hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_correg_mixeff_hier_init (g02jc)

▸▿ Contents

    1  Purpose
    2  Syntax
    3  Description
    4  References
▸▿ 5  Parameters
    6  Error Indicators and Warnings
    7  Accuracy
▸▿ 8  Further Comments
    9  Example

Purpose

nag_correg_mixeff_hier_init (g02jc) preprocesses a dataset prior to fitting a linear mixed effects regression model of the following form via either nag_correg_mixeff_hier_reml (g02jd) or nag_correg_mixeff_hier_ml (g02je).

Syntax

[nff, nlsv, nrf, rcomm, icomm, ifail] = g02jc(dat, levels, y, fixed, rndm, lrcomm, licomm, 'n', n, 'ncol', ncol, 'wt', wt, 'lfixed', lfixed, 'nrndm', nrndm)
[nff, nlsv, nrf, rcomm, icomm, ifail] = nag_correg_mixeff_hier_init(dat, levels, y, fixed, rndm, lrcomm, licomm, 'n', n, 'ncol', ncol, 'wt', wt, 'lfixed', lfixed, 'nrndm', nrndm)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 23: weight was removed from the interface

Description

nag_correg_mixeff_hier_init (g02jc) must be called prior to fitting a linear mixed effects regression model with either nag_correg_mixeff_hier_reml (g02jd) or nag_correg_mixeff_hier_ml (g02je).
The model fitting functions nag_correg_mixeff_hier_reml (g02jd) and nag_correg_mixeff_hier_ml (g02je) fit a model of the following form:
y=Xβ+Zν+ε  
where y is a vector of n observations on the dependent variable,
X is an n by p design matrix of fixed independent variables,
β is a vector of p unknown fixed effects,
Z is an n by 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 ν ε = G 0 0 R  
where R= σ R 2 I , I is the n×n identity matrix and G is a diagonal matrix.
Case weights can be incorporated into the model by replacing X and Z with Wc1/2X and Wc1/2Z respectively where Wc is a diagonal weight matrix.

References

None.

Parameters

Compulsory Input Parameters

1:     datlddatncol – double array
lddat, the first dimension of the array, must satisfy the constraint lddat≥n.
A matrix of data, with datij holding the ith observation on the jth 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 levelsj≠1,1≤datij≤levelsj.
2:     levelsncol – int64int32nag_int array
levelsi contains the number of levels associated with the ith variable held in dat.
If the ith variable is continuous or binary (i.e., only takes the values zero or one) then levelsi must be set to 1. Otherwise the ith variable is assumed to take an integer value between 1 and levelsi, (i.e., the ith variable is discrete with levelsi levels).
Constraint: levelsi≥1, for i=1,2,…,ncol.
3:     yn – double array
y, the vector of observations on the dependent variable.
4:     fixedlfixed – int64int32nag_int array
Defines the structure of the fixed effects design matrix, X.
fixed1
The number of variables, NF, to include as fixed effects (not including the intercept if present).
fixed2
The fixed intercept flag which must contain 1 if a fixed intercept is to be included and 0 otherwise.
fixed2+i
The column of dat holding the ith fixed variable, for i=1,2,…,fixed1.
See Construction of the design matrix, for more details on the construction of X.
Constraints:
  • fixed1≥0;
  • fixed2=0​ or ​1;
  • 1≤fixed2+i≤ncol, for i=1,2,…,fixed1.
5:     rndmldrndmnrndm – int64int32nag_int array
ldrndm, the first dimension of the array, must satisfy the constraint ldrndm≥ max b 3+NRb+NSb .
rndmij defines the structure of the random effects design matrix, Z. The bth column of rndm defines a block of columns in the design matrix Z.
rndm1b
The number of variables, NRb, to include as random effects in the bth block (not including the random intercept if present).
rndm2b
The random intercept flag which must contain 1 if block b includes a random intercept and 0 otherwise.
rndm2+ib
The column of dat holding the ith random variable in the bth block, for i=1,2,…,rndm1b.
rndm3+NRbb
The number of subject variables, NSb, for the bth block. The subject variables define the nesting structure for this block.
rndm3+NRb+ib
The column of dat holding the ith subject variable in the bth block, for i=1,2,…,rndm3+NRbb.
See Construction of design matrix, for more details on the construction of Z.
Constraints:
  • rndm1b≥0;
  • rndm2b=0​ or ​1;
  • at least one random variable or random intercept must be specified in each block, i.e., rndm1b + rndm2b > 0 ;
  • the column identifiers associated with the random variables must be in the range 1 to ncol, i.e., 1≤ rndm2+ib ≤ ncol , for i=1,2,…,rndm1b;
  • rndm3+NRbb ≥0 ;
  • the column identifiers associated with the subject variables must be in the range 1 to ncol, i.e., 1≤ rndm3+ N R b +i b ≤ncol , for i=1,2,…,rndm3+NRbb.
6:     lrcomm – int64int32nag_int scalar
The dimension of the array rcomm.
Constraint: lrcomm≥nrf×nlsv+nff+nff×nlsv+nrf×nlsv+nff+2.
7:     licomm – int64int32nag_int scalar
The dimension of the array icomm.
Constraint: licomm=2 or
licomm≥34+ NF×MFL+1+ nrndm×MNR×MRL+LRNDM+2×nrndm+ ncol+LDID×LB,
where
  • MNR = maxb N R b ,
  • MFL=maxi levels fixed2+i ,
  • MRL=maxb,i levels rndm2+ib ,
  • LDID=maxb NSb ,
  • LB=nff+nrf×nlsv, and
  • LRNDM= max b 3+NRb+NSb  

Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: the dimension of the array y and the first dimension of the array dat. (An error is raised if these dimensions are not equal.)
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≥1.
2:     ncol – int64int32nag_int scalar
Default: the dimension of the arrays dat, levels and the second dimension of the array dat. (An error is raised if these dimensions are not equal.)
The number of columns in the data matrix, dat.
Constraint: ncol≥0.
3:     wt: – double array
The dimension of the array wt must be at least n if weight='W'
If weight='W', wt must contain the diagonal elements of the weight matrix Wc.
If wti=0.0, the ith observation is not included in the model and the effective number of observations is the number of observations with nonzero weights.
If weight='U', wt is not referenced and the effective number of observations is n.
Constraint: if weight='W', wti≥0.0, for i=1,2,…,n.
4:     lfixed – int64int32nag_int scalar
Default: the dimension of the array fixed.
Length of the vector fixed.
Constraint: lfixed≥2+fixed1.
5:     nrndm – int64int32nag_int scalar
Default: the second dimension of the array rndm.
The second dimension of the array rndm.
Constraint: nrndm>0.

Output Parameters

1:     nff – int64int32nag_int scalar
p, the number of fixed effects estimated, i.e., the number of columns in the design matrix X.
2:     nlsv – int64int32nag_int scalar
The number of levels for the overall subject variable (see Construction of design matrix, for a description of what this means). If there is no overall subject variable, nlsv=1.
3:     nrf – int64int32nag_int scalar
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×nlsv.
4:     rcommlrcomm – double array
Communication array as required by the analysis functions nag_correg_mixeff_hier_reml (g02jd) and nag_correg_mixeff_hier_ml (g02je).
5:     icommlicomm – int64int32nag_int array
If licomm=2, icomm1 holds the minimum required value for licomm and icomm2 holds the minimum required value for lrcomm, otherwise icomm is a communication array as required by the analysis functions nag_correg_mixeff_hier_reml (g02jd) and nag_correg_mixeff_hier_ml (g02je).
6:     ifail – int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry, weight had an illegal value.
   ifail=2
Constraint: n≥1.
   ifail=3
Constraint: ncol≥0.
   ifail=4
On entry, variable j of observation i is less than 1 or greater than levelsj.
   ifail=5
Constraint: lddat≥n.
   ifail=6
Constraint: levelsi≥1.
   ifail=8
Constraint: wti≥0.0.
   ifail=9
On entry, number of fixed parameters, _ is less than zero.
   ifail=10
lfixed is too small.
   ifail=11
Constraint: nrndm>0.
   ifail=12
On entry, number of random parameters for random statement i is less than 0.
   ifail=13
ldrndm is too small.
   ifail=18
lrcomm is too small.
   ifail=20
licomm is too small.
   ifail=102
n is too small.
   ifail=108
On entry, no observations due to zero weights.
   ifail=109
On entry, invalid value for fixed intercept flag.
   ifail=112
On entry, invalid value for random intercept flag for random statement i.
   ifail=209
On entry, index of fixed variable j is less than 1 or greater than ncol.
   ifail=212
On entry, must be at least one parameter, or an intercept in each random statement i:
   ifail=312
On entry, index of random variable j in random statement i is less than 1 or greater than ncol.
   ifail=412
On entry, number of subject parameters for random statement i is less than 0.
   ifail=512
On entry, nesting variable j in random statement i has one level.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   ifail=-999
Dynamic memory allocation failed.

Accuracy

Not applicable.

Further Comments

Construction of the fixed effects design matrix, X

Let
The design matrix for the fixed effects, X, is constructed as follows:
The number of columns in the design matrix, X, is therefore given by
p= 1+ ∑ j=1 N F levels fixed 2+j -1 .  
This quantity is returned in nff.
In summary, nag_correg_mixeff_hier_init (g02jc) converts all non-binary categorical variables (i.e., where LFj>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.

Construction of random effects design matrix, Z

Let
The design matrix for the random effects, Z, is constructed as follows:
In summary, each column of rndm defines a block of consecutive columns in Z. nag_correg_mixeff_hier_init (g02jc) converts all non-binary categorical variables (i.e., where LRjb or LSjb>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 Rjb,j=1,2,…,NRb are nested within S1b which in turn is nested within S2b, which in turn is nested within S3b, etc.).
If the last subject variable in each column of rndm are the same (i.e., SNS11 = SNS22 = … = SNSbb ) then all random effects in the model are nested within this variable. In such instances the last subject variable ( SNS11 ) 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 ZTZ 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=LSNS11.
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= ∏ j=NS1-k+1 NS1 LSj1 .  
If there is no overall subject variable then nlsv=1.
The number of columns in the design matrix Z is given by q=nrf×nlsv.

