Integer, Intent (In)	::	n, ncol, lddat, levels(ncol), fixed(lfixed), lfixed, nrndm, rndm(ldrndm,*), ldrndm, lrcomm, licomm
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	nff, nlsv, nrf, icomm(licomm)
Real (Kind=nag_wp), Intent (In)	::	dat(lddat,), y(n), wt()
Real (Kind=nag_wp), Intent (Out)	::	rcomm(lrcomm)
Character (1), Intent (In)	::	weight

C Header Interface

#include <nag.h>

void

g02jcf_ (const char *weight, const Integer *n, const Integer *ncol, const double dat[], const Integer *lddat, const Integer levels[], const double y[], const double wt[], const Integer fixed[], const Integer *lfixed, const Integer *nrndm, const Integer rndm[], const Integer *ldrndm, Integer *nff, Integer *nlsv, Integer *nrf, double rcomm[], const Integer *lrcomm, Integer icomm[], const Integer *licomm, Integer *ifail, const Charlen length_weight)

The routine may be called by the names g02jcf or nagf_correg_mixeff_hier_init.

3 Description

g02jcf must be called prior to fitting a linear mixed effects regression model with either g02jdf or g02jef.

The model fitting routines g02jdf and g02jef fit a model of the following 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,

and

ν

and

ε

are Normally distributed 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.

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.

4 References

None.

5 Arguments

1: $weight$ – Character(1) Input

On entry: indicates if weights are to be used.

$weight ='U'$: No weights are used.
$weight ='W'$: Case weights are used and must be supplied in array wt.

Constraint:

weight ='U'

'W'

2: $n$ – Integer Input

On entry:

n

, the number of observations.

The effective number of observations, that is the number of observations with nonzero weight (see wt for more detail), must be greater than the number of fixed effects in the model (as returned in nff).

Constraint:

n \geq 1

3: $ncol$ – Integer Input

On entry: the number of columns in the data matrix, dat.

Constraint:

ncol \geq 0

4: $dat (lddat, *)$ – Real (Kind=nag_wp) array Input

Note: the second dimension of the array dat must be at least

ncol

On entry: a matrix of data, with

dat (i, j)

holding the

i

th observation on the

j

th variable. The two design matrices

X

and

Z

are constructed from dat and the information given in fixed (for

X

) and rndm (for

Z

Constraint: if

levels (j) \neq 1, 1 \leq dat (i, j) \leq levels (j)

5: $lddat$ – Integer Input

On entry: the first dimension of the array dat as declared in the (sub)program from which g02jcf is called.

Constraint:

lddat \geq n

6: $levels (ncol)$ – Integer array Input

On entry:

levels (i)

contains the number of levels associated with the

i

th variable held in dat.

If the

i

th variable is continuous or binary (i.e., only takes the values zero or one), then

levels (i)

must be set to

1

. Otherwise the

i

th variable is assumed to take an integer value between

1

and

levels (i)

, (i.e., the

i

th variable is discrete with

levels (i)

levels).

Constraint:

levels (i) \geq 1

, for

i = 1, 2, \dots, ncol

7: $y (n)$ – Real (Kind=nag_wp) array Input

On entry:

y

, the vector of observations on the dependent variable.

8: $wt (*)$ – Real (Kind=nag_wp) array Input

Note: the dimension of the array wt must be at least

n

weight ='W'

On entry: if

weight ='W'

, wt must contain the diagonal elements of the weight matrix

W_{c}

wt (i) = 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.

weight ='U'

, wt is not referenced and the effective number of observations is

n

Constraint: if

weight ='W'

wt (i) \geq 0.0

, for

i = 1, 2, \dots, n

9: $fixed (lfixed)$ – Integer array Input

On entry: defines the structure of the fixed effects design matrix,

X

$fixed (1)$: The number of variables, $N_{F}$ , to include as fixed effects (not including the intercept if present).
$fixed (2)$: The fixed intercept flag which must contain $1$ if a fixed intercept is to be included and $0$ otherwise.
$fixed (2 + i)$: The column of dat holding the $i$ th fixed variable, for $i = 1, 2, \dots, fixed (1)$ .

