Generalized linear mixed model

In statistics, a generalized linear mixed model (GLMM) is an extension to the generalized linear model (GLM) in which the linear predictor contains random effects in addition to the usual fixed effects. They also inherit from GLMs the idea of extending linear mixed models to non-normal data. GLMMs provide a broad range of models for the analysis of grouped data, since the differences between groups can be modelled as a random effect. These models are useful in the analysis of many kinds of data, including longitudinal data. GLMMs are generally defined such that, conditioned on the random effects , the dependent variable is distributed according to the exponential family with its expectation related to the linear predictor via a link function Here and are the fixed effects design matrix, and fixed effects respectively; and are the random effects design matrix and random effects respectively. To understand this very brief definition you will first need to understand the definition of a generalized linear model and of a mixed model. Generalized linear mixed models are a special cases of hierarchical generalized linear models in which the random effects are normally distributed. The complete likelihood has no general closed form, and integrating over the random effects is usually extremely computationally intensive. In addition to numerically approximating this integral(e.g. via Gauss–Hermite quadrature), methods motivated by Laplace approximation have been proposed. For example, the penalized quasi-likelihood method, which essentially involves repeatedly fitting (i.e. doubly iterative) a weighted normal mixed model with a working variate, is implemented by various commercial and open source statistical programs. Fitting GLMMs via maximum likelihood (as via AIC) involves integrating over the random effects. In general, those integrals cannot be expressed in analytical form. Various approximate methods have been developed, but none has good properties for all possible models and data sets (e.g.

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Façades and light pattern composition have been shown to influence the spatial experience and physiological responses of humans [1,2]. The present study examines the effect of sunlight penetration and window size on fixations to the floor of the scene, and the relation between visual interest and fixations in an experiment using 360° scenes displayed in Virtual Reality. One hundred participants were shown the same daylit interior space with varying presence of sun patches (based on sky type and time-of-day variations) and window size in a mixed experimental design. Participants' head movements were recorded during the first 25 seconds of silent free-viewing exposure to each scene, after which they rated the visual interest of the scene. Fixation areas were derived from head movement data and were used to extract the percentage of fixations towards different areas in the scene. Linear Mixed Model (LMM) analyses showed that sun patch presence influenced the percentage of fixations towards both the front part of the floor (near the façade) and the whole floor. Pairwise comparisons showed that participants spent more time fixating towards the floor in the presence of small sun patch compared to no sun patch. Adding visual interest as a fixed factor in the LMM did not show a statistically significant relation between fixations towards the floor and visual interest ratings. Although limited to Virtual Reality and thus to its relatively small luminance range, these findings show that the presence of a sun patch in one's field of view elicits visual attraction.

2021

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