In statistics, 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 estimating the parameters of an assumed probability distribution, given some observed data. This is achieved by maximizing a likelihood function so that, under the assumed sta ...

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 parameters have no effect. (see REML) In the case of variance component 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 estimates of variance and covariance parameters. The idea underlying REML estimation was put forward by

M. S. Bartlett Maurice Stevenson Bartlett FRS (18 June 1910 – 8 January 2002) was an English statistician who made particular contributions to the analysis of data with spatial and temporal patterns. He is also known for his work in the theory of statis ...

in 1937. The first description of the approach applied to estimating components of variance in unbalanced data was by Desmond Patterson and Robin Thompson 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 softwa ...

packages, including Genstat (the REML directive),

SAS SAS or Sas may refer to: Arts, entertainment, and media * ''SAS'' (novel series), a French book series by Gérard de Villiers * ''Shimmer and Shine'', an American animated children's television series * Southern All Stars, a Japanese rock ba ...

(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. Cur ...

(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 fiel ...

(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 sys ...

, and R (especially th
lme4
and olde

packages), as well as in more specialist packages such as MLwiN
HLM
ASReml, BLUPF90
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 th ...

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, implementa ...

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