See Section 9.1 for more details on the construction of

X

Constraints:

$fixed (1) \geq 0$ ;
$fixed (2) = 0$ or $1$ ;
$1 \leq fixed (2 + i) \leq ncol$ , for $i = 1, 2, \dots, fixed (1)$ .

10: $lfixed$ – Integer Input

On entry: length of the vector fixed.

Constraint:

lfixed \geq 2 + fixed (1)

11: $nrndm$ – Integer Input

On entry: the number of columns in rndm.

Constraint:

nrndm > 0

12: $rndm (ldrndm, *)$ – Integer array Input

Note: the second dimension of the array rndm must be at least

nrndm

On entry:

rndm (i, j)

defines the structure of the random effects design matrix,

Z

. The

b

th column of rndm defines a block of columns in the design matrix

Z

$rndm (1, b)$: The number of variables, $N_{R_{b}}$ , to include as random effects in the $b$ th block (not including the random intercept if present).
$rndm (2, b)$: The random intercept flag which must contain $1$ if block $b$ includes a random intercept and $0$ otherwise.
$rndm (2 + i, b)$: The column of dat holding the $i$ th random variable in the $b$ th block, for $i = 1, 2, \dots, rndm (1, b)$ .
$rndm (3 + N_{R_{b}}, b)$: The number of subject variables, $N_{S_{b}}$ , for the $b$ th block. The subject variables define the nesting structure for this block.
$rndm (3 + N_{R_{b}} + i, b)$: The column of dat holding the $i$ th subject variable in the $b$ th block, for $i = 1, 2, \dots, rndm (3 + N_{R_{b}}, b)$ .

See Section 9.2 for more details on the construction of

Z

Constraints:

$rndm (1, b) \geq 0$ ;
$rndm (2, b) = 0$ or $1$ ;
at least one random variable or random intercept must be specified in each block, i.e., $rndm (1, b) + rndm (2, b) > 0$ ;
the column identifiers associated with the random variables must be in the range $1$ to ncol, i.e., $1 \leq rndm (2 + i, b) \leq ncol$ , for $i = 1, 2, \dots, rndm (1, b)$ ;
$rndm (3 + N_{R_{b}}, b) \geq 0$ ;
the column identifiers associated with the subject variables must be in the range $1$ to ncol, i.e., $1 \leq rndm (3 + N_{R_{b}} + i, b) \leq ncol$ , for $i = 1, 2, \dots, rndm (3 + N_{R_{b}}, b)$ .

13: $ldrndm$ – Integer Input

On entry: the first dimension of the array rndm as declared in the (sub)program from which g02jcf is called.

Constraint:

ldrndm \geq max_{b} (3 + N_{R_{b}} + N_{S_{b}})

14: $nff$ – Integer Output

On exit:

p

, the number of fixed effects estimated, i.e., the number of columns in the design matrix

X

15: $nlsv$ – Integer Output

On exit: the number of levels for the overall subject variable (see Section 9.2 for a description of what this means). If there is no overall subject variable,

nlsv = 1

16: $nrf$ – Integer Output

On exit: the number of random effects estimated in each of the overall subject blocks. The number of columns in the design matrix

Z

is given by

q = nrf \times nlsv

17: $rcomm (lrcomm)$ – Real (Kind=nag_wp) array Communication Array

On exit: communication array as required by the analysis routines g02jdf and g02jef.

18: $lrcomm$ – Integer Input

On entry: the dimension of the array rcomm as declared in the (sub)program from which g02jcf is called.

Constraint:

lrcomm \geq (nff \times nrf + nrf \times nrf + nrf) \times nlsv + nff \times nff + nff + 2

19: $icomm (licomm)$ – Integer array Communication Array

On exit: if

licomm = 2

icomm (1)

holds the minimum required value for licomm and

icomm (2)

holds the minimum required value for lrcomm, otherwise icomm is a communication array as required by the analysis routines g02jdf and g02jef.

20: $licomm$ – Integer Input

On entry: the dimension of the array icomm as declared in the (sub)program from which g02jcf is called.

