This course provides a comprehensive introduction to a set of inter-related topics of widespread applicability in the social social sciences: structural equation modelling, path analysis, causal modelling, mediation analysis, latent variable modelling (including factor analysis and latent class analysis), Bayesian networks, graphical models, and other related topics. The course begins with a thorough review, both practical and theoretical, of regression modelling, particularly on general and generalized linear regression. We then turn to the topic of path analysis. At its simplest, path analysis can be seen as an extension of standard (e.g. linear) regression analysis to cases where there are more complex structural relationship between the predictor and outcome variables. More generally, and more usefully, we can view path analysis as specifying and modelling causal relationships between observed variables. In order to fully appreciate path analyses, and especially their role as causal models, we will introduce the concept of directed acyclic graphical models, also known as Bayesian networks, which are a powerful mathematical and conceptual tool for reasoning about causal relationships. We then thoroughly cover the topic of mediation analysis, which can seen as a special case, though still very widely applicable, of path analysis and causal models. We then turn to structural equation modelling, which can be seen as an extension of path analysis, particularly due to the inclusion of unobserved or latent variables. More generally, structural equation models allow for the specification and testing of more complex theoretical models of the observed data. In order to properly introduce structural equation models, we first explore latent variable models, particularly factor analysis and latent class models. In our coverage of structural equation models we deal with the general concepts of model identification, inference, and evaluation, and then explore special topics such as categorical, nonlinear, and non-normal structural equation models, multilevel structural equation models, and latent growth curve modelling.

Intended Audience

This course is aimed at anyone who is interested to learn and apply this powerful and flexible set of statistical modelling methods that have widespread application across the social, medical, and biological sciences.

Teaching Format

This course will be hands-on and workshop based. Throughout each day, there will be some lecture style presentation, i.e., using slides, introducing and explaining key concepts. However, even in these cases, the topics being covered will include practical worked examples that will work through together.

Assumed quantitative knowledge

We assume familiarity with linear regression analysis, and the major concepts of classical inferential statistics (p-values, hypothesis testing, confidence intervals, model comparison, etc). Some passing familiarity with models such as logistic regression will also be assumed.

Assumed computer background

R experience is desirable but not essential. Although we will be using R extensively, all the code that we use will be made available, and so attendees will just to add minor modifications to this code. Attendees should install R and RStudio on their own computers before the workshops, and have some minimal familiarity with the R environment.

Equipment and software requirements

A laptop computer with a working version of R or RStudio is required. R and RStudio are both available as free and open source software for PCs, Macs, and Linux computers. R may be downloaded by following the links here https://www.r-project.org/. RStudio may be downloaded by following the links here: https://www.rstudio.com/. All the R packages that we will use in this course will be possible to download and install during the workshop itself as and when they are needed. The major R packages will include lavaan, blavaan, sem, brms, but the full list of required packages will be made available to all attendees prior to the course. In some cases, some additional open-source software will need to be installed to use some R packages. In particular, these include Stan and Jags for probabilistic modelling. Directions on how to install this software will also be provided before and during the course.

Course programme

Monday, September 16

• Topic 1: General and generalized linear regression, including multilevel models. In order to provide a solid foundation for the remainder of the course, we begin by providing a comprehensive overview of general linear models, also covering their multilevel (or hierarchical) counterparts. For this topic, we will use R tools such as lm, lme4::lmer.

• Topic 2: Path analysis. Having covered regression, we proceed to path analysis, which can be viewed as a straightforward extension of standard regression analysis. The primary R package that we will use for this introduction to path analysis is lavaan.

• Topic 3: Graph theory and causal models. Path analysis can, and should, be seen more than just an extension of regression analysis and be seen as a type of causal model. In order to explore this in depth, we will introduce the concepts of directed acyclic graphs and Bayesian networks, originally developed in computer science and artificial intelligence research, and show how they provide a powerful framework of reasoning about and with causal models.

Tuesday, September 17

• Topic 4: Mediation analysis. A special case of path analysis is mediation analysis. This is where the causal effect of one or more variables on some outcome is by way of their effect on an intermediary variable. For example, we say the effect of smoking on lung cancer is mediated by the tar content of cigarettes, (smoking causes tar build-up in the lungs, tar build-up in the lungs causes lung cancer). In this section, we will provide an introduction to mediation analysis, and pay particular attention to how it has been traditionally analysed.

• Topic 5: Causal mediation analysis. Traditional mediation analysis, although useful, does not extend easily to situations where there are interactions (moderations) between predictor and moderator variables, or where there are nonlinear effects between variables, and other variants. Causal mediation analysis is a more general framework for mediation modelling based on modelling counterfactuals and using graphical models. Here, we will also discuss the relationship between causal mediation analysis and traditional mediation analysis, and also how causal mediation analysis is related to the concept of instrumental variables.

Wednesday, September 18

• Topic 6: Factor analysis: Latent variable models assume the presence of variables that are not directly observed but are assumed to affect other variables that are observed. One of the most commonly used latent variable models, and one that can be seen as a special case of structural equation models that we will explore on Day 3, is factor analysis. Here, we will describe factor analysis, distinguishing between what are known as exploratory and confirmatory factor analysis models. We will also discuss how factor analysis relates to other widely used latent variable modelling techniques such as principal components analysis and independent components analysis.

• Topic 7: Latent class models. Latent class models, also known as probabilistic mixture models, are another widely used latent variable modelling technique. They differ from factor analyses and related models by the fact that they assume the latent variable is a categorical variable. What this entails is that latent class models assumes that observed data has emerged from a set of categorically distinct underlying unobserved component.

Thursday, September 19

Days 4 and 5 will cover structural models in depth. On Day 4, we will provide a comprehensive introduction to all the major and general concepts of structural equation models. On Day 5, we will explore different variants of the structural equation models.

• Topic 8: Structural equation modelling: General concepts. Structural equation models can be seen as an extension of path analysis, particularly due to the use of latent variables. In this introduction to the topic, we will first explore different examples of structural equation models using real world example data sets, and consider the standard or typical types of assumed models. We will also cover the major and general topics of model identification, model inference, and model evaluation. Here, we will also describe traditional and more modern, and more flexible, approaches to identification, inference, and evaluation. The R packages that we will use here will primarily include lavaan, blavaan, sem, piecewiseSEM, brms, and we will also use probabilistic programming languages such as Jags and Stan.

Friday, September 20

• Topic 9: Nonlinear, non-normal and categorical structural equation models: The standard, or typical, structural equation model assumes that variables are continuous, have normal distributions, and that there are linear relationships between these variables. While this is often a useful default or starting assumption, more powerful and flexible structural equation models are possible if we allow for continuous variables that have non-normal distributions, nonlinear relationships between these variables, and

• Topic 10: Multilevel structural equation models: Multilevel structural equation models allow us to model variation across different groups. As an example from the context of education studies, we could model phenomena modelled by structural equation models vary across different schools and across different regions, etc.

• Topic 11: Latent growth curve modelling: A special case of structural equation models is a latent growth curve model. These are widely used with data from longitudinal or other repeated measures studies, and their primary purpose is to model change or development trajectories over time.

GitHub resources

Further resources for this training course can be found on Github at mark-andrews/pssem01.