naginterfaces.library.correg.mixeff_​reml

naginterfaces.library.correg.mixeff_reml(dat, levels, yvid, cwid, fvid, fint, rvid, nvpr, vpr, rint, svid, gamma, lb, maxit=- 1, tol=0.0, io_manager=None)[source]

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

Deprecated since version 27.0.0.0: mixeff_reml is deprecated. Please use lmm_init() followed by lmm_fit() instead. See also the Replacement Calls document.

For full information please refer to the NAG Library document for g02ja

https://www.nag.com/numeric/nl/nagdoc_27.1/flhtml/g02/g02jaf.html

Parameters
datfloat, array-like, shape

Array containing all of the data. For the th observation:

holds the dependent variable, ;

if , holds the case weights;

if , holds the subject variable.

The remaining columns hold the values of the independent variables.

levelsint, array-like, shape

contains the number of levels associated with the th variable of the data matrix . If this variable is continuous or binary (i.e., only takes the values zero or one) then should be ; if the variable is discrete then is the number of levels associated with it and is assumed to take the values to , for .

yvidint

The column of holding the dependent, , variable.

cwidint

The column of holding the case weights.

If , no weights are used.

fvidint, array-like, shape

The columns of the data matrix holding the fixed independent variables with holding the column number corresponding to the th fixed variable.

fintint

Flag indicating whether a fixed intercept is included ().

rvidint, array-like, shape

The columns of the data matrix holding the random independent variables with holding the column number corresponding to the th random variable.

nvprint

If and , is the number of variance components being , (), else .

If , is not referenced.

vprint, array-like, shape

holds a flag indicating the variance of the th random variable. The variance of the th random variable is , where if and and otherwise. Random variables with the same value of are assumed to be taken from the same distribution.

rintint

Flag indicating whether a random intercept is included ().

If , is not referenced.

svidint

The column of holding the subject variable.

If , 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 denote the interaction between variables and , fitting a model with , random-effects and subject variable is equivalent to fitting a model with random-effects and no subject variable.

If the model is equivalent to fitting and no subject variable.

gammafloat, array-like, shape

Holds the initial values of the variance components, , with the initial value for , for . If and , , else .

If , the remaining elements of are ignored and the initial values for the variance components are estimated from the data using MIVQUE0.

lbint

The size of the array .

maxitint, optional

The maximum number of iterations.

If , the default value of is used.

If , the parameter estimates and corresponding standard errors are calculated based on the value of supplied in .

tolfloat, optional

The tolerance used to assess convergence.

If , the default value of is used, where is the machine precision.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns
gammafloat, ndarray, shape

, for , holds the final estimate of and holds the final estimate for .

nffint

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

nrfint

The number of random effects estimated (i.e., the number of columns, , in the design matrix ).

dfint

The degrees of freedom.

remlfloat

where is the log of the restricted maximum likelihood calculated at , the estimated variance components returned in .

bfloat, ndarray, shape

The parameter estimates, , with the first elements of containing the fixed effect parameter estimates, and the next elements of containing the random effect parameter estimates, .

Fixed effects

If , contains the estimate of the fixed intercept.

Let denote the number of levels associated with the th fixed variable, that is .

Define

if , else if , ;

, .

Then for :

if , contains the parameter estimate for the th level of the th fixed variable, for ;

if , contains the parameter estimate for the th fixed variable.

Random effects

Redefining to denote the number of levels associated with the th random variable, that is .

Define

if , else if , ;

, .

Then for :

if ,

if , contains the parameter estimate for the th level of the th random variable, for ;

if , contains the parameter estimate for the th random variable;

if ,

let denote the number of levels associated with the subject variable, that is ;

if , contains the parameter estimate for the interaction between the th level of the subject variable and the th level of the th random variable, for , for ;

if , contains the parameter estimate for the interaction between the th level of the subject variable and the th random variable, for ;

if , contains the estimate of the random intercept.

sefloat, ndarray, shape

The standard errors of the parameter estimates given in .

warnint

Is set to if a variance component was estimated to be a negative value during the fitting process. Otherwise is set to .

If , the negative estimate is set to zero and the estimation process allowed to continue.

Raises
NagValueError
(errno )

On entry, .

Constraint: .

(errno )

On entry, .

Constraint: number of observations with nonzero weights must be greater than one.

(errno )

On entry, .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: and .

(errno )

On entry, and .

Constraint: and ( or ).

(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: .

(errno )

On entry, and .

Constraint: and any supplied weights must be .

(errno )

On entry, .

Constraint: or .

(errno )

On entry, .

Constraint: or .

(errno )

On entry, too small: .

(errno )

On entry, , for at least one .

(errno )

On entry, .

Constraint: , for all .

(errno )

On entry, .

Constraint: , for all .

(errno )

On entry, .

Constraint: , for all .

(errno )

On entry, invalid data: categorical variable with value greater than that specified in .

(errno )

On entry, , for at least one .

(errno )

Degrees of freedom : .

(errno )

Routine failed to converge to specified tolerance: .

(errno )

Routine failed to converge in iterations: .

Notes

mixeff_reml fits a model of the form:

where

is a vector of observations on the dependent variable,

is a known design matrix for the fixed independent variables,

is a vector of length of unknown fixed effects,

is a known design matrix for the random independent variables,

is a vector of length of unknown random effects,

and

is a vector of length of unknown random errors.

Both and are assumed to have a Gaussian distribution with expectation zero and

where , is the identity matrix and is a diagonal matrix. It is assumed that the random variables, , can be subdivided into groups with each group being identically distributed with expectations zero and variance . The diagonal elements of matrix , therefore, take one of the values , depending on which group the associated random variable belongs to.

The model, therefore, contains three sets of unknowns, the fixed effects, , the random effects and a vector of variance components, , where . Rather than working directly with , mixeff_reml uses an iterative process to estimate . Due to the iterative nature of the estimation a set of initial values, , for is required. mixeff_reml 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).

mixeff_reml fits the model using a quasi-Newton algorithm to maximize the restricted log-likelihood function:

where

Once the final estimates for have been obtained, the value of is given by:

Case weights, , can be incorporated into the model by replacing and with and respectively, for a diagonal weight matrix .

The log-likelihood, , is calculated using the sweep algorithm detailed in Wolfinger et al. (1994).

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