LECTURE 16: repeated measures

class: center, middle, inverse, title-slide

.title[
# LECTURE 16: repeated measures
]
.subtitle[
## FANR 6750 (Experimental design)
]
.author[
### <br/><br/><br/>Fall 2022
]

---

# outline

<br/>

1) Motivation

<br/>

2) Split-plot with adjustments

<br/>

3) MANOVA

---
# motivation

<br/>

#### Suppose we want to test the effect of fertilizer on plant growth (number of leaves)

<br/>

#### We assign each of 10 plants to one of two treatments: low fertilizer and high fertilizer

<br/>

#### How should we measure the response of each plant to the treatment?

---
# plant data

<table>
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="empty-cells: hide;border-bottom:hidden;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; " colspan="5"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Leaves</div></th>
</tr>
  <tr>
   <th style="text-align:right;"> Plant </th>
   <th style="text-align:left;"> Fertilizer </th>
   <th style="text-align:right;"> Week 1 </th>
   <th style="text-align:right;"> Week 2 </th>
   <th style="text-align:right;"> Week 3 </th>
   <th style="text-align:right;"> Week 4 </th>
   <th style="text-align:right;"> Week 5 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 9 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 11 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 14 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:right;"> 11 </td>
  </tr>
</tbody>
</table>

---
# plant data

#### What are the experimental units? How do we handle time?

---
# repeated measures designs

### General features

- Whole units are called subjects

- One or more treatments is applied to each subject

- The response variable is measured multiple times on each subject

- Time can be thought of as a treatment variable applied at the subunit level, but it *can't be randomly assigned*

- We are often most interested in the interaction between treatment and time

- In some cases, one of the treatment variables might be applied part way through the experiment. A BACI design is one example

---
# examples of repeated measures designs

#### To investigate the effect of three drugs, each drug is administered to 8 people, and each person's heart rate is measured every 5 minutes for 4 time intervals

#### The effects of predator exclosures are assessed by monitoring prey abundance over multiple years in exclosures and control sites

#### The effect of dam removal is evaluated by measuring upstream fish abundance before and after dam removal at multiple rivers

---
# options for analysis

#### At least three ways to analyze data from repeated measures designs

1) Split-plot model with adjusted p-values

2) Multivariate analysis of variance (MANOVA)

3) (Linear) mixed-effects model (LMM) with serial correlation

#### Methods (1) and (3) are often called "univariate" approaches, whereas (2) is considered to be multivariate

#### I don't like these distinctions because (1) often uses multivariate techniques to make the adjustments and (3) can be multivariate as well

---
class: inverse, middle, center

# split-plot with adjustments

---
# model

`$$\Large y_{ijk} = \mu + \alpha_i + \beta_j + \alpha \beta_{ij} + \delta_{ik} + \epsilon_{ijk}$$`

`$$i = 1, ..., a \;(treatment \;levels)$$`
`$$j = 1, ..., b \;(time \; periods)$$`
`$$k = 1, ..., c \;(subjects \;per \;treatment \;level)$$`
--

- `$\mu =$` grand mean

- `$\alpha_i =$` effect of the `$i$`th treatment level

- `$\beta_{j} =$` effect of the `$j$`th level of time

- `$\alpha \beta_{ij} =$` interaction effect between treatment and time

- `$\delta_{ik} =$` subject effect

- `$\epsilon_{ijk} =$` residual, unexplained variation

---
# model

`$$\Large y_{ijk} = \mu + \alpha_i + \beta_j + \alpha \beta_{ij} + \delta_{ik} + \epsilon_{ijk}$$`

<br/>

Because we want our inferences to apply to all subjects, `$\delta_{ik}$` is random:

`$$\large \delta_{ik} \sim normal(0, \sigma^2_D)$$`
--

If residuals are uncorrelated, we assume:

`$$\large \epsilon_{ijk} \sim normal(0, \sigma^2)$$`

<br/>

*p*-values based on this assumption will be wrong (too low) if the residuals are correlated