The rndm argument

To illustrate some additional points about the rndm argument, we assume that we have a dataset with three discrete variables, V1, V2 and V3, with 2,4 and 3 levels respectively, and that V1 is in the first column of dat, V2 in the second and V3 the third. Also assume that we wish to fit a model containing V1 along with V2 nested within V3, as random effects. In order to do this the rndm matrix requires two columns:
rndm= 1 1 0 0 1 2 0 1 0 3  
The first column, 1,0,1,0,0, indicates one random variable (rndm11=1), no intercept (rndm21=0), the random variable is in the first column of dat (rndm31=1), there are no subject variables; as no nesting is required for V1 (rndm41=0). The last element in this column is ignored.
The second column, 1,0,2,1,3, indicates one random variable (rndm12=1), no intercept (rndm22=0), the random variable is in the second column of dat rndm32=2, there is one subject variable (rndm42=1), and the subject variable is in the third column of dat rndm52=3.
The corresponding Z matrix would have 14 columns, with 2 coming from V1 and 12 (4×3) from V2 nested within V3. The, symmetric, ZTZ matrix has the form
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - -  
where 0 indicates a structural zero, i.e., it always takes the value 0, irrespective of the data, and - a value that is not a structural zero. The first two rows and columns of ZTZ correspond to V1. The block diagonal matrix in the 12 rows and columns in the bottom right correspond to V2 nested within V3. With the 4×4 blocks corresponding to the levels of V2. There are three blocks as the subject variable (V3) has three levels.
The model fitting functions, nag_correg_mixeff_hier_reml (g02jd) and nag_correg_mixeff_hier_ml (g02je), use the sweep algorithm to calculate the log-likelihood function for a given set of variance components. This algorithm consists of moving down the diagonal elements (called pivots) of a matrix which is similar in structure to ZTZ, and updating each element in that matrix. When using the k diagonal element of a matrix A, an element a i j ,i≠k,j≠k , is adjusted by an amount equal to a i k a i j / a k k . This process can be referred to as sweeping on the kth pivot. As there are no structural zeros in the first row or column of the above ZTZ, sweeping on the first pivot of ZTZ would alter each element of the matrix and therefore destroy the structural zeros, i.e., we could no longer guarantee they would be zero.
Reordering the rndm matrix to
rndm= 1 1 0 0 2 1 1 0 3 0  
i.e., the swapping the two columns, results in a ZTZ matrix of the form
- - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 - - - - 0 0 0 0 - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - 0 0 0 0 0 0 0 0 - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -  
This matrix is identical to the previous one, except the first two rows and columns have become the last two rows and columns. Sweeping a matrix, A=aij, of this form on the first pivot will only affect those elements aij, where ai1≠0​ and ​a1j≠0, which is only the 13th and 14th row and columns, and the top left hand block of 4 rows and columns. The block diagonal nature of the first 12 rows and columns therefore greatly reduces the amount of work the algorithm needs to perform.
nag_correg_mixeff_hier_init (g02jc) constructs the ZTZ as specified by the rndm matrix, and does not attempt to reorder it to improve performance. Therefore for best performance some thought is required on what ordering to use. In general it is more efficient to structure rndm in such a way that the first row relates to the deepest level of nesting, the second to the next level, etc..

Example

See Example in nag_correg_mixeff_hier_reml (g02jd) and nag_correg_mixeff_hier_ml (g02je).

Open in the MATLAB editor: g02jc_example

function g02jc_example


fprintf('g02jc example results\n\n');

% Problem size
n = 90;
ncol = 12;
nrndm = 3;
nvpr = int64(7);

% The number of levels associated with each of the independent variables
levels = [int64(2); 3; 2; 3; 2; 3; 1; 4; 5; 2; 3; 3];

% The Fixed part of the model
fixed = [int64(2); 1; 1; 2];

% The Random part of the model
rndm = [int64(2), 2,  3;
                0,  0,  0;
                3,  5,  7;
                4,  6,  8;
                3,  2,  9;
               10, 11,  1;
               11, 12, 12;
               12,  0,  0];

% Dependant data
y = [ 3.1100;   2.8226;   7.4543;   4.4313;   6.1543;  -0.1783;
      4.6748;   7.0667;   1.4262;   7.7290;  -2.1806;   6.8419;
      1.2590;   8.8405;   6.1657;  -4.5605;  -1.2367; -12.2932;
     -2.3374;   0.0716;   0.1895;   1.5608;  -0.8529;  -4.1169;
      3.9977;  -8.1277;  -4.9656;  -0.6428;  -5.5152;  -5.5657;
     14.8177;  16.9783;  13.8966;  14.8166;  19.3640;   9.5299;
     12.0102;   6.1551;  -1.7048;   2.7640;   2.8065;   0.0974;
     -7.8080; -18.0450;  -2.8199;   8.9893;   3.7978;  -6.3493;
      8.1411;  -7.5483;  -0.4600;  -3.2135;  -6.6562;   5.1267;
      3.5592;  -4.4420;  -8.5965;  -6.3187;  -7.8953; -10.1383;
     -7.8850;  23.2001;   5.5829;  -4.3698;   2.1274;  -2.7184;
    -17.9128;  -1.2708; -24.2735; -14.7374;   0.1713;   8.0006;
      1.2100;   3.3307; -22.6713;   7.5562;  -7.0694;   3.7159;
     -4.3135; -14.5577; -12.5107;   4.7708;  13.2797;  -6.3243;
     -7.0549;  -9.2713; -18.7788;  -7.7230; -22.7230; -11.6609];

