NAG CL Interface
g02jac (mixeff_​reml)

Note: this function is deprecated. Replaced by g02jfc followed by g02jhc.
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1 Purpose

g02jac fits a linear mixed effects regression model using restricted maximum likelihood (REML).

2 Specification

#include <nag.h>
void  g02jac (Integer n, Integer ncol, const double dat[], Integer tddat, const Integer levels[], Integer yvid, Integer cwid, Integer nfv, const Integer fvid[], Integer fint, Integer nrv, const Integer rvid[], Integer nvpr, const Integer vpr[], Integer rint, Integer svid, double gamma[], Integer *nff, Integer *nrf, Integer *df, double *reml, Integer lb, double b[], double se[], Integer maxit, double tol, Integer *warn, NagError *fail)
The function may be called by the names: g02jac, nag_correg_mixeff_reml or nag_reml_mixed_regsn.

3 Description

g02jac fits a model of the form:
y=Xβ+Zν+ε  
where and
Both ν and ε are assumed to have a Gaussian distribution with expectation zero and
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. It is assumed that the random variables, Z , can be subdivided into g q groups with each group being identically distributed with expectations zero and variance σi2 . The diagonal elements of matrix G , therefore, take one of the values {σi2:i=1,2,,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 = {σ12,σ22,, σ g-1 2 ,σg2,σR2} . Rather than working directly with γ , g02jac uses an iterative process to estimate γ* = { σ12 / σR2 , σ22 / σR2 ,, σg-12 / σR2 , σg2 / σR2 ,1} . Due to the iterative nature of the estimation a set of initial values, γ0 , for γ* is required. g02jac 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).
g02jac fits the model using a quasi-Newton algorithm to maximize the restricted log-likelihood function:
−2 l R = log(|V|) + (n-p) log( r V-1r) + log| X V-1X| + (n-p) (1+log(2π/(n-p)))  
where
V = ZG Z + R,   r=y-Xb   and   b = ( X V-1X) −1 X V-1 y .  
Once the final estimates for γ * have been obtained, the value of σR2 is given by:
σR2 = (rV-1r) / (n-p) .  
Case weights, Wc , can be incorporated into the model by replacing XX and ZZ with XWcX and ZWcZ respectively, for a diagonal weight matrix Wc .
The log-likelihood, lR, is calculated using the sweep algorithm detailed in Wolfinger et al. (1994).

4 References

Goodnight J H (1979) A tutorial on the SWEEP operator The American Statistician 33(3) 149–158
Harville D A (1977) Maximum likelihood approaches to variance component estimation and to related problems JASA 72 320–340
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 Disciplines Southern 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

