0

5: $hrndm [nrndm]$ – void * Input

On entry: a series of G22 handles to internal data structures containing a description of the random part of the model

M_{r}

as returned in hform by g22yac. If

nrndm = 0

, hrndm is not referenced and may be NULL.

6: $n$ – Integer Input

On entry:

n

, the number of observations in the dataset,

D

Constraint:

1 \leq n \leq n_{d}

, where

n_{d}

is the value supplied in nobs when hddesc was created.

7: $y [n]$ – const double Input

On entry:

y

, the vector of observations on the dependent variable.

Constraint:

y [i - 1] \neq 0.0

for at least one

i = 1, 2, \dots, n

8: $wt [n]$ – const double Input

On entry: optionally, the diagonal elements of the weight matrix

W_{c}

wt [i - 1] = 0.0

, the

i

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

n

Constraint: if

wt is not NULL

wt [i - 1] \geq 0.0

, for

i = 1, 2, \dots, n

9: $dat [pddat \times sddat]$ – const double Input

Note: the

(i, j)

th element of the matrix is stored in

dat [(j - 1) \times pddat + i - 1]

On entry: the data matrix,

D

. By default,

D_{i j}

, the

i

th value for the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m_{d}

, should be supplied in

dat [(j - 1) \times pddat + i - 1]

If the optional parameter

Storage Order

, described in g22ybc, is set to

VAROBS

D_{i j}

should be supplied in

dat [(i - 1) \times pddat + j - 1]

If either

y_{i}

w_{i}

D_{i j}

, for a variable

j

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: $pddat$ – Integer Input

On entry: the stride separating matrix row elements in the array dat.

Constraints:

if the optional parameter $Storage Order$ , described in g22ybc, is set to $VAROBS$ , $pddat \geq m_{d}$ ;
otherwise $pddat \geq n$ .

11: $sddat$ – Integer Input

On entry: the secondary dimension of dat.

Constraints:

if the optional parameter $Storage Order$ , described in g22ybc, is set to $VAROBS$ , $sddat \geq n$ ;
otherwise $sddat \geq m_{d}$ .

12: $fnlsv$ – Integer * Output

On exit: the number of levels for the overall subject variable in

M_{f}

. If there is no overall subject variable,

fnlsv = 1

13: $nff$ – Integer * Output

On exit: the number of fixed effects estimated in each of the fnlsv subject blocks. The number of columns,

p

, in the design matrix

X

is given by

p = nff \times fnlsv

14: $rnlsv$ – Integer * Output

On exit: the number of levels for the overall subject variable in

M_{r}

. If there is no overall subject variable,

rnlsv = 1

15: $nrf$ – Integer * Output

On exit: the number of random effects estimated in each of the rnlsv subject blocks. The number of columns,

q

, in the design matrix

Z

is given by

q = nrf \times rnlsv

16: $nvpr$ – Integer * Output

On exit:

g

, the number of variance components being estimated (excluding the overall variance,

σ_{R}^{2}

). This is defined by the number of terms in the random part of the model,

M_{r}

(see Section 11 for details).

17: $rcomm [lrcomm]$ – double Communication Array

On exit: a communication array as required by the functions g02jgc or g02jhc.

18: $lrcomm$ – Integer Input

On entry: the dimension of the array rcomm.

19: $icomm [licomm]$ – Integer Communication Array

On exit: a communication array as required by the functions g02jgc or g02jhc.

If licomm or lrcomm are too small and

licomm \geq 2

, then

fail . code =

NE_ARRAY_SIZE and

icomm [0]

holds the minimum required value for licomm and

icomm [1]

holds the minimum required value for lrcomm.

20: $licomm$ – Integer Input

On entry: the dimension of the array icomm.

21: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ARRAY_SIZE: On entry, $licomm = ⟨ value ⟩$ and $lrcomm = ⟨ value ⟩$ .
Constraint: $licomm \geq ⟨ value ⟩$ and $lrcomm \geq ⟨ value ⟩$ . icomm is not large enough to hold the minimum array sizes.

On entry, $m_{d} = ⟨ value ⟩$ and $pddat = ⟨ value ⟩$ .
Constraint: $pddat \geq m_{d}$ .

