
Multilevel models (also hierarchical linear models, nested models, mixed models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level.[1] These models can be seen as generalizations of linear models (in particular, linear regression), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.
xxM is a package for multilevel structural equation modeling (ML-SEM) with complex dependent data structures. xxM implements a modeling framework called n-Level Structural Equation Modeling (NL-SEM) and can estimate models with any number of levels. Observed and latent variables are allowed at all levels.
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.
LISREL (which stands for “linear structural relations”) involves eight matrices that organize the causal paths, loadings, correlations, and error terms in any model. Although this makes a cumbersome syntax, it is used in the mathematical description of SEM in the vast majority of statistical articles. A shortened, simpler four-matrix version can be used instead for specifying models in LISREL and is presented in some articles.
The lavaan package is developed to provide useRs, researchers and teachers a free open-source, but commercial-quality package for latent variable modeling. You can use lavaan to estimate a large variety of multivariate statistical models, including path analysis, confirmatory factor analysis, structural equation modeling and growth curve models.
This tutorial introduces core concepts of xxM as a modeling framework and as a software package. Key elements of model specification in xxM are introduced in the context of fitting a bivariate random-intercepts model (Mehta, Neale, & Flay, 2005). Although the example is relatively trivial, once you understand the building blocks presented in this tutorial, you should be able to construct complex models easily.
The presentation is in three sections:
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 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.
In statistics, confirmatory factor analysis (CFA) is a special form of factor analysis, most commonly used in social research. It is used to test whether measures of a construct are consistent with a researcher's understanding of the nature of that construct (or factor). As such, the objective of confirmatory factor analysis is to test whether the data fit a hypothesized measurement model. This hypothesized model is based on theory and/or previous analytic research.
Explore constructing a multilevel CFA in xxM by specifying pattern matrices, estimating theta, psi, and lambda for level 1 and level 2, and mapping within- and between-level relations.
Explore multilevel CFA fit and estimation with the XXL library, including handling S4 data objects, creating level-specific data frames, and defining lambda and psi matrices for teacher-student data.
Latent growth modeling is a statistical technique used in the structural equation modeling (SEM) framework to estimate growth trajectory. It is a longitudinal analysis technique to estimate growth over a period of time. It is widely used in the field of behavioral science, education and social science. It is also called latent growth curve analysis. The latent growth model was derived from theories of SEM. General purpose SEM software, such as OpenMx, lavaan (both open source packages based in R), AMOS, Mplus, LISREL, or EQS among others may be used to estimate the trajectory of growth. The R xxM package can estimate the trajectory of growth using multilevel models comprised of both latent (SEM-like) and observed (directly-measured) variables.
Explore the long format LGC example in xxM, defining level-specific matrices, assessing deviance and convergence, and interpreting parameter estimates with confidence intervals.
Multilevel models (also hierarchical linear models, nested models, mixed models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level.[1] These models can be seen as generalizations of linear models (in particular, linear regression), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.
Multilevel modeling is a term alternately used to describe hierarchical linear models, nested models, mixed-effects models, random-effects models, and split-plot designs. They are statistical models for estimating parameters that vary at more than one level and which may contain both observed and latent variables at any level. They are generalizations of linear models, particularly linear regression, although they may be extended to non-linear models.
xxM is an R package which can estimate multilevel SEM models characterized by complex level-dependent data structures containing both observed and latent variables. The package was developed at the University of Houston by a collaborative team headed by Dr. Paras Mehta. xxM implements a modeling framework called n-Level Structural Equation Modeling (NL-SEM) which allows the specification of models with any number of levels. Because observed and latent variables are allowed at all levels, a conventional SEM model may be specified for each level and across any levels. Also, the random-effects of observed variables are allowed both within and across levels. Mehta claims that xxM is the only software tool in the world that is capable of estimating the effects of both observed and latent variables in a SEM nomological network across an unlimited number of levels.
Some of the complex dependent data structures that can be effectively modeled and estimated with xxM include:
⦁ Hierarchically nested data (e.g. students, classrooms, schools)
⦁ Longitudinal data (long or wide)
⦁ Longitudinal data with switching classification (e.g. students changing classrooms)
⦁ Cross-classified data (e.g. students nested within primary and secondary schools)
⦁ Partial nesting (e.g. underperforming students in a classroom receive tutoring)
Model specification with xxM uses a “LEGO-like building block” approach for model construction. With an understanding of these basic building blocks, very complex multilevel models may be constructed by repeating the same key building steps.
This six-session Multilevel SEM Modeling with xxM course is an overview and tutorial of how to perform these key basic building block steps using xxM. To convey a practical understanding of implementing the core model specification and construction concepts of xxM, seven complete illustrative examples are detailed over the six class sessions. One who completes this course will then be able to construct more complex multilevel models tied to their own research projects. The seven complete examples detailed in the course begin with: (1) a streamlined two-level bivariate random-intercepts model; and (2) a two-level random-slopes model. Then a (3) multilevel confirmatory factor analysis (CFA) and a (4) random-slopes multilevel CFA are detailed, followed by random-slopes (5) 'wide' and (6) 'long' latent growth curve model examples. Finally, a (7) three-level hierarchical model containing both observed and latent variables is fully demonstrated. All of the necessary software, data, manuals, slides and course materials to productively specify and estimate all seven of the course model examples are provided and included in 'resources' folders associated with the video lessons.