% Independent data
dat = [1, 3, 2, 1, 2, 2,-0.3160, 4, 2, 1, 1, 1;
       1, 1, 1, 3, 1, 2,-1.3377, 1, 4, 1, 1, 1;
       1, 3, 1, 3, 1, 3,-0.7610, 4, 2, 1, 1, 1;
       2, 3, 2, 1, 1, 3,-2.2976, 4, 2, 1, 1, 1;
       2, 2, 1, 3, 2, 3,-0.4263, 2, 1, 1, 1, 1;
       2, 1, 2, 3, 1, 3, 1.4067, 3, 3, 2, 1, 1;
       2, 3, 2, 1, 2, 1,-1.4669, 1, 2, 2, 1, 1;
       1, 1, 1, 3, 2, 3, 0.4717, 2, 4, 2, 1, 1;
       1, 3, 2, 3, 2, 1, 0.4436, 1, 3, 2, 1, 1;
       1, 1, 1, 2, 2, 3,-0.5950, 3, 4, 2, 1, 1;
       1, 3, 1, 3, 1, 1,-1.7981, 4, 2, 1, 2, 1;
       2, 3, 1, 2, 1, 1, 0.2397, 1, 4, 1, 2, 1;
       1, 2, 2, 1, 2, 3, 0.4742, 1, 1, 1, 2, 1;
       2, 2, 2, 2, 2, 3, 0.6888, 3, 1, 1, 2, 1;
       2, 1, 2, 3, 1, 3,-1.0616, 3, 5, 1, 2, 1;
       1, 2, 2, 2, 2, 1,-0.5356, 1, 3, 2, 2, 1;
       1, 3, 2, 2, 1, 1,-1.2963, 2, 5, 2, 2, 1;
       1, 2, 2, 1, 2, 2,-1.5389, 3, 2, 2, 2, 1;
       2, 3, 1, 1, 2, 2,-0.6408, 2, 1, 2, 2, 1;
       1, 2, 2, 2, 1, 1, 0.6574, 1, 1, 2, 2, 1;
       2, 1, 1, 1, 1, 3, 0.9259, 1, 2, 1, 3, 1;
       2, 2, 2, 1, 2, 2, 1.5080, 3, 1, 1, 3, 1;
       2, 3, 1, 1, 1, 3, 2.5821, 2, 3, 1, 3, 1;
       1, 2, 2, 1, 2, 3, 0.4102, 1, 4, 1, 3, 1;
       2, 1, 2, 3, 2, 2, 0.7839, 2, 5, 1, 3, 1;
       1, 2, 2, 3, 2, 1,-1.8812, 4, 2, 2, 3, 1;
       1, 2, 1, 3, 2, 3, 0.7770, 4, 1, 2, 3, 1;
       2, 2, 1, 2, 1, 3, 0.2590, 3, 1, 2, 3, 1;
       2, 3, 2, 2, 2, 3,-0.9250, 3, 3, 2, 3, 1;
       2, 2, 1, 3, 2, 3,-0.4831, 1, 5, 2, 3, 1;
       2, 2, 1, 3, 1, 3, 0.5046, 3, 3, 1, 1, 2;
       2, 1, 1, 2, 2, 1,-0.6903, 2, 1, 1, 1, 2;
       1, 3, 2, 2, 2, 1, 1.6166, 2, 5, 1, 1, 2;
       2, 2, 2, 2, 1, 3, 0.2778, 2, 3, 1, 1, 2;
       2, 3, 2, 2, 1, 2, 1.9586, 4, 2, 1, 1, 2;
       1, 3, 1, 1, 1, 3, 1.0506, 2, 5, 2, 1, 2;
       2, 1, 1, 3, 2, 3, 0.4871, 1, 1, 2, 1, 2;
       2, 1, 2, 3, 2, 1, 2.0891, 4, 4, 2, 1, 2;
       1, 2, 1, 1, 2, 2, 1.4338, 4, 3, 2, 1, 2;
       1, 1, 2, 3, 1, 2,-1.1196, 3, 4, 2, 1, 2;
       1, 3, 1, 1, 2, 1, 0.3367, 3, 2, 1, 2, 2;
       2, 2, 1, 3, 1, 1, 0.1092, 2, 2, 1, 2, 2;
       1, 1, 1, 2, 2, 2, 0.4007, 4, 1, 1, 2, 2;
       2, 3, 1, 1, 1, 2, 0.1460, 3, 5, 1, 2, 2;
       2, 1, 2, 3, 1, 3,-0.3877, 3, 4, 1, 2, 2;
       1, 1, 1, 2, 2, 1, 0.6957, 4, 3, 2, 2, 2;
       2, 1, 1, 1, 2, 1,-0.4664, 3, 3, 2, 2, 2;
       1, 1, 1, 1, 2, 3, 0.2067, 2, 4, 2, 2, 2;
       2, 1, 2, 1, 1, 2, 0.4112, 1, 4, 2, 2, 2;
       2, 2, 1, 1, 1, 2,-1.3734, 3, 3, 2, 2, 2;
       2, 1, 2, 3, 1, 3, 0.7065, 1, 3, 1, 3, 2;
       1, 2, 2, 2, 1, 2, 1.3628, 4, 2, 1, 3, 2;
       2, 1, 2, 2, 2, 3,-0.5052, 4, 5, 1, 3, 2;
       2, 1, 1, 1, 2, 1,-1.3457, 2, 5, 1, 3, 2;
       1, 1, 2, 1, 2, 3,-1.8022, 3, 4, 1, 3, 2;
       2, 3, 1, 2, 1, 1, 0.0116, 2, 4, 2, 3, 2;
       2, 2, 1, 3, 2, 3,-0.9075, 1, 3, 2, 3, 2;
       2, 2, 2, 2, 2, 3,-1.4707, 1, 1, 2, 3, 2;
       2, 2, 1, 1, 2, 1,-1.2938, 2, 3, 2, 3, 2;
       1, 3, 1, 3, 2, 2,-1.1660, 4, 4, 2, 3, 2;
       1, 2, 1, 1, 2, 3, 0.0397, 4, 4, 1, 1, 3;
       1, 3, 1, 2, 1, 3,-0.5987, 3, 2, 1, 1, 3;
       2, 3, 2, 2, 1, 1, 0.6683, 3, 3, 1, 1, 3;
       2, 2, 1, 1, 2, 2,-0.0106, 1, 3, 1, 1, 3;
       1, 2, 1, 3, 2, 2, 0.5885, 1, 3, 1, 1, 3;
       1, 1, 1, 1, 1, 2, 0.4555, 1, 5, 2, 1, 3;
       2, 2, 2, 1, 1, 2, 0.6502, 4, 3, 2, 1, 3;
       1, 1, 1, 3, 1, 1,-0.1601, 1, 3, 2, 1, 3;
       2, 2, 1, 3, 2, 3, 1.6910, 1, 1, 2, 1, 3;
       2, 2, 2, 3, 1, 2, 0.1053, 4, 4, 2, 1, 3;
       2, 1, 2, 3, 2, 2,-0.4037, 3, 4, 1, 2, 3;
       1, 3, 2, 3, 1, 3,-0.5853, 3, 2, 1, 2, 3;
       2, 3, 2, 1, 1, 1,-0.3037, 1, 3, 1, 2, 3;
       1, 3, 1, 1, 2, 2,-0.0774, 1, 4, 1, 2, 3;
       2, 3, 1, 2, 2, 1, 0.4733, 4, 5, 1, 2, 3;
       1, 3, 2, 2, 1, 2,-0.0354, 4, 2, 2, 2, 3;
       1, 3, 2, 2, 1, 1,-0.6640, 2, 1, 2, 2, 3;
       2, 3, 1, 3, 1, 1, 0.0335, 4, 4, 2, 2, 3;
       1, 2, 2, 2, 1, 3, 0.1351, 1, 1, 2, 2, 3;
       1, 1, 2, 1, 2, 3,-0.5951, 3, 4, 2, 2, 3;
       2, 2, 2, 3, 1, 3, 0.2735, 3, 2, 1, 3, 3;
       2, 2, 1, 1, 1, 3, 0.3157, 1, 2, 1, 3, 3;
       2, 2, 2, 1, 1, 1,-1.0843, 2, 3, 1, 3, 3;
       1, 2, 2, 1, 2, 2,-0.0836, 4, 2, 1, 3, 3;
       2, 1, 2, 1, 1, 2,-0.2884, 2, 1, 1, 3, 3;
       2, 3, 2, 3, 2, 3,-0.1006, 1, 2, 2, 3, 3;
       1, 3, 1, 2, 2, 3, 0.5710, 1, 3, 2, 3, 3;
       1, 1, 2, 1, 1, 2, 0.2776, 2, 3, 2, 3, 3;
       2, 3, 2, 2, 1, 3,-0.7561, 4, 4, 2, 3, 3;
       1, 2, 2, 2, 1, 2, 1.5549, 1, 4, 2, 3, 3];