On entry, $m_{d} = ⟨ value ⟩$ and $sddat = ⟨ value ⟩$ .
Constraint: $sddat \geq m_{d}$ .

On entry, $n = ⟨ value ⟩$ and $pddat = ⟨ value ⟩$ .
Constraint: $pddat \geq n$ .

On entry, $n = ⟨ value ⟩$ and $sddat = ⟨ value ⟩$ .
Constraint: $sddat \geq n$ .
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_FIELD_UNKNOWN: A variable name used when creating hfixed is not present in hddesc.
Variable name: $⟨ value ⟩$ .

A variable name used when creating hrndm is not present in hddesc.
Variable name: $⟨ value ⟩$ .
NE_HANDLE: hddesc has not been initialized or is corrupt.

hddesc is not a G22 handle as generated by g22ybc.

hfixed has not been initialized or is corrupt.

hfixed is not a G22 handle as generated by g22yac.

$i = ⟨ value ⟩$ .
$hrndm [i - 1]$ has not been initialized or is corrupt.

$i = ⟨ value ⟩$ .
$hrndm [i - 1]$ is not a G22 handle as generated by g22yac.

On entry, hlmm is not NULL or a recognised G22 handle.
NE_INT: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

On entry, $n = ⟨ value ⟩$ and $n_{d} = ⟨ value ⟩$ .
Constraint: $n \leq n_{d}$ , where $n_{d}$ is the value supplied in nobs when hddesc was created.

On entry, no observations due to zero weights or missing values.

On entry, $nrndm = ⟨ value ⟩$ .
Constraint: $nrndm \geq 0$ .
NE_INTERNAL_ERROR: 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.
NE_NO_LICENCE: 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.
NE_REAL_ARRAY: On entry, column $j$ of the data matrix, $D$ , is not consistent with information supplied in hddesc, $j = ⟨ value ⟩$ .

On entry, $i = ⟨ value ⟩$ and $wt [i - 1] = ⟨ value ⟩$ .
Constraint: $wt [i - 1] \geq 0.0$ .
NE_ZERO_VARS: No model has been specified.
NW_ARRAY_SIZE: On entry, $licomm = ⟨ value ⟩$ and $lrcomm = ⟨ value ⟩$ .
Constraint: $licomm \geq ⟨ value ⟩$ and $lrcomm \geq ⟨ value ⟩$ . The minimum array sizes for licomm and lrcomm are held in the first two elements of icomm repectively.
NW_POTENTIAL_PROBLEM: Column $j$ of the data matrix, $D$ , required rounding more than expected when being treated as a categorical variable, $j = ⟨ value ⟩$ .
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.

7 Accuracy

Not applicable.

8 Parallelism 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.

9 Further Comments

None.

10 Example

This example fits a random effects model with three random submodels and two fixed effects to a simulated dataset with

90

observations and

12

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.

11 Algorithmic Details

11.1 Fixed Effects Design Matrix, $X$

The fixed effects design matrix,

X

, is constructed from the data matrix

D

and

M_{f}

, 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,

X^{T} X

in a block diagonal form if

M_{f}

contains an overall subject effect,

S_{f}

. In this context

S_{f}

is defined as a main effect or interaction term that is contained in all other terms. For example, if

M_{f}

simplifies to

V_{1} . V_{4} + V_{1} . V_{2} . V_{4} + V_{1} . V_{2} . V_{3} . V_{4}

, then

S_{f} = V_{1} . V_{4}

. 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

S_{f}

, otherwise

fnlsv = 1

11.2 Random Effects Design Matrix, $Z$

The random effects design matrix,

Z

, is constructed from the data matrix

D

and

M_{r}

which is made up of nrndm submodels,

M_{r i}

, where

M_{r i}

is encoded in

hrndm [i - 1]

. 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

Subject

. The design matrix

Z

is constructed as described in Section 3 in g22ycc using a model constructed from the nrndm submodels. As an example, if there were

3

submodels:

```
-1+V07+V08+V09 / SUBJECT = V13
```
```
-1+V05+V06     / SUBJECT = V11.V12
```
```
V03+V04        / SUBJECT = V10.V11.V12
```

then

Z

would be constructed as if g22ycc was called using the model

(V07+V08+V09).V13 + (V05+V06).V11.V12 + (V10.V11.V12 + (V03+V04).V10.V11.V12)

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).

