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. They are particularly useful in settings where repeated measurements are made on the same statistical units, or where measurements are made on clusters of related statistical units.


History and current status

Ronald Fisher introduced random effects models to study the correlations of trait values between relatives.[1] In the 1950s, Charles Roy Henderson provided best linear unbiased estimates (BLUE) of fixed effects and best linear unbiased predictions (BLUP) of random effects.[2][3][4][5] Subsequently, mixed modeling has become a major area of statistical research, including work on computation of maximum likelihood estimates, non-linear mixed effect models, missing data in mixed effects models, and Bayesian estimation of mixed effects models. Mixed models are applied in many disciplines where multiple correlated measurements are made on each unit of interest. They are prominently used in research involving human and animal subjects in fields ranging from genetics to marketing, and have also been used in industrial statistics.[citation needed]


In matrix notation a mixed model can be represented as

\ y = X \beta + Zu + \epsilon\,\!


  • y is a vector of observations, with mean E(y) = Xβ
  • β is a vector of fixed effects
  • \epsilon is a vector of IID random error terms with mean E(\epsilon)=0 and variance \operatorname{var}(\epsilon)=R
  • X and Z are matrices of regressors relating the observations y to β and u


Henderson's "mixed model equations" (MME) are:[2][4][citation needed]

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

The solutions to the MME, \textstyle\tilde{\beta} and \textstyle\tilde{u} are best linear unbiased estimates (BLUE) and predictors for β and u, respectively. This is a consequence of the Gauss-Markov theorem when the conditional variance of the outcome is not scalable to the identity matrix. When the conditional variance is known, then the inverse variance weighted least squares estimate is BLUE. However, the conditional variance is rarely, if ever, known. So it is desirable to jointly estimate the variance and weighted parameter estimates when solving MMEs.

One method used to fit such mixed models is that of the EM algorithm[6] where the variance components are treated as unobserved nuisance parameters in the joint likelihood. Currently, this is the implemented method for the major statistical software packages R (lme in the nlme library) and SAS (proc mixed). The solution to the mixed model equations is a maximum likelihood estimate when the distribution of the errors is normal. [7]

See also


  1. ^ Fisher, RA (1918). "The correlation between relatives on the supposition of Mendelian inheritance". Transactions of the Royal Society of Edinburgh 52: 399–433. 
  2. ^ a b Robinson, G.K. (1991). "That BLUP is a Good Thing: The Estimation of Random Effects". Statistical Science 6 (1): 15–32. doi:10.1214/ss/1177011926. JSTOR 2245695. 
  3. ^ C. R. Henderson, Oscar Kempthorne, S. R. Searle and C. M. von Krosigk (1959). "The Estimation of Environmental and Genetic Trends from Records Subject to Culling". Biometrics (International Biometric Society) 15 (2): 192–218. doi:10.2307/2527669. JSTOR 2527669. 
  4. ^ a b L. Dale Van Vleck. "Charles Roy Henderson, April 1, 1911 – March 14, 1989". United States National Academy of Sciences. http://books.nap.edu/html/biomems/chenderson.pdf. 
  5. ^ McLean, Robert A.; Sanders, William L.; Stroup, Walter W. (1991). "A Unified Approach to Mixed Linear Models". The American Statistician (American Statistical Association) 45 (1): 54–64. doi:10.2307/2685241. JSTOR 2685241. 
  6. ^ Lindstrom, ML; Bates, DM (1988). "Newton-Raphson and EM algorithms for linear mixed-effects models for repeated-measures data". JASA 83 (404): 1014–1021. 
  7. ^ Laird, Nan M.; Ware, James H. (1982). "Random-Effects Models for Longitudinal Data". Biometrics (International Biometric Society) 38 (4): 963–974. doi:10.2307/2529876. JSTOR 2529876. PMID 7168798. 

Further reading

  • Milliken, G. A., & Johnson, D. E. (1992). Analysis of messy data: Vol. I. Designed experiments. New York: Chapman & Hall.
  • West, B. T., Welch, K. B., & Galecki, A. T. (2007). Linear mixed models: A practical guide to using statistical software. New York: Chapman & Hall/CRC.

Wikimedia Foundation. 2010.

Игры ⚽ Поможем решить контрольную работу

Look at other dictionaries:

  • Generalized linear mixed model — In statistics, a generalized linear mixed model (GLMM) is a particular type of mixed model (multilevel model). It is an extension to the generalized linear model in which the linear predictor contains random effects in addition to the usual fixed …   Wikipedia

  • Mixed logit — is a fully general statistical model for examining discrete choices. The motivation for the mixed logit model arises from the limitations of the standard logit model. The standard logit model has three primary limitations, which mixed logit… …   Wikipedia

  • Mixed reality — (MR) (encompassing both augmented reality and augmented virtuality) refers to the merging of real and virtual worlds to produce new environments and visualisations where physical and digital objects co exist and interact in real time. A mix of… …   Wikipedia

  • Mixed — is the past tense of mix. It may also refer to: Mixed breed, an animal whose parents are from different breeds or species Mixed anomaly, in theoretical physics, an example of an anomaly Mixed data sampling, an econometric model developed by… …   Wikipedia

  • Mixed member proportional representation — Mixed member proportional representation, also termed mixed member proportional voting and commonly abbreviated to MMP, is an additional member voting system used to elect representatives to numerous legislatures around the world. MMP is similar… …   Wikipedia

  • Model selection — is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre existing set of data is considered. However, the task can also involve the design of experiments such that the data collected is …   Wikipedia

  • Mixed raster content — or MRC is a process of using image segmentation methods to improve the contrast resolution of a raster image composed of pixels. The decomposition of an image using segmentation approaches separates image objects on a foreground and background… …   Wikipedia

  • Mixed data sampling — (MIDAS) is an econometric regression or filtering method developed by Ghysels et al. A simple regression example has the regressor appearing at a higher frequency than the regressand: where y is the regressand, x is the regressor, m denotes the… …   Wikipedia

  • Mixed-member proportional representation — Part of the Politics series Electoral methods Single winner …   Wikipedia

  • Mixed mating model — The mixed mating model is a mathematical model that describes the mating system of a plant population in terms of the degree of self fertilisation present. It is a fairly simplistic model, employing several simplifying assumptions, most notably… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”