Linear Mixed-Effects Modeling (LMEM) Algorithm
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A Linear Mixed-Effects Modeling (LMEM) Algorithm is a linear regression model that is a mixed-effects model for data that are collected and summarized in groups.
- Context:
- It can be implemented in an LMEM System (to solve an LMEM task).
- Example(s):
- See: Generalized LMEM, Group-Summarized Data.
References
2016
- http://www.statsmodels.org/dev/mixed_linear.html
- QUOTE: Linear Mixed Effects models are used for regression analyses involving dependent data. Such data arise when working with longitudinal and other study designs in which multiple observations are made on each subject. Two specific mixed effects models are random intercepts models, where all responses in a single group are additively shifted by a value that is specific to the group, and random slopes models, where the values follow a mean trajectory that is linear in observed covariates, with both the slopes and intercept being specific to the group. The Statsmodels MixedLM implementation allows arbitrary random effects design matrices to be specified for the groups, so these and other types of random effects models can all be fit.
The Statsmodels LME framework currently supports post-estimation inference via Wald tests and confidence intervals on the coefficients, profile likelihood analysis, likelihood ratio testing, and AIC. Some limitations of the current implementation are that it does not support structure more complex on the residual errors (they are always homoscedastic), and it does not support crossed random effects. We hope to implement these features for the next release.
- QUOTE: Linear Mixed Effects models are used for regression analyses involving dependent data. Such data arise when working with longitudinal and other study designs in which multiple observations are made on each subject. Two specific mixed effects models are random intercepts models, where all responses in a single group are additively shifted by a value that is specific to the group, and random slopes models, where the values follow a mean trajectory that is linear in observed covariates, with both the slopes and intercept being specific to the group. The Statsmodels MixedLM implementation allows arbitrary random effects design matrices to be specified for the groups, so these and other types of random effects models can all be fit.
2014
- https://www.mathworks.com/help/stats/linear-mixed-effects-models.html
- QUOTE: Linear mixed-effects models are extensions of linear regression models for data that are collected and summarized in groups. These models describe the relationship between a response variable and independent variables, with coefficients that can vary with respect to one or more grouping variables. A mixed-effects model consists of two parts, fixed effects and random effects. Fixed-effects terms are usually the conventional linear regression part, and the random effects are associated with individual experimental units drawn at random from a population. The random effects have prior distributions whereas fixed effects do not. Mixed-effects models can represent the covariance structure related to the grouping of data by associating the common random effects to observations that have the same level of a grouping variable. The standard form of a linear mixed-effects model is: [math]\displaystyle{ y=Xβ}fixed+Zb}random+ε}error, }[/math] where …
2013
- (Barr et al., 2013) ⇒ Dale J. Barr, Roger Levy, Christoph Scheepers, and Harry J. Tily. (2013). “Random Effects Structure for Confirmatory Hypothesis Testing: Keep It Maximal.” In: Journal of memory and language, 68(3).
- QUOTE: Linear mixed-effects models (LMEMs) have become increasingly prominent in psycholinguistics and related areas. However, many researchers do not seem to appreciate how random effects structures affect the generalizability of an analysis. Here, we argue that researchers using LMEMs for confirmatory hypothesis testing should minimally adhere to the standards that have been in place for many decades. Through theoretical arguments and Monte Carlo simulation, we show that LMEMs generalize best when they include the maximal random effects structure justified by the design.