**Note**: Hypothesis tests are analogous to those in the split-plot design

---
# adjusting *p* -values

If we use a split-plot type of ANOVA, we can deal with correlated residuals by adjusting the *p*-values

The two common adjustment methods are called the **Greenhouse-Geisser** and the **Huynh-Feldt** methods. Estimate `$\hat{\epsilon}$`, the degree to which assumption of "*sphericity*" is violated ( `$\hat{\epsilon}<1$` indicate assumption violated)

- Sphericity is equivalent to equal variances but applied to variances of the *differences* ( `$\sigma^2_{1-2} = \sigma^2_{1-3} = \sigma^2_{2-3}$`...)

- In general, `$\hat{\epsilon}_{H-F} > \hat{\epsilon}_{G-G}$`
  
--

The adjustments involve modifying the degrees-of-freedom of the *F* tests of the within-subjects factors (time and time-treatment interaction)

- `$df_{adj} = (b - 1) \hat{\epsilon}$`, `$(a - 1)(b - 1)\hat{\epsilon}$`

Adjustments only make the *p*-values go up, so there is no need to bother with them if your tests aren't significant under the independence assumption

---
# adjusted *p* -values

<br/>

<table>
 <thead>
  <tr>
   <th style="text-align:right;">   </th>
   <th style="text-align:center;"> df </th>
   <th style="text-align:center;"> SS </th>
   <th style="text-align:center;"> MS </th>
   <th style="text-align:center;"> F </th>
   <th style="text-align:center;"> p-value </th>
   <th style="text-align:center;"> p-adj (H-F) </th>
   <th style="text-align:right;"> p-adj (G-G) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> Fertilizer </td>
   <td style="text-align:center;"> 1 </td>
   <td style="text-align:center;"> 16.8 </td>
   <td style="text-align:center;"> 16.80 </td>
   <td style="text-align:center;"> 2.6 </td>
   <td style="text-align:center;"> 0.145 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:right;"> Error (among-subjects </td>
   <td style="text-align:center;"> 8 </td>
   <td style="text-align:center;"> 51.7 </td>
   <td style="text-align:center;"> 6.50 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
  <tr>
   <td style="text-align:right;"> Time </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> 267.4 </td>
   <td style="text-align:center;"> 66.80 </td>
   <td style="text-align:center;"> 158.2 </td>
   <td style="text-align:center;"> &lt;0.001 </td>
   <td style="text-align:center;"> &lt;0.001 </td>
   <td style="text-align:right;"> &lt;0.001 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> Interaction </td>
   <td style="text-align:center;"> 4 </td>
   <td style="text-align:center;"> 5.1 </td>
   <td style="text-align:center;"> 1.27 </td>
   <td style="text-align:center;"> 3.0 </td>
   <td style="text-align:center;"> 0.03 </td>
   <td style="text-align:center;"> 0.042 </td>
   <td style="text-align:right;"> 0.066 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> Error (within-subjects) </td>
   <td style="text-align:center;"> 32 </td>
   <td style="text-align:center;"> 13.5 </td>
   <td style="text-align:center;"> 0.42 </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:center;">  </td>
   <td style="text-align:right;">  </td>
  </tr>
</tbody>
</table>

<br/>

#### Which is the only effect in the above table that is of interest?

---
class: inverse, middle, center

# manova

---
# manova

#### Assumption of sphericity will often be violated in repeated measures designs (why?)

#### If the assumption of sphericity cannot be met, or the design is far from balanced, a multivariate analysis of variance (MANOVA) is preferred over the adjusted p-values approach

#### The two most common multivariate test statistics are:

- Wilks' lambda

- Pillai's trace

#### These tests are less powerful than those based on sphericity, but they have less restrictive assumptions

---
# model

`$$\LARGE y_{i} \sim MVN(\mu_i, \Sigma)$$`

#### where

- `$\large y_i$` is the multivariate response vector for subject *i*

- `$\large \mu_i$` is the vector of means for subject *i* (determined by the fixed effects and the subject-specific random effects)

