Trying to Understand Mixed Models (from an Experimental Perspective)

This repository contains a Quarto report documenting my attempt to better understand mixed models.

I did not start from statistical theory.

I started from a practical question in analytical chemistry:

how much of the variability I observe is due to the method, and how much comes from the matrix?

Motivation

In many experiments, especially in analytical chemistry, data are not truly independent.

Measurements are often:

• repeated within the same system
• influenced by hidden structure (batch, day, instrument, matrix)

Ignoring this structure can lead to:

• overestimating the amount of information
• underestimating uncertainty
• drawing misleading conclusions

This project explores these issues through a concrete example.

The example

The report is based on a simulated experiment:

• 6 soil samples
• 3 temperatures
• 4 extraction solvents
• 2 replicates per condition

The response variable is extraction yield.

The key feature of the experiment is that:

observations within the same soil are more similar than observations across soils.

What this project tries to do

This is not a tutorial.

It is a structured attempt to answer a few questions:

• What is the real experimental unit?
• When are observations not independent?
• What changes when I move from lm() to lmer()?
• What am I actually estimating with a random effect?
• How does the structure of the experiment affect uncertainty?

Key ideas explored

• The model must reflect how the experiment was physically performed
• Variability is not a single quantity; it has structure
• More data does not necessarily mean more information
• Mixed models are not about complexity, but about alignment with reality

Contents

• report.qmd
  Main Quarto document containing the full analysis and narrative

• R/create_dataset.R
  Script used to generate the dataset

Tools

The analysis is intentionally kept minimal:

• data.table for data manipulation
• ggplot2 for visualization
• lme4 and lmerTest for mixed models

How to run

1. Clone the repository:

git clone https://github.com/andreabz/mixed_models.git
cd mixed_models

2. Reproduce the R environment:

install.packages("renv")
renv::restore()

3. Render the report

quarto render report.qmd

Notes

This work reflects an ongoing learning process, some interpretations may evolve as my understanding improves. The goal is not to provide definitive answers, but to make the reasoning explicit.

Why this might be useful

If you work with experimental data, especially in applied contexts:

• chemistry
• biology
• engineering

you may recognize situations where observations are treated as independent by default. This repository is an attempt to question that assumption.

Next steps

Areas I am still trying to understand:

• how many groups are needed to estimate variability reliably
• when mixed models materially change decisions
• how to connect variance components to experimental design choices