Restricted maximum likelihood
Encyclopedia
In statistics
, the restricted (or residual, or reduced) maximum likelihood (REML) approach is a particular form of maximum likelihood
estimation which does not base estimates on a maximum likelihood fit of all the information, but instead uses a likelihood function
calculated from a transformed set of data, so that nuisance parameters have no effect.
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
s. In contrast to the earlier maximum likelihood
estimation, REML can produce unbiased 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 and Robin Thompson
of the University of Edinburgh
, 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 packages, including Genstat
(the REML directive), SAS (the MIXED procedure), SPSS
(the MIXED command), Stata
(the xtmixed command), and R
(the lme4 and older nlme packages),
as well as in more specialist packages such as MLwiN
, HLM
, ASReml
, Statistical Parametric Mapping
and CropStat.
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
, the restricted (or residual, or reduced) maximum likelihood (REML) approach is a particular form of maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
estimation which does not base estimates on a maximum likelihood fit of all the information, but instead uses a likelihood function
Likelihood function
In statistics, a likelihood function is a function of the parameters of a statistical model, defined as follows: the likelihood of a set of parameter values given some observed outcomes is equal to the probability of those observed outcomes given those parameter values...
calculated from a transformed set of data, so that nuisance parameters have no effect.
In the case of variance component estimation, the original data set is replaced by a set of contrasts
Contrast (statistics)
In statistics, particularly analysis of variance, a contrast is a linear combination of two or more factor level means whose coefficients add up to zero. A simple contrast is the difference between two means...
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
Mixed model
A mixed model is a statistical model containing both fixed effects and random effects, that is mixed effects. These models are useful in a wide variety of disciplines in the physical, biological and social sciences....
s. In contrast to the earlier maximum likelihood
Maximum likelihood
In statistics, maximum-likelihood estimation is a method of estimating the parameters of a statistical model. When applied to a data set and given a statistical model, maximum-likelihood estimation provides estimates for the model's parameters....
estimation, REML can produce unbiased estimates of variance and covariance parameters.
The idea underlying REML estimation was put forward by M. S. Bartlett
M. S. Bartlett
Maurice Stevenson Bartlett FRS was an English statistician who made particular contributions to the analysis of data with spatial and temporal patterns...
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
University of Edinburgh
The University of Edinburgh, founded in 1583, is a public research university located in Edinburgh, the capital of Scotland, and a UNESCO World Heritage Site. The university is deeply embedded in the fabric of the city, with many of the buildings in the historic Old Town belonging to the university...
, 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 packages, including Genstat
GenStat
GenStat is a general statistical package. Early versions were developed for large mainframe computers. Up until version 5, there was a Unix binary available, and this continues to be used by many universities and research institutions...
(the REML directive), SAS (the MIXED procedure), SPSS
SPSS
SPSS is a computer program used for survey authoring and deployment , data mining , text analytics, statistical analysis, and collaboration and deployment ....
(the MIXED command), Stata
Stata
Stata is a general-purpose statistical software package created in 1985 by StataCorp. It is used by many businesses and academic institutions around the world...
(the xtmixed command), and R
R (programming language)
R is a programming language and software environment for statistical computing and graphics. The R language is widely used among statisticians for developing statistical software, and R is widely used for statistical software development and data analysis....
(the lme4 and older nlme packages),
as well as in more specialist packages such as MLwiN
MLwiN
MLwiN is a statistical software package for fitting multilevel models. It uses both maximum likelihood estimation and Markov Chain Monte Carlo methods...
, HLM
HLM
HLM , French for "housing at moderated rents" or "rent-controlled housing", is a form of subsidised housing in France. There are approximately four million such residences, housing an estimated 12 million people — nearly one-fifth of the population of France...
, ASReml
ASReml
ASReml is a statistical software package for fitting linear mixed models using restricted maximum likelihood, a technique commonly used in plant and animal breeding and quantitative genetics as well as other fields...
, Statistical Parametric Mapping
Statistical parametric mapping
Statistical parametric mapping or SPM is a statistical technique created by Karl Friston for examining differences in brain activity recorded during functional neuroimaging experiments using neuroimaging technologies such as fMRI or PET...
and CropStat.