The above model expands out further to:

V07.V13 + V08.V13 + V09.V13 + V05.V11.V12 + V06.V11.V12 + V10.V11.V12 + V03.V10.V11.V12 + V04.V10.V11.V12

Each term in the expanded model corresponds to a variance component, so in this case,

g = 8

When constructing

Z

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,

Z^{T} Z

in a block diagonal form if

M_{r}

contains an overall subject effect,

S_{r}

. In this context

S_{r}

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

3

submodels with subject terms

V_{1} . V_{4}

V_{1} . V_{2} . V_{4}

and

V_{1} . V_{2} . V_{3} . V_{4}

, then

S_{r} = V_{1} . V_{4}

and rnlsv will be set to the number of levels in

S_{r}

, otherwise

rnlsv = 1

12 Optional Parameters

As well as the optional parameters common to all G22 handles described in g22zmc and g22znc, 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.

12.1 Description of the Optional Parameters

For each option, we give a summary line, a description of the optional parameter and details of constraints.

The summary line contains:

a parameter value, where the letters $a$ , $i$ and $r$ denote options that take character, integer and real values respectively;
the default value.

Keywords and character values are case and white space insensitive.

Gamma Lower Bound

r

Default

= \sqrt{machine precision} / 100

A lower bound for the elements of

γ^{*}

, where

γ^{*} = γ / σ_{R}^{2}

Gamma Upper Bound

r

Default

= 10^{20}

An upper bound for the elements of

γ^{*}

, where

γ^{*} = γ / σ_{R}^{2}

Initial Distance

r

Default

= 100000.0

The initial distance from the solution.

When $Solver = E04LB$ , g02jhc passes $Initial Distance$ to the solver as stepmx.
When $Solver = E04UC$ , this option is ignored.

Initial Value Strategy

i

Default

= special

Controls how g02jhc will choose the initial values for the variance components,

γ

, if not supplied.

$Initial Value Strategy = 0$: The MIVQUE0 estimates of the variance components based on the likelihood specified by $Likelihood$ are used.
$Initial Value Strategy = 1$: The MIVQUE0 estimates based on the maximum likelihood are used, irrespective of the value of $Likelihood$ .

See Rao (1972) for a description of the minimum variance quadratic unbiased estimators (MIVQUE0).

By default, for small problems,

Initial Value Strategy = 0

and for large problems

Initial Value Strategy = 1

Constraint:

Initial Value Strategy = 0

1

Likelihood

a

Default

= REML

Likelihood

defines whether g02jhc will use the restricted maximum likelihood (REML) or the maximum likelihood (ML) when fitting the model.

Constraint:

Likelihood = REML

ML

Linear Minimization Accuracy

r

Default

= 0.9

The accuracy of the linear minimizations.

When $Solver = E04LB$ , g02jhc passes $Linear Minimization Accuracy$ to the solver as eta.
When $Solver = E04UC$ , this option is ignored.

Line Search Tolerance

r

Default

= 0.9

The line search tolerance.

When $Solver = E04LB$ , this option is ignored.
When $Solver = E04UC$ , g02jhc passes $Line Search Tolerance$ to the solver as $Line Search Tolerance$ .

List

NoList

Default

Optional parameter

List

enables printing of each optional parameter specification as it is supplied.

NoList

suppresses this printing.

Major Iteration Limit

i

Default

= special

The number of major iterations.

When $Solver = E04LB$ , g02jhc passes $Major Iteration Limit$ to the solver as maxcal. In this case, the default value used is $1000$ .
When $Solver = E04UC$ , g02jhc passes $Major Iteration Limit$ to the solver as $Major Iteration Limit$ . In this case, the default value used is $\max (50, 3 \times g)$ , where $g$ is the number of variance components being estimated (excluding the overall variance, $σ_{R}^{2}$ ).

