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Linear Mixed-Effects Models with R is a 7-session course that teaches the requisite knowledge and skills necessary to fit, interpret and evaluate the estimated parameters of linear mixed-effects models using R software. Alternatively referred to as nested, hierarchical, longitudinal, repeated measures, or temporal and spatial pseudo-replications, linear mixed-effects models are a form of least-squares model-fitting procedures. They are typically characterized by two (or more) sources of variance, and thus have multiple correlational structures among the predictor independent variables, which affect their estimated effects, or relationships, with the predicted dependent variables. These multiple sources of variance and correlational structures must be taken into account in estimating the "fit" and parameters for linear mixed-effects models.
The structure of mixed-effects models may be additive, or non-linear, or exponential or binomial, or assume various other ‘families’ of modeling relationships with the predicted variables. However, in this "hands-on" course, coverage is restricted to linear mixed-effects models, and especially, how to: (1) choose an appropriate linear model; (2) represent that model in R; (3) estimate the model; (4) compare (if needed), interpret and report the results; and (5) validate the model and the model assumptions. Additionally, the course explains the fitting of different correlational structures to both temporal, and spatial, pseudo-replicated models to appropriately adjust for the lack of independence among the error terms. The course does address the relevant statistical concepts, but mainly focuses on implementing mixed-effects models in R with ample R scripts, ‘real’ data sets, and live demonstrations. No prior experience with R is necessary to successfully complete the course as the first entire course section consists of a "hands-on" primer for executing statistical commands and scripts using R.
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|Section 1: Introduction to R as a Statistical Environment|
Introduction to the CoursePreview
RStudio Integrated Development Environment (IDE) is a powerful and productive user interface for R. It’s free and open source, and works great on Windows, Mac, and Linux.
Basic Quantitative Operations in R (part 1)Preview
Basic Quantitative Operations in R (part 2)
More R Scripting and Plotting (part 1)
More R Scripting and Plotting (part 2)
One of the great strengths of R is the user's ability to add functions. In fact, many of the functions in Rare actually functions of functions. The structure of a function is given below.
Functions in R (part 2)Preview
R has a wide variety of data types including scalars, vectors (numerical, character, logical).
All columns in a matrix must have the same mode(numeric, character, etc.) and the same length.
A data frame is more general than a matrix, in that different columns can have different modes (numeric, character, factor, etc.). This is similar to SAS and SPSS datasets.
Exercises: Getting Started with R as a Statistical Environment
|Section 2: Basic Linear Mixed-Effects Concepts|
Solution to Getting Started with R as a Statistical Environment Exercise
Some LME Background from MVA Package (scripts, part 1)
Some LME Background (scripts, part 2)
Some LME Background (scripts, part 3)
Some LME Background (scripts, part 4)
Temporal Pseudoreplication Fertilizer Exercise
Solution to Temporal Pseudoreplication Fertilizer Exercise
A mixed model is a statistical model containing both fixed effects and random 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 (longitudinal study), or where measurements are made on clusters of related statistical units. Because of their advantage in dealing with missing values, mixed effects models are often preferred over more traditional approaches such as repeated measures ANOVA.
Basic LME Concepts (slides, part 2)Preview
Basic LME Concepts (slides, part 3)
Mixed-Effects Fertilized Plants Example using nlme Package (part 1)Preview
Finish Fertilized Plants Example and Begin Zuur Material
Two-Stage Beaches Example (part 1)Preview
Two-Stage Beaches Example (part 2)
Random Intercepts Model Example
Random Intercepts and Slopes Model Example
|Section 3: Timber and Plasma Data Examples|
Split-plot designs result when a particular type of restricted randomization has occurred during the experiment. A simple factorial experiment can result in a split-plot type of design because of the way the experiment was actually executed.
Split-Plot Exercise Solution (part 2)
Split-Plot Exercise Solution (part 3)
A random intercepts model is a model in which intercepts are allowed to vary, and therefore, the scores on the dependent variable for each individual observation are predicted by the intercept that varies across groups. This model assumes that slopes are fixed (the same across different contexts). In addition, this model provides information about intraclass correlations, which are helpful in determining whether multilevel models are required in the first place.
A random slopes model is a model in which slopes are allowed to vary, and therefore, the slopes are different across groups. This model assumes that intercepts are fixed (the same across different contexts).
A model that includes both random intercepts and random slopes is likely the most realistic type of model, although it is also the most complex. In this model, both intercepts and slopes are allowed to vary across groups, meaning that they are different in different contexts.