vpr = [int64(1):7];
gamma = zeros(8, 1);
gamma(1) = -1; % Estimate initial values for the variance components

% Call the initialisation routine once to get lrcomm and licomm
lrcomm = int64(0);
licomm = int64(2);
[nff, nlsv, nrf, rcomm, icomm, ifail] = ...
  g02jc( ...
         dat, levels, y, fixed, rndm, lrcomm, licomm);
licomm = icomm(1);
lrcomm = icomm(2);

% Pre-process the data
[nff, nlsv, nrf, rcomm, icomm, ifail] = ...
  g02jc( ...
         dat, levels, y, fixed, rndm, lrcomm, licomm);

% Use default options
iopt = zeros(0, 0, 'int64');
ropt = zeros(0, 0);

lb = nff + nrf*nlsv;

% Perform the analysis
[gamma, effn, rnkx, ncov, lnlike, id, b, se, czz, cxx, cxz, ifail] = ...
  g02jd( ...
         vpr, nvpr, gamma, lb, rcomm, icomm, iopt, ropt);

% Print results
fprintf('Number of observations (n)                    = %d\n', n);
fprintf('Number of random factors (nrf)                = %d\n', nrf);
fprintf('Number of fixed factors (nff)                 = %d\n', nff);
fprintf('Number of subject levels (nlsv)               = %d\n', nlsv);
fprintf('Rank of X (rnkx)                              = %d\n', rnkx);
fprintf('Effective n (effn)                            = %d\n', effn);
fprintf('Number of non-zero variance components (ncov) = %d\n', ncov);

fprintf('\nParameter Estimates\n');


if nrf > 0
  fprintf('\nRandom Effects\n');
end
pb = -999;
pfmt = ' ';
for k = 1:nrf*nlsv
  tb = id(1,k);
  if tb ~= -999
    vid = id(2,k);
    nv = rndm(1,tb);
    ns = rndm(3+nv,tb);
    tfmt = sprintf('%d ', id(3+1:3+ns,k));
    if ( (pb ~= tb) || (strcmp(tfmt, pfmt) == 0) )
      if (k ~= 1)
        fprintf('\n');
      end
      fprintf('  Subject: ');
      for l=1:ns
        fprintf(' Variable %2d (Level %2d)',rndm(3+nv+l,tb), id(3+l,k));
      end
      fprintf('\n');
    end
    if (vid==0)
      % Intercept
      fprintf('    Intercept%18s%10.4f %10.4f\n', ' ', b(k), se(k));
    else
      % variable vid specified in rndm
      aid = rndm(2+vid,tb);
      if (id(3,k)==0)
        fprintf('    Variable %2d%16s%10.4f %10.4f\n', aid, ' ', b(k), se(k));
      else
        fprintf('    Variable %2d (Level %2d)     %10.4f %10.4f\n', ...
                aid, id(3,k), b(k), se(k));
      end
    end
    pfmt = tfmt;
  end
  pb = tb;
end

