LECTURE 12: fixed effects and random effects

class: center, middle, inverse, title-slide

.title[
# LECTURE 12: fixed effects and random effects
]
.subtitle[
## FANR 6750 (Experimental design)
]
.author[
### <br/><br/><br/>Fall 2022
]

---

# outline

<br/>

1) Motivation

<br/>

2) Random effects in one-way ANOVA

<br/>

3) Random effects in blocked ANOVA

---
# motivation

#### The completely randomized ANOVA we have discussed can be applied to many different experiments. For example, it could be used to:

- Assess the best option from two different pesticides (plus no spray) used for gypsy moth control

- Assess if there is significant variability among the test scores of introductory statistics classes when taught by different instructors

#### In the first example, we would model the effect of each pesticide as a fixed effect

#### In the second example, the effect of each instructor would be modeled as a random effect

---
# fixed vs random effects

<br/>

#### A fixed effects model is appropriate when the treatment levels included in the experiment are exhaustive

<br/>

#### A random effects model is appropriate when the treatment levels in the experiment can be considered a sample from a larger population of interest

- In the previous example, we are interested in all instructors, not just the ones included in the experiment

---
# models

### Fixed-effects model

`$$\Large y_{ij} = \mu + \alpha_i + \epsilon_{ij}$$`
`$$\Large  \epsilon_{ij} \simnormal(0, \sigma^2)$$`
--

### Random-effects model

`$$\Large y_{ij} = \mu + \alpha_i + \epsilon_{ij}$$`
`$$\Large \alpha_{i} \simnormal(0, \sigma_A^2)$$`

`$$\Large \epsilon_{ij} \simnormal(0, \sigma^2)$$`

In a random effects model, the effects are assumed to be random variables following some probability distribution

---
# hypotheses

#### In a random-effects model, our interest is in assessing how much variation there is among all the effects in the population, not just the ones in our sample

#### As a result, our hypotheses must be written differently:

#### Fixed-effects model

`$$\large H_0 : \alpha_1 = \alpha_2 = ... = \alpha_a = 0$$`

`$$\large H_a : At\;least\;one\;inequality$$`

#### Random-effects model

`$$\large H_0 : \sigma^2_A = 0$$`

`$$\large H_a : \sigma^2_A > 0$$`

---
# implications?

#### Oddly enough, in the context of a one-way ANOVA, the analysis procedure is exactly the same for the fixed-effects and random-effects models

#### In other words, we can construct the ANOVA table and test the null hypothesis using the same procedure for the random-effects model as we used for the fixed-effects model, even though the interpretation of the results is slightly different

#### A comprehensive list of the distinctions between the two approaches is given in section 8.2 of Quinn et al.

#### The distinction between random and fixed effects becomes much more important in the more complicated models that we will cover soon

---
class: inverse, center, middle

# random effects in blocked anova

---
# blocked anova

<br/>

### Should we treat block effects as fixed or random?

<br/>

### As before, we need to answer the question:

> Can we view the blocks as samples of a larger population, or do the blocks represent all possible levels?

---
# random effects model for blocked design

`$$\huge y_{ij} = \mu + \alpha_i + \beta_j + \epsilon_{ij}$$`
`$$\Large \beta_{j} \sim normal(0, \sigma^2_B)$$`

`$$\Large \epsilon_{ij} \sim normal(0, \sigma^2)$$`

<br/>

#### This is often called a mixed effects model because it includes fixed and random effects

#### If we treated the `$\alpha$`'s as random too, we might call the model a *variance components model*

---
# hypotheses

.pull-left[
#### Main hypothesis

`$$H_0 : \alpha_1 = \alpha_2 = ... = \alpha_a = 0$$`

`$$H_a : At\;least\;one\;inequality$$`

`$$H_0 : \sigma^2_A = 0$$`

`$$H_a : \sigma^2_A > 0$$`
]
--

.pull-right[

#### Secondary hypothesis

`$$H_0 : \beta_1 = \beta_2 = ... = \beta_b = 0$$`

`$$H_a : At\;least\;one\;inequality$$`
OR

`$$H_0 : \sigma^2_B = 0$$`

`$$H_a : \sigma^2_B > 0$$`
]

---
# implications

#### Once again, treating block effects as random doesn't affect our calculations for the ANOVA table<sup>1</sup>  - it only affects the interpretation

.footnote[<sup>1</sup> Technically, this is only true for balanced designs]

---
# random-effects model in `r`

```r
aov1 <- aov(caterpillar ~ Treatment + Region, mothData)
summary(aov1)
```

<table class="table table-condensed" style="font-size: 10px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> sumsq </th>
   <th style="text-align:right;"> meansq </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Treatment </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 223.2 </td>
   <td style="text-align:right;"> 111.58 </td>
   <td style="text-align:right;"> 5.830 </td>
   <td style="text-align:right;"> 0.0392 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Region </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 430.9 </td>
   <td style="text-align:right;"> 143.64 </td>
   <td style="text-align:right;"> 7.505 </td>
   <td style="text-align:right;"> 0.0187 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 114.8 </td>
   <td style="text-align:right;"> 19.14 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>

```r
aov2 <- aov(caterpillar ~ Treatment + Error(Region), mothData)
summary(aov2)
```

<table class="table table-condensed" style="font-size: 10px; width: auto !important; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:left;"> stratum </th>
   <th style="text-align:left;"> term </th>
   <th style="text-align:right;"> df </th>
   <th style="text-align:right;"> sumsq </th>
   <th style="text-align:right;"> meansq </th>
   <th style="text-align:right;"> statistic </th>
   <th style="text-align:right;"> p.value </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Region </td>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 430.9 </td>
   <td style="text-align:right;"> 143.64 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Within </td>
   <td style="text-align:left;"> Treatment </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 223.2 </td>
   <td style="text-align:right;"> 111.58 </td>
   <td style="text-align:right;"> 5.83 </td>
   <td style="text-align:right;"> 0.0392 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Within </td>
   <td style="text-align:left;"> Residuals </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 114.8 </td>
   <td style="text-align:right;"> 19.14 </td>
   <td style="text-align:right;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>

---
# summary

- We usually model the main treatment effects as fixed, but sometimes we are interested in effects not in the sample

- Random effects models allow for inference about entire population of effects

- Random effects models make additional distributional assumptions

- Usually, you need many treatment levels (5-10) to get a decent estimate of the variance parameter associated with the random effects

- In spite of the big conceptual differences, the procedures aren't much different for one-way and blocked ANOVAs