• Understand the difference between fixed and random effects

• Understand reasons to use random effects models

• Understand the benefits & costs of random effects models

Mixed effects models are known by many names

• Variance components models
  
• Random effects models
  
• Varying coefficients models
  
• Hierarchical linear models
  
• Multilevel models

• Ecological data are often messy, complex, and incomplete

• Data are often grouped by location, species, etc

• May have multiple samples from the same individual

• Often small sample sizes for some locations, species, etc

fixed factor: qualitative predictor (eg, sex)

fixed effect: quantitative change (“slope”)

fixed factor: qualitative predictor (eg, sex)

fixed effect: quantitative change (“slope”)

random factor: qualitative predictor whose levels are randomly sampled from a population (eg, age)

random effect: quantitative change whose levels are randomly sampled from a population

Fixed effects describe specific levels of factors that are not part of a larger group

Fixed effects describe specific levels of factors that are not part of a larger group

Random effects describe varying levels of factors drawn from a larger group

Random effects occur in 3 circumstances

1. nested (hierarchical) studies

(eg, fish within lakes, multiple lakes within a state)

Random effects occur in 3 circumstances

1. nested (hierarchical) studies

2. time series (longitudinal) studies

(eg, repeated measurements from the same place or individual)

Random effects occur in 3 circumstances

1. nested (hierarchical) studies

2. time series (longitudinal) studies

3. spatial studies

(eg, multiple trees within a plot)

Fixed effects influence only the mean of \(y\)

Random effects influence only the variance of \(y\)

Fish mass as a function of its length and specific lake

\[ y_{i,j} = \underbrace{\alpha + \beta x_{i,j} + \delta_{j}}_{\text{fixed}} + \underbrace{\epsilon_{i,j}}_{\text{random}} \]

\(y_i\) is the log(mass) for fish i in lake \(j\)

\(x_i\) is the log(length) for fish i in lake \(j\)

\(\delta_j\) is the mean log(mass) of fish in lake \(j\)

\(\epsilon_{i,j} \sim \text{N}(0,\sigma_\epsilon)\)

Fish mass as a function of its length and general lake

\[ y_{i,j} = \underbrace{\alpha + \beta x_{i,j}}_{\text{fixed}} + \underbrace{\delta_{j} + \epsilon_{i,j}}_{\text{random}} \]

\(y_i\) is the log(mass) for fish i in lake \(j\)

\(x_i\) is the log(length) for fish i in lake \(j\)

\(\delta_j\) is the mean log(mass) of fish in lake \(j\)

\(\epsilon_{i,j} \sim \text{N}(0,\sigma_\epsilon) ~ \text{and} ~ \delta_{j} \sim \text{N}(0,\sigma_\delta)\)

• Within-group errors are independent with mean zero and variance \(\sigma^2\)

• Within-group errors are independent of the random effects

• Random effects are normally distributed with mean zero and covariance \(\Psi\)

• Covariance matrix \(\Psi\) does not depend on the level

• Random effects are independent among different levels

In many cases, we can have multiple levels of random effects

trees within plots within forests within regions within states

• learning which variables are random effects

• correctly specifying the fixed and random effects in a model

• getting the nesting structure correct

Where does most of the variation occur & where would increased replication help?

What are the different levels of variation?

To qualify as true replicates, measurements must

• be independent

• not be part of a time series

• not be grouped in together in one place

• not be repeated on the same subject

Imagine a field experiment to test insecticide effects on plant2

• 20 plots: 10 sprayed & 10 unsprayed

• 50 plants within each plot

• each plant is measured 5 times

Imagine a field experiment to test insecticide effects on plants

• 20 plots: 10 sprayed & 10 unsprayed

• 50 plants within each plot

• each plant is measured 5 times

What are the degrees of freedom?

20 \(\times\) 50 \(\times\) 5 = 5000 (?)

Imagine a field experiment to test insecticide effects on plants

• 20 plots: 10 sprayed & 10 unsprayed

• 50 plants within each plot

• each plant is measured 5 times

What are the degrees of freedom?

20 \(\times\) 50 \(\times\) 5 = 5000 (?)

2 \(\times\) 9 = 18 (!)

Consider a simple one-way ANOVA model

\[ y_{ij} = \mu + \alpha_j + \epsilon_{ij} \\ \epsilon_{ij} \sim \text{N}(0, \sigma^2_{\sigma}) \]

where the group-level means \(\alpha_j\) are fixed

Now consider this one-way ANOVA model

\[ y_{ij} = \mu + \alpha_j + \epsilon_{ij} \\ \epsilon_{ij} \sim \text{N}(0, \sigma^2_{\sigma}) \\ \alpha_j \sim \text{N}(0, \sigma^2_{\alpha}) \]

where the group-level means \(\alpha_j\) are random

The means in the fixed effect model are independent

The means in the random effects model are correlated

The means in the fixed effect model are independent

The means in the random effects model are correlated

\[ \rho = \frac{\sigma^2_{\alpha}}{\sigma^2_{\alpha} + \sigma^2_{\epsilon}} \]

The correlation depends on the relative size of \(\sigma^2_{\alpha}\) vs \(\sigma^2_{\epsilon}\)

In fixed effects models, the group means are

\[ \alpha_j = \bar{y} - \mu \]

In fixed effects models, the group means are

\[ \alpha_j = \bar{y} - \mu \]

In random effects models, the group means “shrink” towards one another

\[ \alpha_j = (\bar{y} - \mu) \left( \frac{\sigma^2_{\alpha}}{\sigma^2_{\alpha} + \sigma^2_{\epsilon}} \right) \]

Consider what happens to \(\alpha_j\) as \(\sigma^2_{\alpha} \rightarrow \infty\)

\[ \alpha_j = (\bar{y} - \mu) \left( \frac{\sigma^2_{\alpha}}{\sigma^2_{\alpha} + \sigma^2_{\epsilon}} \right) \\ \Downarrow \\ \begin{aligned} \alpha_j &= (\bar{y} - \mu) \left( \frac{\infty}{\infty + \sigma^2_{\epsilon}} \right) \\ &= \bar{y} - \mu \end{aligned} \]

As \(\sigma^2_{\alpha} \rightarrow \infty\), our random effects become increasingly independent

\[ \alpha_j \sim \text{N}(0, \sigma^2_{\alpha}) \\ \Downarrow \\ \alpha_j \sim \text{Unif}(-\infty, \infty) \]

• Broadens our inference to a larger population

• Larger groups inform smaller groups (“Robin Hood Effect”)

• Broadens our inference to a larger population

• Larger groups inform smaller groups (“Robin Hood Effect”)

• Represents a “compromise” in terms of information used
  • fixed effects: no grouping (“no pooling”)
  • random effects: some grouping (“partial pooling”)
  • none: all one group (“complete pooling”)

• Precision decreases with number of levels

• Random effects perhaps more difficult to explain