if nff>0
  fprintf('\nFixed Effects\n');
end
for k = (nrf*nlsv+1):(nrf*nlsv+nff)
  if vid~=-999
    vid = id(2,k);
    if vid==0
      % Intercept
      fprintf('    Intercept%18s%10.4f %10.4f\n', ' ', b(k), se(k));
    else
      % vid'th variable specified in fixed
      aid = fixed(2+vid);
      if (id(3,k)==0)
        fprintf('    Variable %2d%16s%10.4f %10.4f\n', aid, ' ', b(k), se(k));
      else
        fprintf('    Variable %2d (Level %2d)     %10.4f %10.4f\n', ...
                aid, id(3,k), b(k), se(k));
      end
    end
  end
end

fprintf('\nVariance Components\n');
fprintf('  Estimate     Parameter       Subject\n');
for k = 1:nvpr
  fprintf('%10.5f     ', gamma(k));
  p = 0;
  for tb = 1:nrndm
    nv = rndm(1,tb);
    ns = rndm(3+nv,tb);
    if (rndm(2,tb)==1)
      p = p + 1;
      if (vpr(p)==k)
        fprintf('Intercept       Variables ');
        fprintf('%2d ', rndm(3+nv+1:3+nv+ns, tb));
        fprintf('\n');
      end
    end
    for i = 1:nv
      p = p + 1;
      if (vpr(p)==k)
       fprintf('Variable %2d     Variables %2d ', rndm(2+i,tb));
       fprintf('%2d ', rndm(3+nv+1:3+nv+ns, tb));
      end
    end
  end
  fprintf('\n');
end

fprintf('\nsigma^2          = %15.5f\n', gamma(nvpr+1));
fprintf('-2log likelihood = %15.5f\n', lnlike);



g02jc example results

Number of observations (n)                    = 90
Number of random factors (nrf)                = 55
Number of fixed factors (nff)                 = 4
Number of subject levels (nlsv)               = 3
Rank of X (rnkx)                              = 4
Effective n (effn)                            = 90
Number of non-zero variance components (ncov) = 7

Parameter Estimates

Random Effects
  Subject:  Variable 10 (Level  1) Variable 11 (Level  1) Variable 12 (Level  1)
    Variable  3 (Level  1)         2.1561     3.7946
    Variable  3 (Level  2)         1.8951     3.9284
    Variable  4 (Level  1)         0.6496     3.1617

  Subject:  Variable 10 (Level  1) Variable 11 (Level  1) Variable 12 (Level  1)
    Variable  4 (Level  3)         0.7390     3.1424

  Subject:  Variable 10 (Level  2) Variable 11 (Level  1) Variable 12 (Level  1)
    Variable  3 (Level  1)         1.4216     3.3773
    Variable  3 (Level  2)        -2.8921     3.3953
    Variable  4 (Level  1)         3.6789     2.3162
    Variable  4 (Level  2)        -1.9742     2.3887
    Variable  4 (Level  3)        -2.2088     2.0697

  Subject:  Variable 10 (Level  1) Variable 11 (Level  2) Variable 12 (Level  1)
    Variable  3 (Level  1)        -2.9659     3.9127
    Variable  3 (Level  2)         2.7951     4.7183
    Variable  4 (Level  1)        -4.7330     2.3094
    Variable  4 (Level  2)         5.5161     2.2330
    Variable  4 (Level  3)        -0.8417     2.3826

  Subject:  Variable 10 (Level  2) Variable 11 (Level  2) Variable 12 (Level  1)
    Variable  3 (Level  1)         4.2202     3.6675
    Variable  3 (Level  2)        -4.3883     3.4424
    Variable  4 (Level  1)        -1.1391     3.2187
    Variable  4 (Level  2)         1.0814     3.0654

  Subject:  Variable 10 (Level  1) Variable 11 (Level  3) Variable 12 (Level  1)
    Variable  3 (Level  1)         0.3391     4.0647
    Variable  3 (Level  2)         0.1502     3.4787
    Variable  4 (Level  1)        -1.0026     2.4363

  Subject:  Variable 10 (Level  1) Variable 11 (Level  3) Variable 12 (Level  1)
    Variable  4 (Level  3)         1.1703     2.6365

  Subject:  Variable 10 (Level  2) Variable 11 (Level  3) Variable 12 (Level  1)
    Variable  3 (Level  1)         1.2658     3.4819
    Variable  3 (Level  2)        -1.5356     3.9097

  Subject:  Variable 10 (Level  2) Variable 11 (Level  3) Variable 12 (Level  1)
    Variable  4 (Level  2)         0.7992     2.7902
    Variable  4 (Level  3)        -0.8916     2.8763

  Subject:  Variable 11 (Level  1) Variable 12 (Level  1)
    Variable  5 (Level  1)        -0.4885     2.8206
    Variable  5 (Level  2)         1.8829     2.7530
    Variable  6 (Level  1)         0.9249     3.7747
    Variable  6 (Level  2)        -2.3568     3.1624
    Variable  6 (Level  3)         4.3117     3.1474