5 Arguments

1: n Integer Input
On entry: n, the number of observations.
Constraint: n1.
2: ncol Integer Input
On entry: the number of columns in the data matrix, DAT.
Constraint: ncol1.
3: dat[n×tddat] const double Input
Note: where DAT(i,j) appears in this document, it refers to the array element dat[(i-1)×tddat+j-1].
On entry: array containing all of the data. For the ith observation:
  • DAT(i,yvid) holds the dependent variable, y;
  • if cwid0, DAT(i,cwid) holds the case weights;
  • if svid0, DAT(i,svid) holds the subject variable.
The remaining columns hold the values of the independent variables.
Constraints:
  • if cwid0, DAT(i,cwid)0.0;
  • if levels[j-1]1, 1DAT(i,j)levels[j-1].
4: tddat Integer Input
On entry: the stride separating matrix column elements in the array dat.
Constraint: tddatncol.
5: levels[ncol] const Integer Input
On entry: levels[i-1] contains the number of levels associated with the ith variable of the data matrix DAT. If this variable is continuous or binary (i.e., only takes the values zero or one) then levels[i-1] should be 1; if the variable is discrete then levels[i-1] is the number of levels associated with it and DAT(j,i) is assumed to take the values 1 to levels[i-1], for j=1,2,,n.
Constraint: levels[i-1]1, for i=1,2,,ncol.
6: yvid Integer Input
On entry: the column of DAT holding the dependent, y, variable.
Constraint: 1yvidncol.
7: cwid Integer Input
On entry: the column of DAT holding the case weights.
If cwid=0, no weights are used.
Constraint: 0cwidncol.
8: nfv Integer Input
On entry: the number of independent variables in the model which are to be treated as being fixed.
Constraint: 0nfv<ncol.
9: fvid[nfv] const Integer Input
On entry: the columns of the data matrix DAT holding the fixed independent variables with fvid[i-1] holding the column number corresponding to the i th fixed variable.
Constraint: 1fvid[i-1]ncol, for i=1,2,,nfv.
10: fint Integer Input
On entry: flag indicating whether a fixed intercept is included (fint=1).
Constraint: fint=0 or 1.
11: nrv Integer Input
On entry: the number of independent variables in the model which are to be treated as being random.
Constraints:
  • 0nrv<ncol;
  • nrv+rint>0.
12: rvid[nrv] const Integer Input
On entry: the columns of the data matrix dat holding the random independent variables with rvid[i-1] holding the column number corresponding to the i th random variable.
Constraint: 1rvid[i-1]ncol, for i=1,2,,nrv.
13: nvpr Integer Input
On entry: if rint=1 and svid0, nvpr is the number of variance components being estimated-2, (g-1), else nvpr=g.
If nrv=0, nvpr is not referenced.
Constraint: if nrv0, 1nvprnrv.
14: vpr[nrv] const Integer Input
On entry: vpr[i-1] holds a flag indicating the variance of the i th random variable. The variance of the i th random variable is σ j 2 , where j = vpr[i-1] + 1 if rint=1 and svid0 and j = vpr[i-1] otherwise. Random variables with the same value of j are assumed to be taken from the same distribution.
Constraint: 1vpr[i-1]nvpr, for i=1,2,,nrv.
15: rint Integer Input
On entry: flag indicating whether a random intercept is included (rint=1).
If svid=0, rint is not referenced.
Constraint: rint=0 or 1.
16: svid Integer Input
On entry: the column of DAT holding the subject variable.
If svid=0, no subject variable is used.
Specifying a subject variable is equivalent to specifying the interaction between that variable and all of the random-effects. Letting the notation Z1 × ZS denote the interaction between variables Z1 and ZS , fitting a model with rint = 0 , random-effects Z1 + Z2 and subject variable ZS is equivalent to fitting a model with random-effects Z1 × ZS + Z2 × ZS and no subject variable. If rint = 1 the model is equivalent to fitting ZS + Z1 × ZS + Z2 × ZS and no subject variable.
Constraint: 0svidncol.
17: gamma[nvpr+2] double Input/Output
On entry: holds the initial values of the variance components, γ0 , with gamma[i-1] the initial value for σi2/σR2, for i=1,2,,g. If rint=1 and svid0, g=nvpr+1, else g=nvpr.
If gamma[0]=-1.0, the remaining elements of gamma are ignored and the initial values for the variance components are estimated from the data using MIVQUE0.
On exit: gamma[i-1], for i=1,2,,g, holds the final estimate of σi2 and gamma[g] holds the final estimate for σR2.
Constraint: gamma[0]=-1.0 or gamma[i-1]0.0, for i=1,2,,g.
18: nff Integer * Output
On exit: the number of fixed effects estimated (i.e., the number of columns, p, in the design matrix X).
19: nrf Integer * Output
On exit: the number of random effects estimated (i.e., the number of columns, q, in the design matrix Z).
20: df Integer * Output
On exit: the degrees of freedom.
21: reml double * Output
On exit: - 2 lR (γ^) where lR is the log of the restricted maximum likelihood calculated at γ^ , the estimated variance components returned in gamma.
22: lb Integer Input
On entry: the size of the array b.
Constraint: lb fint + i=1 nfv max(levels[fvid[i-1]-1]-1,1) + LS × (rint+ i=1 nrv levels[rvid[i-1]-1]) where LS = levels[svid-1] if svid0 and 1 otherwise.
23: b[lb] double Output
On exit: the parameter estimates, (β,ν), with the first nff elements of b containing the fixed effect parameter estimates, β and the next nrf elements of b containing the random effect parameter estimates, ν.
Fixed effects
If fint=1, b[0] contains the estimate of the fixed intercept. Let Li denote the number of levels associated with the ith fixed variable, that is Li = levels[fvid[i-1]-1] . Define
  • if fint=1, F1 = 2 else if fint=0, F1=1 ;
  • F i+1 = Fi + max(Li-1,1) , i1 .
Then for i=1,2,,nfv:
  • if Li > 1 , b[Fi+j-3] contains the parameter estimate for the jth level of the ith fixed variable, for j=2,3,,Li;
  • if Li 1 , b[Fi-1] contains the parameter estimate for the ith fixed variable.
Random effects
Redefining Li to denote the number of levels associated with the ith random variable, that is Li = levels[rvid[i-1]-1] . Define
  • if rint=1, R1 = 2 else if rint=0, R1=1 ;
    R i+1 = Ri + Li , i1 .
Then for i = 1 , 2 , , nrv :
  • if svid=0,
    • if Li > 1 , b[nff+Ri+j-2] contains the parameter estimate for the jth level of the ith random variable, for j=1,2,,Li;
    • if Li 1 , b[nff+Ri-1] contains the parameter estimate for the ith random variable;
  • if svid 0 ,
    • let LS denote the number of levels associated with the subject variable, that is LS = levels[svid-1] ;
    • if Li > 1 , b[nff+(s-1)LS+Ri+j-2] contains the parameter estimate for the interaction between the sth level of the subject variable and the jth level of the ith random variable, for s=1,2,,LS and j=1,2,,Li;
    • if Li 1 , b[nff+(s-1)LS+Ri-1] contains the parameter estimate for the interaction between the sth level of the subject variable and the ith random variable, for s=1,2,,LS;
    • if rint=1, b[nff] contains the estimate of the random intercept.
24: se[lb] double Output
On exit: the standard errors of the parameter estimates given in b.
25: maxit Integer Input
On entry: the maximum number of iterations.
If maxit < 0 , the default value of 100 is used.
If maxit=0, the parameter estimates (β,ν) and corresponding standard errors are calculated based on the value of γ0 supplied in gamma.
26: tol double Input
On entry: the tolerance used to assess convergence.
If tol0.0, the default value of ε0.7 is used, where ε is the machine precision.
27: warn Integer * Output
On exit: is set to 1 if a variance component was estimated to be a negative value during the fitting process. Otherwise warn is set to 0 .
If warn=1, the negative estimate is set to zero and the estimation process allowed to continue.
28: 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_BAD_PARAM
On entry, argument value had an illegal value.
On entry, invalid data: categorical variable with value greater than that specified in levels.
NE_CONV
Routine failed to converge in maxit iterations: maxit=value.
See Section 10 for advice.
NE_FAIL_TOL
Routine failed to converge to specified tolerance: tol=value.
See Section 10 for advice.
NE_INT
On entry, fint=value.
Constraint: fint=0 or 1.
On entry, lb too small: lb=value.
On entry, levels[i]<1, for at least one i.
On entry, n=value.
Constraint: number of observations with nonzero weights must be greater than one.
On entry, n=value.
Constraint: n1.
On entry, ncol=value.
Constraint: 1fvid[i]ncol, for all i.
On entry, ncol=value.
Constraint: 1rvid[i]ncol, for all i.
On entry, ncol=value.
Constraint: ncol1.
On entry, nvpr=value.
Constraint: 1vpr[i]nvpr, for all i.
On entry, rint=value.
Constraint: rint=0 or 1.
NE_INT_2
On entry, cwid=value and ncol=value.
Constraint: 0cwidncol and any supplied weights must be 0.0.
On entry, nfv=value and ncol=value.
Constraint: 0nfv<ncol.
On entry, nrv=value and ncol=value.
Constraint: 0nrv<ncol and nrv+rint>0.
On entry, nvpr=value and nrv=value.
Constraint: 0nvprnrv and (nrv0 or nvpr1).
On entry, svid=value and ncol=value.
Constraint: 0svidncol.
On entry, tddat=value and ncol=value.
Constraint: tddatncol.
On entry, yvid=value and ncol=value.
Constraint: 1yvidncol.
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
On entry, gamma[i]<0.0, for at least one i.
NE_ZERO_DOF_ERROR
Degrees of freedom <1: df=value.
This is due to the number of parameters exceeding the effective number of observations.

