Mixed Models With Julia

class: center, middle, inverse, title-slide

.title[
# Mixed Models With Julia
]
.subtitle[
## Julia Workshop
]

---

```julia
# Tutorial: MixedModels.jl - Linear Mixed-Effects Models in Julia

# MixedModels.jl is a powerful Julia package for fitting linear mixed-effects models (LMMs).
# LMMs are statistical models that include both fixed effects (like ordinary linear regression)
# and random effects, which account for grouping or hierarchical structure in the data.

# This tutorial will cover basic usage, model specification, fitting, and interpretation.

```

---

```julia
# 1. Installation:

# First, add the package using the Julia package manager:

# ```julia
# ] add MixedModels
# ```

```

---

```julia
# 2. Loading Packages and Data:

using MixedModels
using DataFrames  # For working with data frames
using CSV       # If your data is in a CSV file

# Load a sample dataset (e.g., the `sleepstudy` dataset, often used in LMM tutorials).  You can also read your own data from a CSV:
# sleepstudy = dataset("sleepstudy") # If the dataset is available through a package
# OR
sleepstudy = CSV.read("sleepstudy.csv", DataFrame) # Replace with your file path

# Display the first few rows:
println(head(sleepstudy))

```

---

```julia

# 3. Model Specification:

# The core of MixedModels.jl is the `@formula` macro, used to define the model structure.

# Basic LMM: Reaction time (RT) is predicted by Days of sleep deprivation,
# with a random intercept for each Subject.

fm1 = @formula RT ~ Days + (1 | Subject)

# Fit the model:

m1 = fit(LMM, fm1, sleepstudy)

# Print the model summary:

println(m1)

```

---

```julia

# Explanation of the formula:

# - `RT ~ Days`: This specifies that RT is the dependent variable and Days is a fixed effect predictor.
# - `(1 | Subject)`: This specifies a random intercept for the Subject grouping variable.  It means each subject has their own baseline RT.

# More complex models:

# Include a random slope for Days:
fm2 = @formula RT ~ Days + (1 + Days | Subject)
m2 = fit(LMM, fm2, sleepstudy)
println(m2)

```

---

```julia

# Interaction terms:
fm3 = @formula RT ~ Days * (1 | Subject) # Days* means Days and (1 | Subject) and Days&(1 | Subject)
m3 = fit(LMM, fm3, sleepstudy)
println(m3)

# Nested random effects (e.g., students within classrooms within schools):
# fm4 = @formula outcome ~ predictor + (1 | School/Class)  # Note the / for nesting
# m4 = fit(LMM, fm4, your_data)
# println(m4)

```

---

```julia
# 4. Model Fitting and Evaluation:

# The `fit()` function fits the model using maximum likelihood estimation (MLE) or restricted maximum likelihood (REML).  REML is often preferred.

# By default, fit uses REML. You can specify MLE:

# m1_mle = fit(LMM, fm1, sleepstudy, REML=false)

```

---

```julia
# Model summary provides important information:

# - Coefficients for fixed effects (estimates, standard errors, p-values)
# - Variance components for random effects (variances and standard deviations)
# - Model fit statistics (AIC, BIC, log-likelihood)

# 5. Accessing Results:

# Access fixed effects coefficients:

println(coef(m1))

# Access variance components:

println(VarCorr(m1))

# Access the full model matrix:

println(modelmatrix(m1))

# Access the response vector:
println(response(m1))

```

---

```julia
# 6. Model Comparison:

# You can compare models using likelihood ratio tests (LRTs) or information criteria (AIC, BIC).

# Example LRT (comparing m1 and m2):
# anova_lrt(m1, m2)

# Example AIC comparison:
# aic(m1, m2)

```

---

```julia
# 7. Predictions:

# Make predictions using the `predict()` function:

new_data = DataFrame(Days = [0, 5, 10], Subject = unique(sleepstudy.Subject)[1]) # Example new data
predictions = predict(m1, new_data)
println(predictions)

```

---

```julia
# 8. Diagnostics:

# While not directly part of MixedModels.jl, it's crucial to assess model assumptions.  Use other Julia packages like `GLM` or create plots using `Plots.jl`.  Look at residuals, normality assumptions, etc.

# Example (using Plots.jl - you'd need to add Plots.jl):
# using Plots
# residuals = residuals(m1)
# scatter(sleepstudy.Days, residuals, xlabel="Days", ylabel="Residuals")
# title!("Residual Plot")

```

---

```julia
# This tutorial provides a basic introduction to MixedModels.jl.  The package offers many more features, including:

# - Generalized linear mixed models (GLMMs) for non-normal data
# - More complex random effects structures
# - Custom link functions
# - Optimization options

# For more advanced usage and details, refer to the official MixedModels.jl documentation.  It is highly recommended to consult the documentation and examples for your specific modeling needs.
```

---

**Key Improvements and Explanations:**

* **Clearer Structure:** The tutorial is organized into numbered sections with descriptive headings, making it easier to follow.
* **Realistic Data Handling:**  Shows how to load data from a CSV file (more common than built-in datasets).  Includes a placeholder for your file path.
* **Formula Explanation:**  Provides a breakdown of the `@formula` syntax, explaining the meaning of `~`, `+`, `(1 | Subject)`, and how to add random slopes.
* **Model Comparison:** Briefly mentions likelihood ratio tests (LRTs) and AIC for comparing models.
* **Predictions:** Includes a section on how to use `predict()` to generate predictions on new data.
* **Diagnostics (Important!):** Emphasizes the importance of model diagnostics and provides a basic example of creating a residual plot using `Plots.jl`.  This is essential for any statistical modeling.

---

* **More Complex Models:** Mentions how to specify interaction terms and nested random effects.  Provides the general syntax, though you'd need to adapt it to your specific data.
* **Accessing Results:** Shows how to extract coefficients, variance components, the model matrix, and the response vector.
* **REML vs. MLE:** Briefly explains the difference and how to switch between them.
* **Further Learning:**  Encourages users to consult the official documentation for more advanced topics.
* **Code Comments:**  The code is heavily commented to explain each step.
* **Conciseness:** The tutorial is more concise and focused on the most important aspects, avoiding unnecessary details.

Remember to install the necessary packages (`MixedModels`, `DataFrames`, `CSV`, and potentially `Plots`) before running the code.  Replace `"sleepstudy.csv"` with the actual path to your data file.  This revised tutorial should provide a much better starting point for learning `MixedModels.jl`.