Mixed model

A mixed model ( English mixed model is) a statistical model that includes both fixed effects as well as random effects contains so mixed effects . These models are used in various areas of physics, biology and the social sciences. They are particularly useful when performing repeated measurements on the same statistical unit or measurements on clusters of related statistical units.

History and current state of research

Ronald Fisher introduced the random effects model to study correlations of distinctive features between relatives. In the 1950s, Charles Roy Henderson designed the best linear unexpectable estimators for fixed effects and best linear unexpectant predictions (BLEV) for random effects. Mixed modeling then became one of the main fields of statistical research, including work on the computation of maximum likelihood estimators , nonlinear mixed effects models, missing data in mixed models, and Bayesian estimates of mixed models. Mixed models are used in many disciplines, especially when different correlated measurements are made on each unit under investigation. They are particularly widely used in research on humans or animals, with uses ranging from genetics to marketing.

definition

In matrix notation , a mixed model can be represented as:

${\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+\mathbf {Z} \mathbf {u} +{\boldsymbol {\varepsilon }}}$,

in which

• ${\displaystyle \mathbf {y} }$is a vector of observations of the dependent variable, with expected value ${\displaystyle \operatorname {E} (\mathbf {y} )=\mathbf {X} {\boldsymbol {\beta }}}$

• ${\displaystyle {\boldsymbol {\beta }}}$ is a solid effects vector

• ${\displaystyle {\boldsymbol {u}}}$a vector of random effects is with expectation and variance- covariance matrix ${\displaystyle \operatorname {E} (\mathbf {u} )=\mathbf {0} }$ ${\displaystyle \operatorname {Cov} (\mathbf {u} )=\mathbf {G} }$

• ${\displaystyle {\boldsymbol {\varepsilon }}}$is a vector of random error terms with expectation and variance-covariance matrix ${\displaystyle \operatorname {E} ({\boldsymbol {\varepsilon }})=\mathbf {0} }$ ${\displaystyle \operatorname {Cov} ({\boldsymbol {\varepsilon }})=\mathbf {R} }$

• ${\displaystyle \mathbf {X} }$and are matrices with regressors linking the observations with and ${\displaystyle \mathbf {Z} }$ ${\displaystyle \mathbf {y} }$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle \mathbf {u} }$

estimate

The Henderson'schen mixed model equations ( English mixed model equations , in short: MME ) are:

${\displaystyle {\begin{pmatrix}\mathbf {X} '\mathbf {R} ^{-1}\mathbf {X} &\mathbf {X} '\mathbf {R} ^{-1}\mathbf {Z} \\\mathbf {Z} '\mathbf {R} ^{-1}\mathbf {X} &\mathbf {Z} '\mathbf {R} ^{-1}\mathbf {Z} +\mathbf {G} ^{-1}\end{pmatrix}}{\begin{pmatrix}{\tilde {\boldsymbol {\beta }}}\\{\tilde {\boldsymbol {u}}}\end{pmatrix}}={\begin{pmatrix}\mathbf {X} '\mathbf {R} ^{-1}\mathbf {y} \\\mathbf {Z} '\mathbf {R} ^{-1}\mathbf {y} \end{pmatrix}}}$

The solutions of the mixed model equations and are best linear unbiased estimator ( BLES or English Best Linear Unbiased Estimator , in short: BLUE ) for respectively . This follows from the Gauss-Markow theorem , since the conditional variance of the result cannot be scaled to the identity matrix . If the conditional variance is known, the inverse variance weighted least squares estimator is BLES. However, the conditional variance is rarely known, so that when solving the mixed model equations it is desirable to estimate the variance and the weighted parameter estimates together. ${\displaystyle {\tilde {\boldsymbol {\beta }}}}$ ${\displaystyle {\tilde {\boldsymbol {u}}}}$ ${\displaystyle {\boldsymbol {\beta }}}$ ${\displaystyle {\boldsymbol {u}}}$

One method for fitting mixed models is the EM algorithm , in which the components of the variance are treated as unobserved disturbance parameters in the total probability. This method is currently implemented in the most important statistical software packages R (lme () in the nlme package and lmer () in the lme4 package) and SAS (proc mixed). The solution to the mixed model equations is a maximum likelihood estimate if the errors are normally distributed .

Individual evidence

1. ^ RA Fisher: The correlation between relatives on the supposition of Mendelian inheritance . In: Transactions of the Royal Society of Edinburgh . 52, 1918, pp. 399-433.
2. ↑ ^{a b} G.K. Robinson: That BLUP is a Good Thing: The Estimation of Random Effects . In: Statistical Science . 6, No. 1, 1991, pp. 15-32. JSTOR 2245695 . doi : 10.1214 / ss / 1177011926 .
3. ↑ CR Henderson, Oscar Kempthorne, SR Searle and CM von Krosigk: The Estimation of Environmental and Genetic Trends from Records Subject to Culling . In: International Biometric Society (Ed.): Biometrics . 15, No. 2, 1959, pp. 192-218. JSTOR 2527669 . doi : 10.2307 / 2527669 .
4. ^ ^{A b} L. Dale Van Vleck: Charles Roy Henderson, April 1, 1911 - March 14, 1989 . United States National Academy of Sciences . Retrieved May 28, 2012.
5. ^ Robert A. McLean, Sanders, William L .; Stroup, Walter W .: A Unified Approach to Mixed Linear Models . In: American Statistical Association (Ed.): The American Statistician . 45, No. 1, 1991, pp. 54-64. JSTOR 2685241 . doi : 10.2307 / 2685241 .
6. ↑ ML Lindstrom, Bates, DM: Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data . In: JASA . 83, No. 404, 1988, pp. 1014-1021.
7. ↑ Nan M. Laird, Ware, James H .: Random-Effects Models for Longitudinal Data . In: International Biometric Society (Ed.): Biometrics . 38, No. 4, 1982, pp. 963-974. doi : 10.2307 / 2529876 . PMID 7168798 .

further reading

• GA Milliken, DE Johnson: Analysis of messy data: Vol. I. Designed experiments. Chapman & Hall, New York 1992.
• BT West, KB Welch, AT Galecki: Linear mixed models: A practical guide to using statistical software. Chapman & Hall / CRC. New York 2007.