Major Print Level

i

Default

= special

The frequency that monitoring information is output to

Unit Number

When $Solver = E04LB$ , g02jhc passes $Major Print Level$ to the solver as iprint. In this case, the default value used is $- 1$ and hence no monitoring information will be output.
When $Solver = E04UC$ , g02jhc passes $Major Print Level$ to the solver as $Major Print Level$ . In this case, the default value used is $0$ and hence no monitoring information will be output.

Maximum Number of Threads

i

Default

= special

Controls the maximum number of threads used by g02jhc in a multithreaded library. By default, the maximum number of available threads are used.

In a library that is not multithreaded, this option has no effect.

Constraint:

Maximum Number of Threads \geq 0

Minor Iteration Limit

i

Default

= \max (50, 3 \times g)

The number of minor iterations.

When $Solver = E04LB$ , this option is ignored.
When $Solver = E04UC$ , g02jhc passes $Minor Iteration Limit$ to the solver as $Minor Iteration Limit$ . In this case, the default value used is $\max (50, 3 \times g)$ , where $g$ is the number of variance components being estimated (excluding the overall variance, $σ_{R}^{2}$ ).

Minor Print Level

i

Default

= 0

The frequency that additional monitoring information is output to

Unit Number

When $Solver = E04LB$ , this option is ignored.
When $Solver = E04UC$ , g02jhc passes $Minor Print Level$ to the solver as $Minor Print Level$ . The default value of $0$ means that no additional monitoring information will be output.

Optimality Tolerance

r

Default

= {machine precision}^{0.72}

The optimality tolerance.

When $Solver = E04LB$ , this option is ignored.
When $Solver = E04UC$ , g02jhc passes $Optimality Tolerance$ to the solver as $Optimality Tolerance$ .

Parallelisation Strategy

i

Default

= special

Maximum Number of Threads > 0

then

Parallelisation Strategy

controls how g02jhc is parallelised in a multithreaded library.

$Parallelisation Strategy = 1$: g02jhc will attempt to parallelise operations involving $Z$ , even if $rnlsv = 1$ .
$Parallelisation Strategy = 2$: g02jhc will only attempt to parallelise operations involving $Z$ , if $rnlsv > 1$ .

By default,

Parallelisation Strategy = 1

, however, for some models / datasets, this may be slower than using

Parallelisation Strategy = 2

when

rnlsv = 1

In a library that is not multithreaded, this option has no effect.

Constraint:

Parallelisation Strategy = 1

2

Solution Accuracy

r

Default

= 0.0

The accuracy to which the solution is required.

When $Solver = E04LB$ , g02jhc passes $Solution Accuracy$ to the solver as xtol.
When $Solver = E04UC$ , this option is ignored.

Solver

a

Default

= special

Controls which solver g02jhc will use when fitting the model. By default,

Solver = E04LB

is used for small problems and

Solver = E04UC

, otherwise.

Solver = E04LB

, then the solver used is the one implemented in e04lbc and if

Solver = E04UC

, then the solver used is the one implemented in e04ucc.

Constraint:

Solver = E04LB

E04UC

Sweep Tolerance

r

Default

= special

The sweep tolerance used by g02jhc when performing the sweep operation Wolfinger et al. (1994). The default value used is

Sweep Tolerance = \max (ε, ε \times (\max_{i} {(Z^{T})}_{i i}))

, where

ε = \sqrt{machine precision}

Unit Number

i

Default

= Nag_FileID number associated with stdout

The monitoring Nag_FileID number (as returned from x04acc, stdout as the default) to which g02jhc will send any monitoring information.

NAG Library Manual, Mark 30.1

Interfaces: FL CL CPP AD PY MB

NAG CL Interface Introduction

G02 (Correg) Chapter Contents

G02 (Correg) Chapter Introduction

g02jf: FL CL CPP AD PY MB

NAG CL Interfaceg02jfc (lmm_​init)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

11 Algorithmic Details

11.1 Fixed Effects Design Matrix, X

11.2 Random Effects Design Matrix, Z

12 Optional Parameters

12.1 Description of the Optional Parameters

NAG CL Interface
g02jfc (lmm_init)

11.1 Fixed Effects Design Matrix, $X$

11.2 Random Effects Design Matrix, $Z$