  Subject:  Variable 11 (Level  2) Variable 12 (Level  1)
    Variable  5 (Level  1)         1.3898     2.9362
    Variable  5 (Level  2)        -1.5729     2.8909
    Variable  6 (Level  1)         0.2111     3.9967
    Variable  6 (Level  2)        -3.7083     4.2866
    Variable  6 (Level  3)         3.1190     4.7983

  Subject:  Variable 11 (Level  3) Variable 12 (Level  1)
    Variable  5 (Level  1)         1.7352     3.1370
    Variable  5 (Level  2)        -1.6165     3.1713
    Variable  6 (Level  1)        -1.1102     3.9374
    Variable  6 (Level  2)         4.4877     3.6980
    Variable  6 (Level  3)        -3.1325     3.1966

  Subject:  Variable 12 (Level  1)
    Variable  7                    0.6827     0.5060
    Variable  8 (Level  1)         1.5964     1.3206
    Variable  8 (Level  2)        -0.7533     1.5663
    Variable  8 (Level  3)         0.4035     1.6840
    Variable  8 (Level  4)        -0.8523     1.7518
    Variable  9 (Level  1)         0.5699     1.6236
    Variable  9 (Level  2)         0.0012     1.9111
    Variable  9 (Level  3)        -0.2850     1.9245
    Variable  9 (Level  4)         0.4468     2.0329
    Variable  9 (Level  5)         0.0030     2.1390

  Subject:  Variable 10 (Level  1) Variable 11 (Level  1) Variable 12 (Level  2)
    Variable  3 (Level  1)         6.2551     3.3595
    Variable  3 (Level  2)         5.6085     3.4127

  Subject:  Variable 10 (Level  1) Variable 11 (Level  1) Variable 12 (Level  2)
    Variable  4 (Level  2)         2.6922     2.7542
    Variable  4 (Level  3)         1.3742     2.8068

  Subject:  Variable 10 (Level  2) Variable 11 (Level  1) Variable 12 (Level  2)
    Variable  3 (Level  1)         1.5647     3.8353
    Variable  3 (Level  2)        -2.7565     3.9041
    Variable  4 (Level  1)        -0.8621     2.8257

  Subject:  Variable 10 (Level  2) Variable 11 (Level  1) Variable 12 (Level  2)
    Variable  4 (Level  3)         0.4536     2.8070

  Subject:  Variable 10 (Level  1) Variable 11 (Level  2) Variable 12 (Level  2)
    Variable  3 (Level  1)       -10.1544     3.3433
    Variable  3 (Level  2)         3.2446     4.1221
    Variable  4 (Level  1)        -2.9419     2.3508
    Variable  4 (Level  2)         0.2510     3.0675
    Variable  4 (Level  3)         0.3224     2.9710

  Subject:  Variable 10 (Level  2) Variable 11 (Level  2) Variable 12 (Level  2)
    Variable  3 (Level  1)        -1.3577     3.1925
    Variable  3 (Level  2)         8.1277     3.9975
    Variable  4 (Level  1)        -0.4290     2.4578
    Variable  4 (Level  2)         2.7495     2.5821

  Subject:  Variable 10 (Level  1) Variable 11 (Level  3) Variable 12 (Level  2)
    Variable  3 (Level  1)         4.8432     4.0069
    Variable  3 (Level  2)         0.0370     3.6006
    Variable  4 (Level  1)         3.0713     2.2706
    Variable  4 (Level  2)        -1.8899     2.4756
    Variable  4 (Level  3)         0.4914     2.2914

  Subject:  Variable 10 (Level  2) Variable 11 (Level  3) Variable 12 (Level  2)
    Variable  3 (Level  1)        -4.4766     3.3355
    Variable  3 (Level  2)        -3.7936     4.0759
    Variable  4 (Level  1)        -0.5459     2.7097
    Variable  4 (Level  2)        -1.5619     2.7412
    Variable  4 (Level  3)        -0.7269     2.9735

  Subject:  Variable 11 (Level  1) Variable 12 (Level  2)
    Variable  5 (Level  1)         4.8653     3.0706
    Variable  5 (Level  2)         0.9011     3.0696
    Variable  6 (Level  1)         6.9277     3.8411
    Variable  6 (Level  2)        -1.3108     3.1667
    Variable  6 (Level  3)         6.2916     3.5327

  Subject:  Variable 11 (Level  2) Variable 12 (Level  2)
    Variable  5 (Level  1)        -0.4047     3.0956
    Variable  5 (Level  2)         0.3291     3.0784
    Variable  6 (Level  1)         6.9096     3.3073
    Variable  6 (Level  2)        -1.0680     3.6213
    Variable  6 (Level  3)        -5.9977     3.7299

  Subject:  Variable 11 (Level  3) Variable 12 (Level  2)
    Variable  5 (Level  1)        -1.0925     3.0994
    Variable  5 (Level  2)        -0.7392     2.9900
    Variable  6 (Level  1)         2.7758     3.8748
    Variable  6 (Level  2)        -6.3526     3.3014
    Variable  6 (Level  3)        -0.2060     3.6481

