Search

The default procedures of the software programs Mplus and lavaan tend to yield an inadmissible solution (also called a Heywood case) when the sample is small or the parameter is close to the boundary of the parameter space. In factor models, a negatively estimated variance does often occur. One strategy to deal with this is fixing the variance to zero and then estimating the model again in order to obtain the estimates of the remaining model parameters. In the present article, we present one possible approach for justifying this strategy. Specifically, using a simple one-factor model as an example, we show that the maximum likelihood (ML) estimate of the variance of the latent factor is zero when the initial solution to the optimization problem (i.e., the solution provided by the default procedure) is a negative value. The basis of our argument is the very definition of ML estimation, which requires that the log-likelihood be maximized over the parameter space. We present the results of a small simulation study, which was conducted to evaluate the proposed ML procedure and compare it with Mplus’ default procedure. We found that the proposed ML procedure increased estimation accuracy compared to Mplus’ procedure, rendering the ML procedure an attractive option to deal with inadmissible solutions.

Small sample sizes pose a severe threat to convergence and accuracy of between-group level parameter estimates in multilevel structural equation modeling (SEM). However, in certain situations, such as pilot studies or when populations are inherently small, increasing samples sizes is not feasible. As a remedy, we propose a two-stage regularized estimation approach designed for scenarios with both a small number of groups and small group sizes, and a low ICC. The method employs the wide format approach to multilevel SEM, where, at first, the sample covariance matrix is replaced by a shrinkage estimate, and then, this estimate is used to fit the SEM. By means of a simulation study, we evaluated the effectiveness of our two-stage approach. Our findings demonstrate that this method consistently ensures model convergence, provides more accurate between-level estimates, and even improves accuracy of within-level estimates in cases of very small group sizes.

A two-level data set can be structured in either long format (LF) or wide format (WF), and both have corresponding SEM approaches for estimating multilevel models. Intuitively, one might expect these approaches to perform similarly. However, the two data formats yield data matrices with different numbers of columns and rows, and their "cols : rows" is related to the magnitude of eigenvalue bias in sample covariance matrices. Previous studies have shown similar performance for both approaches, but they were limited to settings where "cols << rows" in both data formats. We conducted a Monte Carlo study to investigate whether varying "cols : rows" result in differing performances. Specifically, we examined the p:N ("cols : rows") effect on convergence and estimation accuracy in multilevel settings. Our findings suggest that (1) the LF approach is more likely to achieve convergence, but for the models that converged in both; (2) the LF and WF approach yield similar estimation accuracy; which is related to (3) differential "cols : rows" effects in both approaches; and (4) smaller ICC values lead to less accurate between-group parameter estimates.

Planning longitudinal studies can be challenging as various design decisions need to be made. Often, researchers are in search for the optimal design that maximizes statistical power to test certain parameters of the employed model. We provide a user-friendly Shiny app OptDynMo available at https://shiny.psychologie.hu-berlin.de/optdynmo that helps to find the optimal number of persons (N) and the optimal number of time points (T) for which the power of the likelihood ratio test (LRT) for a model parameter is maximal given a fixed budget for conducting the study. The total cost of the study is computed from two components: the cost to include one person in the study and the cost for measuring one person at one time point. Currently supported models are the cross-lagged panel model (CLPM), factor CLPM, random intercepts cross-lagged panel model (RI-CLPM), stable trait autoregressive trait and state model (STARTS), latent curve model with structured residuals (LCM-SR), autoregressive latent trajectory model (ALT), and the latent change score model (LCS).