Lecture 8

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# Lecture 8
## Introduction to hierarchical models
### WILD6900 (Spring 2021)

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## Readings

> Kéry & Schaub 73-82

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class: inverse, center, middle

# What are random effects?

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## What are random effects?

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

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- 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

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

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- Random effects are the realized values of a random variable

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

???

Definitions from Gelman & Hill (2007) pg. 245

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## A simple linear model

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

`$$\Large \epsilon_i \sim normal(0, \tau)$$`

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

--
```
model {
 # Priors
 beta0 ~ dnorm(0, 0.33)
 tau ~ dgamma(0.25, 0.25)
 
 # Likelihood
 for (i in 1:N){
 y[i] ~ dnorm(mu[i], tau) 
 mu[i] <- beta0
 } #i
}
```

---
## A simple linear model

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

`$$\Large \epsilon_i \sim normal(0, \tau)$$`

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

--
```
model {
 # Priors
 for(j in 1:J){
 beta0[j] ~ dnorm(0, 0.33)
 }
 tau ~ dgamma(0.25, 0.25)
 
 # Likelihood
 for (i in 1:N){
 y[i] ~ dnorm(mu[i], tau) 
 mu[i] <- beta0[group[j]]
 } #i
}
```
???

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

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

---
## A simple linear model

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

`$$\Large \epsilon_i \sim normal(0, \tau)$$`

--
- If `$\beta_{[j]} \sim normal(\mu_{\beta}, \tau_{\beta})$`

---
## A simple linear model

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

`$$\Large \epsilon_i \sim normal(0, \tau)$$`

--
- If `$\beta_{[j]} \sim normal(\mu_{\beta}, \tau_{\beta})$`

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

---
## Random effects

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- Only apply to factors

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- Imply grouped effects

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- Can include intercept, slope, and variance parameters

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- Assume exchangeability

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

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

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## Random effects = hierarchical model

`$$\Large [\beta_{[j]}, \mu_{\beta}, \tau_{\beta}, \tau|y_{ij}] = [y_{ij}|\beta_{[j]}, \tau][\beta_{[j]}|\mu_{\beta}, \tau_{\beta}][\tau][\mu_{\beta}][\tau_{\beta}]$$`

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- Can include multiple random effects

- Can include fixed and random effects (mixed-models)

- Can include multiple levels of hierarchy