  Subject:  Variable 12 (Level  2)
    Variable  7                    0.1711     0.5785
    Variable  8 (Level  1)         1.7186     1.9143
    Variable  8 (Level  2)        -0.6768     1.7352
    Variable  8 (Level  3)        -0.0439     1.6395
    Variable  8 (Level  4)         0.1463     1.5358
    Variable  9 (Level  1)         0.9761     2.3930
    Variable  9 (Level  2)         6.5436     1.8193
    Variable  9 (Level  3)        -1.5504     1.8527
    Variable  9 (Level  4)         0.1047     2.0244
    Variable  9 (Level  5)        -3.9386     1.7937

  Subject:  Variable 10 (Level  1) Variable 11 (Level  1) Variable 12 (Level  3)
    Variable  3 (Level  1)        10.6802     3.2596
    Variable  3 (Level  2)        -1.0290     3.7842
    Variable  4 (Level  1)        -2.8612     2.2917
    Variable  4 (Level  2)         3.9265     2.8934
    Variable  4 (Level  3)         2.2427     2.3737

  Subject:  Variable 10 (Level  2) Variable 11 (Level  1) Variable 12 (Level  3)
    Variable  3 (Level  1)        -6.2076     3.3642
    Variable  3 (Level  2)        -8.7670     3.8463
    Variable  4 (Level  1)        -2.9251     2.4657

  Subject:  Variable 10 (Level  2) Variable 11 (Level  1) Variable 12 (Level  3)
    Variable  4 (Level  3)        -2.2077     2.3743

  Subject:  Variable 10 (Level  1) Variable 11 (Level  2) Variable 12 (Level  3)
    Variable  3 (Level  1)        -3.3334     3.4665
    Variable  3 (Level  2)        -0.3111     3.2650
    Variable  4 (Level  1)         1.5131     2.4890
    Variable  4 (Level  2)        -3.0345     3.0562
    Variable  4 (Level  3)         0.2722     2.8300

  Subject:  Variable 10 (Level  2) Variable 11 (Level  2) Variable 12 (Level  3)
    Variable  3 (Level  1)         6.5905     4.0386
    Variable  3 (Level  2)        -5.3168     3.4549
    Variable  4 (Level  1)        -3.5280     2.9663
    Variable  4 (Level  2)         1.7056     2.9293
    Variable  4 (Level  3)         2.2590     3.1780

  Subject:  Variable 10 (Level  1) Variable 11 (Level  3) Variable 12 (Level  3)
    Variable  3 (Level  1)         8.1889     4.1429
    Variable  3 (Level  2)        -1.5388     3.3333
    Variable  4 (Level  1)         3.4338     2.6376

  Subject:  Variable 10 (Level  1) Variable 11 (Level  3) Variable 12 (Level  3)
    Variable  4 (Level  3)        -1.1544     2.9885

  Subject:  Variable 10 (Level  2) Variable 11 (Level  3) Variable 12 (Level  3)
    Variable  3 (Level  1)        -4.4243     4.0049
    Variable  3 (Level  2)        -4.1349     3.1248
    Variable  4 (Level  1)         1.0460     2.6550
    Variable  4 (Level  2)        -4.4844     2.2843
    Variable  4 (Level  3)         0.5046     2.6926

  Subject:  Variable 11 (Level  1) Variable 12 (Level  3)
    Variable  5 (Level  1)         5.3030     3.0278
    Variable  5 (Level  2)        -8.1794     3.1335
    Variable  6 (Level  1)        -0.8188     3.7810
    Variable  6 (Level  2)        -2.5078     3.1514
    Variable  6 (Level  3)        -2.6138     3.4600

  Subject:  Variable 11 (Level  2) Variable 12 (Level  3)
    Variable  5 (Level  1)         4.3331     3.1489
    Variable  5 (Level  2)        -5.6142     3.1649
    Variable  6 (Level  1)        -5.8804     3.1770
    Variable  6 (Level  2)         5.4265     3.3006
    Variable  6 (Level  3)        -2.1917     3.2156

  Subject:  Variable 11 (Level  3) Variable 12 (Level  3)
    Variable  5 (Level  1)         0.4305     2.9144
    Variable  5 (Level  2)        -1.4620     3.0119
    Variable  6 (Level  1)        14.3595     3.9254
    Variable  6 (Level  2)        -5.2399     3.3099
    Variable  6 (Level  3)       -11.2498     3.2212

  Subject:  Variable 12 (Level  3)
    Variable  7                   -0.3839     0.6755
    Variable  8 (Level  1)         2.7549     1.6017
    Variable  8 (Level  2)         0.4377     1.8826
    Variable  8 (Level  3)        -0.2261     1.9909
    Variable  8 (Level  4)        -4.5051     1.5398
    Variable  9 (Level  1)        -4.7091     2.1458
    Variable  9 (Level  2)         3.7940     1.9872
    Variable  9 (Level  3)        -1.7994     1.8614
    Variable  9 (Level  4)         0.4480     1.9016
    Variable  9 (Level  5)        -0.6047     2.4729

Fixed Effects
    Intercept                      1.6433     2.4596
    Variable  1 (Level  2)        -1.6224     0.8549
    Variable  2 (Level  2)        -2.4817     1.1414
    Variable  2 (Level  3)         0.4624     1.2133

Variance Components
  Estimate     Parameter       Subject
  36.32491     Variable  3     Variables 10 11 12 
  12.45090     Variable  4     Variables 10 11 12 
  19.62767     Variable  5     Variables 11 12 
  40.53480     Variable  6     Variables 11 12 
   0.56320     Variable  7     Variables 12 
   5.81968     Variable  8     Variables 12 
  10.86069     Variable  9     Variables 12 

sigma^2          =         0.00239
-2log likelihood =       608.19449
PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox
© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015