statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...

, 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 stati ...

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

likelihood function A likelihood function (often simply called the likelihood) measures how well a statistical model explains observed data by calculating the probability of seeing that data under different parameter values of the model. It is constructed from the ...

calculated from a transformed set of data, so that

nuisance parameter In statistics, a nuisance parameter is any parameter which is unspecified but which must be accounted for in the hypothesis testing of the parameters which are of interest. The classic example of a nuisance parameter comes from the normal distri ...

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

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 (, ; abbreviated as ''Edin.'' in Post-nominal letters, post-nominals) is a Public university, public research university based in Edinburgh, Scotland. Founded by the City of Edinburgh Council, town council under th ...

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 The following is a list of statistical software. Open-source * ADaMSoft – a generalized statistical software with data mining algorithms and methods for data management * ADMB – a software suite for non-linear statistical modeling based on C+ ...

packages, including Genstat (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. Versi ...

(the MIXED command),

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

(the mixed command), JMP (statistical software), 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 t ...

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, implementat ...

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