Mixed effect models

Module survey

Recap

• Yor data has structure (Batch, nested, repeated)
• Would you want know the mean (fixed) or don’t you care (random)
• \[ Y_{ij} = \alpha_j + \beta_j \,\text{bodylengths}_{ij} + \epsilon_{ij}, \quad \epsilon_{ij} \sim \mathcal{N}(0, \sigma^2) \]
• \[ Y_{ij} = \alpha + a_j + \beta \,\text{bodylengths}_{ij} + b_j \,\text{bodylengths}_{ij} + \epsilon_{ij}, \quad a_j \sim \mathcal{N}(0, \sigma_a^2), \quad b_j \sim \mathcal{N}(0, \sigma_b^2), \quad \epsilon_{ij} \sim \mathcal{N}(0, \sigma^2) \]

Identify Fixed vs Random Effects

Answers

Build the Models

A researcher studies the effect of temperature on metabolic rate in insects.

• Metabolic rate is measured for multiple individuals
  
• Each individual belongs to one of several colonies
  
• Body size is also recorded for each individual

Task

1. Write a linear model where the relationship between body size and metabolic rate can differ between colonies

2. Write a mixed effects model where colonies are treated as random effects, allowing both intercepts and slopes to vary

Linear model

• \[ Y_{ij} = \alpha_j + \beta_j \,\text{bodysize}_{ij} + \epsilon_{ij}, \quad \epsilon_{ij} \sim \mathcal{N}(0, \sigma^2) \]

Mixed effect model

• \[ Y_{ij} = \alpha + a_j + \beta \,\text{bodysize}_{ij} + b_j \,\text{bodysize}_{ij} + \epsilon_{ij} \]

\[ a_j \sim \mathcal{N}(0, \sigma_a^2), \quad b_j \sim \mathcal{N}(0, \sigma_b^2), \quad \epsilon_{ij} \sim \mathcal{N}(0, \sigma^2) \]