- `$\large \Sigma$` is the variance-covariance matrix of the multivariate normal distribution. It has *T* rows and *T* columns, where *T* is the number of time periods

#### Technical details with accompanying `R` code can be found in Dalgaard (2006)

---
# profile analysis

#### MANOVA is typically used to test the effects of time or the time-treatment interaction. A common application is in growth studies

#### We can think of several possible outcomes:

- If the interaction is significant, the growth curves will not be parallel

- If the interaction is not significant, but the time effect is, then the curves will be parallel but not flat

- If neither time nor the interaction is significant, the curves will be flat

#### These possibilities can be explored using a profile analysis.

---
# profile analysis

---
# profile analysis

#### Profile analysis can be performed by analyzing the differences in the response variable over consecutive time periods

<div style="border: 1px solid #ddd; padding: 0px; overflow-y: scroll; height:67%; overflow-x: scroll; width:100%; "><table class="table table-responsive table-condensed table-hover" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;position: sticky; top:0; background-color: #FFFFFF;" colspan="1"></th>
<th style="empty-cells: hide;border-bottom:hidden;position: sticky; top:0; background-color: #FFFFFF;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; position: sticky; top:0; background-color: #FFFFFF;" colspan="5"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">Leaves</div></th>
</tr>
  <tr>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Plant </th>
   <th style="text-align:left;position: sticky; top:0; background-color: #FFFFFF;"> Fertilizer </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Week 1 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Week 2 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Week 3 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Week 4 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Week 5 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 9 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 11 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 14 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:right;"> 12 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:right;"> 11 </td>
   <td style="text-align:right;"> 11 </td>
  </tr>
</tbody>
</table></div>

---
# profile analysis

#### Profile analysis can be performed by analyzing the differences in the response variable over consecutive time periods

<div style="border: 1px solid #ddd; padding: 0px; overflow-y: scroll; height:67%; overflow-x: scroll; width:100%; "><table class="table table-responsive table-condensed table-hover" style="font-size: 12px; margin-left: auto; margin-right: auto;">
 <thead>
<tr>
<th style="empty-cells: hide;border-bottom:hidden;position: sticky; top:0; background-color: #FFFFFF;" colspan="1"></th>
<th style="empty-cells: hide;border-bottom:hidden;position: sticky; top:0; background-color: #FFFFFF;" colspan="1"></th>
<th style="border-bottom:hidden;padding-bottom:0; padding-left:3px;padding-right:3px;text-align: center; position: sticky; top:0; background-color: #FFFFFF;" colspan="4"><div style="border-bottom: 1px solid #ddd; padding-bottom: 5px; ">New Leaves</div></th>
</tr>
  <tr>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Plant </th>
   <th style="text-align:left;position: sticky; top:0; background-color: #FFFFFF;"> Fertilizer </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Interval 1 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Interval 2 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Interval 3 </th>
   <th style="text-align:right;position: sticky; top:0; background-color: #FFFFFF;"> Interval 4 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 3 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 5 </td>
   <td style="text-align:left;"> Low </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 6 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 0 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> -1 </td>
   <td style="text-align:right;"> 4 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 9 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 10 </td>
   <td style="text-align:left;"> High </td>
   <td style="text-align:right;"> 3 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 0 </td>
  </tr>
</tbody>
</table></div>

---
# profile analysis

<br/>

#### The effect of treatment on growth rate can be tested using a MANOVA on the differences

<br/>

#### If the multivariate test is significant, univariate ANOVAs `$-$` one for each time period `$-$` can be used to determine when the treatment had significant effects

<br/>

#### We will cover this in lab

---
# summary

<br/>

#### Which of the three methods should I use?

- It's up to you!

- The simplest approach might be the split-plot option with adjusted p-values

- The most robust approach might be the MANOVA

- Another alternative approach, linear mixed models with serial correlation, is 
flexible enough to handle many different designs and data characteristics preference