Constraint:

licomm = 2

licomm \geq 34 + N_{F} \times (MFL + 1) + nrndm \times MNR \times MRL + (LRNDM + 2) \times nrndm + ncol + LDID \times LB,

where

$MNR = \max_{b} (N_{R_{b}})$ ,
$MFL = \max_{i} (levels (fixed (2 + i)))$ ,
$MRL = \max_{b, i} (levels (rndm (2 + i, b)))$ ,
$LDID = \max_{b} N_{S_{b}}$ ,
$LB = nff + nrf \times nlsv$ , and
$LRNDM = max_{b} (3 + N_{R_{b}} + N_{S_{b}})$

21: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

- 1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

- 1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

- 1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, weight had an illegal value.
Constraint: $weight ='U'$ or $'W'$ .

$ifail = 2$: On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 1$ .

$ifail = 3$: On entry, $ncol = ⟨ value ⟩$ .
Constraint: $ncol \geq 0$ .

$ifail = 4$: On entry, variable $j$ of observation $i$ is less than $1$ or greater than $levels (j)$ : $i = ⟨ value ⟩$ , $j = ⟨ value ⟩$ , value $= ⟨ value ⟩$ , $levels (j) = ⟨ value ⟩$ .

$ifail = 5$: On entry, $lddat = ⟨ value ⟩$ and $n = ⟨ value ⟩$ .
Constraint: $lddat \geq n$ .

$ifail = 6$: On entry, $levels (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $levels (i) \geq 1$ .

$ifail = 8$: On entry, $wt (⟨ value ⟩) = ⟨ value ⟩$ .
Constraint: $wt (i) \geq 0.0$ .

$ifail = 9$: On entry, number of fixed parameters, $⟨ value ⟩$ is less than zero.

$ifail = 10$: On entry, $lfixed = ⟨ value ⟩$ .
Constraint: $lfixed \geq ⟨ value ⟩$ .

$ifail = 11$: On entry, $nrndm = ⟨ value ⟩$ .
Constraint: $nrndm > 0$ .

$ifail = 12$: On entry, number of random parameters for random statement $i$ is less than $0$ : $i = ⟨ value ⟩$ , number of parameters $= ⟨ value ⟩$ .

$ifail = 13$: On entry, $ldrndm = ⟨ value ⟩$ .
Constraint: $ldrndm \geq ⟨ value ⟩$ .

$ifail = 18$: On entry, $lrcomm = ⟨ value ⟩$ .
Constraint: $lrcomm \geq ⟨ value ⟩$ .

$ifail = 20$: On entry, $licomm = ⟨ value ⟩$ .
Constraint: $licomm \geq ⟨ value ⟩$ .

$ifail = 102$: On entry, more fixed factors than observations, $n = ⟨ value ⟩$ .
Constraint: $n \geq ⟨ value ⟩$ .

$ifail = 108$: On entry, no observations due to zero weights.

$ifail = 109$: On entry, invalid value for fixed intercept flag: value $= ⟨ value ⟩$ .

$ifail = 112$: On entry, invalid value for random intercept flag for random statement $i$ : $i = ⟨ value ⟩$ , value $= ⟨ value ⟩$ .

$ifail = 209$: On entry, index of fixed variable $j$ is less than $1$ or greater than $ncol$ : $j = ⟨ value ⟩$ , index $= ⟨ value ⟩$ and $ncol = ⟨ value ⟩$ .

$ifail = 212$: On entry, must be at least one parameter, or an intercept in each random statement $i$ : $i = ⟨ value ⟩$ .

$ifail = 312$: On entry, index of random variable $j$ in random statement $i$ is less than $1$ or greater than $ncol$ : $i = ⟨ value ⟩$ , $j = ⟨ value ⟩$ , index $= ⟨ value ⟩$ and $ncol = ⟨ value ⟩$ .

$ifail = 412$: On entry, number of subject parameters for random statement $i$ is less than $0$ : $i = ⟨ value ⟩$ , number of parameters $= ⟨ value ⟩$ .

$ifail = 512$: On entry, nesting variable $j$ in random statement $i$ has one level: $j = ⟨ value ⟩$ , $i = ⟨ value ⟩$ .

$ifail = - 99$: 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.