Plasma Data Example (part 1)
Plasma Data Example (part 2)
Vatiance Components Analysis Exercise
|Section 4: Selecting LME Model Structures|
In statistics, a random effect(s) model, also called a variance components model, is a kind of hierarchical linear model. It assumes that the dataset being analysed consists of a hierarchy of different populations whose differences relate to that hierarchy. In econometrics, random effects models are used in the analysis of hierarchical or panel data when one assumes no fixed effects (it allows for individual effects). The random effects model is a special case of the fixed effects model.
Variance Components Analysis Exercise Solution (part 2)
Variance Components Analysis Exercise Solution (part 3)
LME Model Structure Selection (part 1)Preview
LME Model Structure Selection (part 2)
LME Model Structure Selection (part 3)
LME Model Structure Selection (part 4)
LME Model Structure Selection (part 5)
Regression versus Fixed Effects Bias Exercise
|Section 5: Compare LM and LME Parameters|
In econometrics and statistics, a fixed effects model is a statistical model that represents the observed quantities in terms of explanatory variables that are treated as if the quantities were non-random. This is in contrast to random effects models and mixed models in which either all or some of the explanatory variables are treated as if they arise from random causes.
Regression versus Fixed Effects Exercise Solution (part 2)
Regression versus Fixed Effects Exercise Solution (part 3)
Regression versus Fixed Effects Exercise Solution (part 4)
Nestling Barn Owls (part 1)
Nestling Barn Owls (part 2)
Nestling Barn Owls: Find Optimal Structure (part 1)Preview
Nestling Barn Owls: Find Optimal Structure (part 2)
Nestling Barn Owls: Find Optimal Structure (part 3)
Exercise: Beat the Blues I
|Section 6: 10-Step Protocol for Optimal Structure|
Beat the Blues I Exercise Solution (part 1)
Beat the Blues I Exercise Solution (part 2)
Beat the Blues I Exercise Solution (part 3)
10-Step Protocol for Optimal Structure (part 1)
10-Step Protocol for Optimal Structure (part 2)
10-Step Protocol for Optimal Structure (part 3)
10-Step Protocol for Optimal Structure (part 4)
One application of multilevel modeling (MLM) is the analysis of repeated measures data. Multilevel modeling for repeated measures data is most often discussed in the context of modeling change over time (i.e. growth curve modeling for longitudinal designs); however, it may also be used for repeated measures data in which time is not a factor. The issue of subjects leaving the study ("dropouts") midway through the periodic intervals of data collection is a perennial problem with these types of studies.
The Problem of Dropouts in Longitudinal Studies (part 2)
Beat the Blues II Exercises
|Section 7: Violation of Independence Errors|
Beat the Blues II Exercise Solutions
Just as with ordinary least-squared linear regression, the observed distribution of error terms (residuals) is assumed to be normally distributed and characterized by statistical independence from other error terms in the practice of linear mixed-effects modeling.
Time-Based Residual Patterns (part 1)
Time-Based Residual Patterns (part 2)
Independence and Compound Symmetry
In the statistical analysis of time series, autoregressive–moving-average (ARMA) models provide a parsimonious description of a (weakly) stationary stochastic process in terms of two polynomials, one for the auto-regression and the second for the moving average.
AR-1 and ARMA Residual Dependence (part 2)
Introduction to Spatial Dependence
Prepare Data and Run Bubble PlotPreview
Candidate Spatial Correlative Structures
Irish Rivers Acid Sensitivity
Using Spatial Correlational Structures
Dr. Geoffrey Hubona held full-time tenure-track, and tenured, assistant and associate professor faculty positions at 3 major state universities in the Eastern United States from 1993-2010. In these positions, he taught dozens of various statistics, business information systems, and computer science courses to undergraduate, master's and Ph.D. students. He earned a Ph.D. in Business Administration (Information Systems and Computer Science) from the University of South Florida (USF) in Tampa, FL (1993); an MA in Economics (1990), also from USF; an MBA in Finance (1979) from George Mason University in Fairfax, VA; and a BA in Psychology (1972) from the University of Virginia in Charlottesville, VA. He was a full-time assistant professor at the University of Maryland Baltimore County (1993-1996) in Catonsville, MD; a tenured associate professor in the department of Information Systems in the Business College at Virginia Commonwealth University (1996-2001) in Richmond, VA; and an associate professor in the CIS department of the Robinson College of Business at Georgia State University (2001-2010). He is the founder of the Georgia R School (2010-2014) and of R-Courseware (2014-Present), online educational organizations that teach research methods and quantitative analysis techniques. These research methods techniques include linear and non-linear modeling, multivariate methods, data mining, programming and simulation, and structural equation modeling and partial least squares (PLS) path modeling. Dr. Hubona is an expert of the analytical, open-source R software suite and of various PLS path modeling software packages, including SmartPLS. He has published dozens of research articles that explain and use these techniques for the analysis of data, and, with software co-development partner Dean Lim, has created a popular cloud-based PLS software application, PLS-GUI.