statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the restricted (or residual, or reduced) maximum likelihood (REML) approach is a particular form of

maximum likelihood In statistics, maximum likelihood estimation (MLE) is a method of estimation theory, estimating the Statistical parameter, parameters of an assumed probability distribution, given some observed data. This is achieved by Mathematical optimization, ...

estimation that does not base estimates on a maximum likelihood fit of all the information, but instead uses a

likelihood function The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood funct ...

calculated from a transformed set of data, so that

nuisance parameter Nuisance (from archaic ''nocence'', through Fr. ''noisance'', ''nuisance'', from Lat. ''nocere'', "to hurt") is a common law tort. It means that which causes offence, annoyance, trouble or injury. A nuisance can be either public (also "common") ...

s have no effect. (see REML) In the case of

variance component In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are ...

estimation, the original data set is replaced by a set of contrasts calculated from the data, and the likelihood function is calculated from the probability distribution of these contrasts, according to the model for the complete data set. In particular, REML is used as a method for fitting linear

mixed model A mixed model, mixed-effects model or mixed error-component model is a statistical model containing both fixed effects and random effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences. ...

s. In contrast to the earlier

estimation, REML can produce

unbiased Bias is a disproportionate weight ''in favor of'' or ''against'' an idea or thing, usually in a way that is closed-minded, prejudicial, or unfair. Biases can be innate or learned. People may develop biases for or against an individual, a group, ...

estimates of variance and covariance parameters. The idea underlying REML estimation was put forward by M. S. Bartlett in 1937. The first description of the approach applied to estimating components of variance in unbalanced data was by

Desmond Patterson Desmond or Desmond's may refer to: Arts and entertainment * ''Desmond'' (novel), 1792 novel by Charlotte Turner Smith * ''Desmond's'', 1990s British television sitcom Ireland * Kingdom of Desmond, medieval Irish kingdom * Earl of Desmond, Irish a ...

and

Robin Thompson Robin Henderson Thompson (5 May 1931 – 14 August 2003) was a rugby union international for Ireland, a former British Lions captain and rugby league player. Robin Thompson won two Ulster Schools Senior Cup medals with RBAI (Royal Belfast Acad ...

of the

University of Edinburgh The University of Edinburgh ( sco, University o Edinburgh, gd, Oilthigh Dhùn Èideann; abbreviated as ''Edin.'' in post-nominals) is a public research university based in Edinburgh, Scotland. Granted a royal charter by King James VI in 15 ...

in 1971, although they did not use the term REML. A review of the early literature was given by Harville. REML estimation is available in a number of general-purpose

statistical software Statistical software are specialized computer programs for analysis in statistics and econometrics. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a software ...

packages, including

Genstat Genstat (General Statistics) is a statistical software package with data analysis capabilities, particularly in the field of agriculture. It was developed in 1968 by the Rothamsted Research in the United Kingdom and was designed to provide modul ...

(the REML directive), SAS (the MIXED procedure),

SPSS SPSS Statistics is a statistical software suite developed by IBM for data management, advanced analytics, multivariate analysis, business intelligence, and criminal investigation. Long produced by SPSS Inc., it was acquired by IBM in 2009. C ...

(the MIXED command),

Stata Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...

(the mixed command),

JMP (statistical software) JMP (pronounced "jump") is a suite of computer programs for statistical analysis developed by JMP, a subsidiary of SAS Institute. It was launched in 1989 to take advantage of the graphical user interface introduced by the Macintosh operating s ...

, and R (especially th
lme4
and olde

packages), as well as in more specialist packages such as MLwiN
HLM
ASReml,

BLUPF90 The BLUPF90 family of programs is a statistical software package used in quantitative genetics for animal and plant breeding. It can fit mixed models using restricted maximum likelihood as well as Gibbs sampling to estimate variance components, and ...

wombat

Statistical Parametric Mapping Statistical parametric mapping (SPM) is a statistical technique for examining differences in brain activity recorded during functional neuroimaging experiments. It was created by Karl Friston. It may alternatively refer to software created by the ...

an
CropStat
REML estimation is implemented i
Surfstat
a

Matlab MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation ...

toolbox for the statistical analysis of univariate and multivariate surface and volumetric neuroimaging data using linear mixed effects models and random field theory, but more generally in the fitlme package for modeling linear mixed effects models in a domain-general way.

References

Maximum likelihood estimation {{statistics-stub