lecture 7: Random Effects

class: center, middle, inverse, title-slide

.title[
# lecture 7: Random Effects
]
.subtitle[
## FANR 8370
]
.author[
### Spring 2025
]

---

## Readings

> Kéry & Schaub 73-82

---
class: inverse, center, middle

# What are random effects?

---
## What are random effects?

--
- Fixed effects are constant across observational units, random effects vary across units

--
- Fixed effects are used when you are interested in the specific factor levels, random effects are used when you are interested in the underlying population

--
- When factors levels are exhaustive, they are fixed. When they are a sample of the possible levels, they are random

--
- Random effects are the realized values of a random variable

--
- Fixed effects are estimated by maximum likelihood, random effects are estimated with shrinkage

???

Definitions from Gelman & Hill (2007) pg. 245

---
## A simple linear model

`$$\Large y_{ij} = \beta_{[j]} + \epsilon_i$$`

`$$\Large \epsilon_i \sim Normal(0, \sigma^2)$$`

--
- If `$\beta_{[1]} = \beta_{[2]} = \beta_{[3]} = ...=\beta_{[J]}$`

--
```
nimbleCode({
 # Priors
 beta0 ~ dnorm(0, 1)
 sd ~ dgamma(0.25, 0.25)
 
 # Likelihood
 for (i in 1:N){
 y[i] ~ dnorm(mu[i], sd = sd) 
 mu[i] <- beta0
 } #i
})
```

---
## A simple linear model

`$$\Large y_{ij} = \beta_{[j]} + \epsilon_i$$`

`$$\Large \epsilon_i \sim Normal(0, \sigma^2)$$`

--
- If `$\beta_{[1]} \perp \beta_{[2]} \perp \beta_{[3]} \perp ...\perp \beta_{[J]}$`

--
```
nimbleCode({
 # Priors
 for(j in 1:J){
 beta0[j] ~ dnorm(0, 1)
 }
 sd ~ dgamma(0.25, 0.25)
 
 # Likelihood
 for (i in 1:N){
 y[i] ~ dnorm(mu[i], sd = sd) 
 mu[i] <- beta0[group[i]]
 } #i
})
```
???

nb `$\perp$` means "independent of"

This should look familiar - it's the means parameterization of the ANOVA model

In this case, since 'group' is not defined as a random variable, it must be provided as a constant to Nimble

---
## A simple linear model

`$$\Large y_{ij} = \beta_{[j]} + \epsilon_i$$`

`$$\Large \epsilon_i \sim Normal(0, \sigma^2)$$`

--
- If `$\beta_{[j]} \sim Normal(\mu_{\beta}, \sigma^2_{\beta})$`

---
## A simple linear model

`$$\Large y_{ij} = \beta_{[j]} + \epsilon_i$$`

`$$\Large \epsilon_i \sim Normal(0, \sigma^2)$$`

--
- If `$\beta_{[j]} \sim Normal(\mu_{\beta}, \sigma^2_{\beta})$`

--
```
nimbleCode({
 # Priors
 for(j in 1:J){
 beta0[j] ~ dnorm(mu.beta, sd = sd.beta)
 }
 mu.beta ~ dnorm(0, 0.33)
 sd.beta ~ dgamma(0.25, 0.25)
 sd.mu ~ dgamma(0.25, 0.25)
 
 # Likelihood
 for (i in 1:N){
 y[i] ~ dnorm(mu[i], sd = sd.mu) 
 mu[i] <- beta0[group[i]]
 } #i
})
```

---
## Random effects

--
- Only apply to factors

--
- Imply grouped effects

--
- Can include intercept, slope, and variance parameters

--
- Assume exchangeability

--
- Represent a compromise between total pooling `$(\beta_{[1]}=\beta_{[2]}=...=\beta_{[J]})$` and no pooling `$(\beta_{[1]} \perp \beta_{[2]} \perp ...\perp \beta_{[J]})$`

--
- Typically require `$>5-10$` factor levels

---
## Random effects = hierarchical model

`$$\Large [\beta_{[j]}, \mu_{\beta}, \sigma^2_{\beta}, \sigma^2|y_{ij}] = [y_{ij}|\beta_{[j]}, \sigma^2][\beta_{[j]}|\mu_{\beta}, \sigma^2_{\beta}][\sigma^2][\mu_{\beta}][\sigma^2_{\beta}]$$`

--
- Can include multiple random effects

- Can include fixed and random effects (mixed-effect models, multi-level models)

- Can include multiple levels of hierarchy