Mixed effects algebra#

Starter simulations#

We’ll take a simulation-based approach to understanding mixed effects and how they perform. We’ll address questions including:

  • Is the 2-stage summary statistics approach valid?

  • Is is as powerful as a full mixed effects model?

  • What is the power and false positive rate of different approaches and implementations?

  • What happens when we violate some of the assumptions?

First, we’ll generate some data under a simple mixed effects scenario (a simple generative model) and explore some plots of the data. We’ll fit the model using several mixed effects models:

  • Matlab’s fitlme

  • CANlab’s glmfit_multilevel

  • CANlab’s igls and rigls

The live script below also reviews the main elements of the matrix algebra underlying mixed effects, using the notation in [Lindquist et al. 2012](papers/Lindquist_2012_variance components_in_multi-level_GLM.pdf). The model describes the basic framework and Iterative Generalized Least Squares, one way of estimating variance components. Other models have a similar structure but differ in some details (e.g., Bates et al. 2015 for LMER in R).

Download the Matlab live script

Activities#

  1. Identify and compare residual variance terms, slope estimates, t-values across models

  2. Modify the code to generate one or more continuous predictors instead of categorical/experimental ones.

  3. Turn the data-generation script into a function, so that you can efficiently simulate power and false positive rates using repeated simulations – two key properties of any statistical model.

Answer key For a pre-cooked answer (Matlab), see: Download a Matlab data sim function

But…don’t peek! Do it yourself!!!