$ifail = - 399$: 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.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

g02jcf is not threaded in any implementation.

9 Further Comments

9.1 Construction of the fixed effects design matrix, $X$

Let

$N_{F}$ denote the number of fixed variables, that is $fixed (1) = N_{F}$ ;
$F_{j}$ denote the $j$ th fixed variable, that is the vector of values held in the $k$ th column of dat when $fixed (2 + j) = k$ ;
$F_{i j}$ denote the $i$ th element of $F_{j}$ ;
$L (F_{j})$ denote the number of levels for $F_{j}$ , that is $L (F_{j}) = levels (fixed (2 + j))$ ;
$D_{v} (F_{j})$ denoted an indicator function that returns a vector of values whose $i$ th element is $1$ if $F_{i j} = v$ and $0$ otherwise.

The design matrix for the fixed effects,

X

, is constructed as follows:

set $k$ to one and the flag $done_first$ to false;
if a fixed intercept is included, that is $fixed (2) = 1$ ,
- set the first column of $X$ to a vector of $1$ s;
- set $k = k + 1$ ;
- set $done_first$ to true;
loop over each fixed variable, so for each j=1,2,…,NF,
- if $L (F_{j}) = 1$ ,
  - set the $k$ th column of $X$ to be $F_{j}$ ;
  - set $k = k + 1$ ;
- else
  - if $done_first$ is false then
    - set the $L (F_{j})$ columns, $k$ to $k + L (F_{j}) - 1$ , of $X$ to $D_{v} (F_{j})$ , for $v = 1, 2, \dots, L (F_{j})$ ;
    - set $k = k + L (F_{j})$ ;
    - set $done_first$ to true;
  - else
    - set the $L (F_{j}) - 1$ columns, $k$ to $k + L (F_{j}) - 2$ , of $X$ to $D_{v} (F_{j})$ , for $v = 2, 3, \dots, L (F_{j})$ ;
    - set $k = k + L (F_{j}) - 1$ .

The number of columns in the design matrix,

X

, is, therefore, given by

p = 1 + \sum_{j = 1}^{N_{F}} (levels (fixed (2 + j)) - 1) .

This quantity is returned in nff.

In summary, g02jcf converts all non-binary categorical variables (i.e., where

L (F_{j}) > 1

) to dummy variables. If a fixed intercept is included in the model then the first level of all such variables is dropped. If a fixed intercept is not included in the model then the first level of all such variables, other than the first, is dropped. The variables are added into the model in the order they are specified in fixed.

9.2 Construction of random effects design matrix, $Z$

Let

$N_{R_{b}}$ denote the number of random variables in the $b$ th random statement, that is $N_{R_{b}} = rndm (1, b)$ ;
$R_{j b}$ denote the $j$ th random variable from the $b$ th random statement, that is the vector of values held in the $k$ th column of dat when $rndm (2 + j, b) = k$ ;
$R_{i j b}$ denote the $i$ th element of $R_{j b}$ ;
$L (R_{j b})$ denote the number of levels for $R_{j b}$ , that is $L (R_{j b}) = levels (rndm (2 + j, b))$ ;
$D_{v} (R_{j b})$ denoted an indicator function that returns a vector of values whose $i$ th element is $1$ if $R_{i j b} = v$ and $0$ otherwise;
$N_{S_{b}}$ denote the number of subject variables in the $b$ th random statement, that is $N_{S_{b}} = rndm (3 + N_{R_{b}}, b)$ ;
$S_{j b}$ denote the $j$ th subject variable from the $b$ th random statement, that is the vector of values held in the $k$ th column of dat when $rndm (3 + N_{R_{b}} + j, b) = k$ ;
$S_{i j b}$ denote the $i$ th element of $S_{j b}$ ;
$L (S_{j b})$ denote the number of levels for $S_{j b}$ , that is $L (S_{j b}) = levels (rndm (3 + N_{R_{b}} + j, b))$ ;
$I_{b} (s_{1}, s_{2}, \dots, s_{N_{S_{b}}})$ denoted an indicator function that returns a vector of values whose $i$ th element is $1$ if $S_{i j b} = s_{j}$ for all $j = 1, 2, \dots, N_{S_{b}}$ and $0$ otherwise.