7 Accuracy

The accuracy of the results can be adjusted through the use of the tol argument.

8 Parallelism and Performance

g02jac is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02jac 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

Wherever possible any block structure present in the design matrix Z should be modelled through a subject variable, specified via svid, rather than being explicitly entered into dat.
g02jac uses an iterative process to fit the specified model and for some problems this process may fail to converge (see fail.code= NE_CONV or NE_FAIL_TOL). If the function fails to converge then the maximum number of iterations (see maxit) or tolerance (see tol) may require increasing; try a different starting estimate in gamma. Alternatively, the model can be fit using maximum likelihood (see g02jbc) or using the noniterative MIVQUE0.
To fit the model just using MIVQUE0, the first element of gamma should be set to -1.0 and maxit should be set to zero.
Although the quasi-Newton algorithm used in g02jac tends to require more iterations before converging compared to the Newton–Raphson algorithm recommended by Wolfinger et al. (1994), it does not require the second derivatives of the likelihood function to be calculated and consequentially takes significantly less time per iteration.

10 Example

The following dataset is taken from Stroup (1989) and arises from a balanced split-plot design with the whole plots arranged in a randomized complete block-design.
In this example the full design matrix for the random independent variable, Z , is given by:
Z = ( 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 )  
= ( A 0 0 0 0 A 0 0 0 0 A 0 0 0 0 A A 0 0 0 0 A 0 0 0 0 A 0 0 0 0 A ) , (1)
where
A = ( 1 1 0 0 1 0 1 0 1 0 0 1 ) .  
The block structure evident in (1) is modelled by specifying a four-level subject variable, taking the values {1,1,1,2,2,2,3,3,3,4,4,4,1,1,1,2,2,2,3,3,3,4,4,4} . The first column of 1s is added to A by setting rint=1. The remaining columns of A are specified by a three level factor, taking the values, {1,2,3,1,2,3,1,} .

10.1 Program Text

Program Text (g02jace.c)

10.2 Program Data

Program Data (g02jace.d)

10.3 Program Results

Program Results (g02jace.r)