Exploratory Factor Analysis (EFA) identifies a number of latent factors that explain correlations between observed variables. A key issue in the application of EFA is the selection of an adequate number of factors. This is a non-trivial problem because more factors always improve the fit of the model. Most methods for selecting the number of factors fall into two categories: either they analyze the patterns of eigenvalues of the correlation matrix, such as parallel analysis; or they frame the selection of the number of factors as a model selection problem and use approaches such as likelihood ratio tests or information criteria.

Gaussian Mixture Models (GMMs) and its special cases Latent Profile Analysis and k-Means are a popular and versatile tools for exploring heterogeneity in multivariate continuous data. However, they assume that the observed data are continuous, an assumption that is often not met: for example, the severity of symptoms of diseases is often measured in ordinal categories such as not at all, several days, more than half the days, and nearly every day, and survey questions are often assessed using ordinal responses such as strongly agree, agree, neutral, and agree, strongly agree. In this blog post, I summarize a paper which investigates to what extent estimating GMMs is robust against observing ordinal instead of continuous variables.

Interpreting statistical network models typically involves interpreting individual edge parameters. If the network model is a Gaussian Graphical Model (GGM), the interpretation is relatively simple: the pairwise interaction parameters are partial correlations, which indicate conditional linear relationships and vary from -1 to 1. Using the standard deviations of the two involved variables, the partial correlation can also be transformed into a linear regression coefficient (see for example here). However, when studying interactions involving categorical variables, such as in an Ising model or a Mixed Graphical Model (MGM), the parameters are not limited to a certain range and their interpretation is less intuitive. In these situations it may be helpful to report the interactions between variables in terms of odds ratios.

Researchers are often interested in comparing statistical network models across groups. For example, Fritz and colleagues compared the relations between resilience factors in a network model for adolescents who did experience childhood adversity to those who did not. Several methods are already available to perform such comparisons. The Network Comparison Test (NCT) performs a permutation test to decide for each parameter whether it differs across two groups. The Fused Graphical Lasso (FGL) uses a lasso penalty to estimate group differences in Gaussian Graphical Models (GGMs). And the BGGM package allows one to test and estimate differences in GGMs in a Bayesian setting. In a recent preprint, I proposed an additional method based on moderation analysis which has the advantage that it can be applied to essentially any network model and at the same time allows for comparisons across more than two groups.

Models for individual subjects are becoming increasingly popular in psychological research. One reason is that it is difficult to make inferences from between-person data to within-person processes. Another is that time series obtained from individuals are becoming increasingly available due to the ubiquity of mobile devices. The central goal of so-called idiographic modeling is to tap into the within-person dynamics underlying psychological phenomena. With this goal in mind many researchers have set out to analyze the multivariate dependencies in within-person time series. The most simple and most popular model for such dependencies is the first-order Vector Autoregressive (VAR) model, in which each variable at the current time point is predicted by (a linear function of) all variables (including itself) at the previous time point.

Statistical network models have become a popular exploratory data analysis tool in psychology and related disciplines that allow to study relations between variables. The most popular models in this emerging literature are the binary-valued Ising model and the multivariate Gaussian distribution for continuous variables, which both model interactions between pairs of variables. In these pairwise models, the interaction between any pair of variables A and B is a constant and therefore does not depend on the values of any of the variables in the model. Put differently, none of the pairwise interactions is moderated. However, in the highly complex and contextualized fields like psychology, such moderation effects are often plausible. In this blog post, I show how to fit, analyze, visualize and assess the stability of Moderated Network Models for continuous data with the R-package mgm.

Centering predictors in a regression model with only main effects has no influence on the main effects. In contrast, in a regression model including interaction terms centering predictors does have an influence on the main effects. After getting confused by this, I read this nice paper by Afshartous & Preston (2011) on the topic and played around with the examples in R. I summarize the resulting notes and code snippets in this blogpost.

Yesterday, I read ‘Measurement error and the replication crisis’ by Eric Loken and Andrew Gelman, which left me puzzled. The first part of the paper consists of general statements about measurement error. The second part consists of the claim that in the presence of measurement error, we overestimate the true effect when having a small sample size. This sounded wrong enough to ask the authors for their simulation code and spend a couple of hours to figure out what they did in their paper. I am offering a short and a long version.

Network models have become a popular way to abstract complex systems and gain insights into relational patterns among observed variables in many areas of science. The majority of these applications focuses on analyzing the structure of the network. However, if the network is not directly observed (Alice and Bob are friends) but estimated from data (there is a relation between smoking and cancer), we can analyze - in addition to the network structure - the predictability of the nodes in the network. That is, we would like to know: how well can a given node in the network predicted by all remaining nodes in the network?

We use network visualizations to look into the voting patterns in the current German parliament. I downloaded the data here and all figures can be reproduced using the R code available on Github.

In a previous post we estimated a Mixed Graphical Model (MGM) on a dataset of mixed variables describing different aspects of the life of individuals diagnosed with Autism Spectrum Disorder, using the mgm package. For interactions between continuous variables, the weighted adjacency matrix fully describes the underlying interaction parameter. Correspondinly, the parameters are fully represented in the graph visualization: the width of the edges is proportional to the absolute value of the parameter, and the edge color indicates the sign of the parameter. This means that we can clearly interpret an edge between two continuous variables as a positive or negative linear relationship of some strength.

Determining conditional independence relationships through undirected graphical models is a key component in the statistical analysis of complex obervational data in a wide variety of disciplines. In many situations one seeks to estimate the underlying graphical model of a dataset that includes variables of different domains.