The design matrix for the random effects,

Z

, is constructed as follows:

set $k$ to one;
loop over each random statement, so for each b=1,2,…,nrndm,
- loop over each level of the last subject variable, so for each sNSb=1,2,…,L(RNSb b),
  - $⋮$
    - loop over each level of the second subject variable, so for each s2=1,2,…,L(R2b),
      - loop over each level of the first subject variable, so for each $s_{1} = 1, 2, \dots, L (R_{1 b})$ ,
        
        if a random intercept is included, that is $rndm (2, b) = 1$ ,
        
        set the $k$ th column of $Z$ to $I_{b} (s_{1}, s_{2}, \dots, s_{N_{S_{b}}})$ ;
        
        set $k = k + 1$ ;
        
        loop over each random variable in the $b$ th random statement, so for each $j = 1, 2, \dots, N_{R_{b}}$ ,
        
        if $L (R_{j b}) = 1$ ,
        
        set the $k$ th column of $Z$ to $R_{j b} \times I_{b} (s_{1}, s_{2}, \dots, s_{N_{S_{b}}})$ where $\times$ indicates an element-wise multiplication between the two vectors, $R_{j b}$ and $I_{b} (\dots)$ ;
        
        set $k = k + 1$ ;
        
        else
        
        set the $L (R_{b j})$ columns, $k$ to $k + L (R_{b j})$ , of $Z$ to $D_{v} (R_{j b}) \times I_{b} (s_{1}, s_{2}, \dots, s_{N_{S_{b}}})$ , for $v = 1, 2, \dots, L (R_{j b})$ . As before, $\times$ indicates an element-wise multiplication between the two vectors, $D_{v} (\dots)$ and $I_{b} (\dots)$ ;
        
        set $k = k + L (R_{j b})$ .

In summary, each column of rndm defines a block of consecutive columns in

Z

. g02jcf converts all non-binary categorical variables (i.e., where

L (R_{j b})

L (S_{j b}) > 1

) to dummy variables. All random variables defined within a column of rndm are nested within all subject variables defined in the same column of rndm. In addition each of the subject variables are nested within each other, starting with the first (i.e., each of the

R_{j b}, j = 1, 2, \dots, N_{R_{b}}

are nested within

S_{1 b}

which in turn is nested within

S_{2 b}

, which in turn is nested within

S_{3 b}

, etc.).

If the last subject variable in each column of rndm are the same (i.e.,

S_{N_{S_{1}} 1} = S_{N_{S_{2}} 2} = \dots = S_{N_{S_{b}} b}

) then all random effects in the model are nested within this variable. In such instances the last subject variable (

S_{N_{S_{1}} 1}

) is called the overall subject variable. The fact that all of the random effects in the model are nested within the overall subject variable means that

Z^{T} Z

is block diagonal in structure. This fact can be utilised to improve the efficiency of the underlying computation and reduce the amount of internal storage required. The number of levels in the overall subject variable is returned in

nlsv = L (S_{N_{S_{1}} 1})

If the last

k

subject variables in each column of rndm are the same, for

k > 1

then the overall subject variable is defined as the interaction of these

k

variables and

nlsv = \prod_{j = N_{S_{1}} - k + 1}^{N_{S_{1}}} L (S_{j 1}) .

If there is no overall subject variable then

nlsv = 1

The number of columns in the design matrix

Z

is given by

q = nrf \times nlsv

10 Example

See Section 10 in g02jdf and g02jef.

NAG Library Manual, Mark 27.1

Interfaces: FL CL CPP AD

NAG FL Interface Introduction

G02 (Correg) Chapter Contents

G02 (Correg) Chapter Introduction

g02jc: FL CL CPP AD

NAG FL Interfaceg02jcf (mixeff_​hier_​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

9.1 Construction of the fixed effects design matrix, X

9.2 Construction of random effects design matrix, Z

10 Example

NAG FL Interface
g02jcf (mixeff_hier_init)

9.1 Construction of the fixed effects design matrix, $X$

9.2 Construction of random effects design matrix, $Z$