Mixed model

A mixed model (also mixed model ) is a statistical model containing both fixed effects and random effects, ie mixed effects. These models are applied in various fields of physics, biology and the social sciences. They are particularly useful if a repeated measurement on the same statistical measurements or unit of clusters of related statistical units are performed.

History and current state of research

Ronald Fisher introduced the random effects a model to investigate correlations of characteristics between relatives. In the fifties, Charles Roy Henderson BLUE designed for fixed effects and BLUP for random effects. The mixture was then mixed modeling one of the main research areas of statistical research, including work on the calculation of maximum likelihood estimators, non- linear mixed - effects models, missing data in mixed models and Bayesian estimates of mixed - effects models. Mixed models are used in many disciplines, especially when different correlated measurements are made on each unit to be examined. They are most often used in research on humans or animals, the range of applications of genetics sufficient to marketing.

Definition

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

In which

• A vector of observations, with mean

• A vector of fixed effects

• A vector of independent and identically distributed ( IID) random effects with mean and variance -covariance matrix

• A vector of IID random error terms with mean and variance

• And matrices with regressors are consistent with and link the observations

Estimate

Hendersons " mixed model equations" (. MME, English for mixed - model equations ) are:

The solutions of the MME and are minimally variant linear unbiased estimators (BLUE) for or. This follows from the theorem of Gauss-Markov, since the conditional variance of the result ' is not scalable to the unit matrix. If the conditional variance is known, is the weighted by the inverse variance least squares estimator BLUE. However, the conditional variance is rarely known, so it is desirable to solve the MME to estimate the variance and the weighted parameter estimates together.

One way to customize a mixed model is the EM algorithm, in which the components of the variance are treated as unobserved nuisance parameter in the entire probability. Currently, this method is in the major statistical software packages R ( lme in the nlme library) and SAS ( proc mixed) implemented. The solution of the mixed model equations is a maximum likelihood estimate if the errors are normally distributed.

Swell

Further Reading

• Milliken, G. A., and Johnson, D. E. (1992). Analysis of messy data: Designed experiments I. Vol. New York: Chapman & Hall.

• West, BT, Welch, KB, & Galecki, AT ( 2007). Linear mixed models: A practical guide to using statistical software. New York: Chapman & Hall / CRC.

• Regression model