Week 5: Individual Differences, part 1

class: center, middle, inverse, title-slide

# <b>Week 5: Individual Differences, part 1</b>
## Multivariate Statistics and Methodology using R (MSMR)<br><br>
### Dan Mirman
### Department of Psychology<br>The University of Edinburgh

---

# Individual differences

**Individual differences provide an additional level of analysis for understanding phenomena**

* At a group level, a treatment works better than a placebo, but why does it work better for some people than for others?
* People solve easy problems faster than hard problems, but why are some people a lot faster on the easy problems and other people only a little faster?

t-test and ANOVA methods treat individual differences as noise

Multilevel models provide two ways to quantify and analyse individual differences

---
# "External" individual differences can be added as fixed effects

Example: USA National Youth Survey longitudinal data on tolerance for deviant behaviour, exposure to deviant behaviour, and gender. Subset of 16 subjects printed in Willett (1997, Table 11.1). (`DeviantBehavior`)

`Tolerance` was measured by asking whether it is wrong for someone their age to:

* cheat on tests
* purposely destroy property of others
* hit or threaten someone without reason
* use alcohol
* use marijuana
* sell hard drugs
* steal something worth less than $5
* steal something worth more than $50
* break into a building or vehicle to steal

---
# Tolerance for deviant behaviour

```r
load("./data/DeviantBehavior.RData")
summary(DeviantBehavior)
```

```
##      SubjID      Gender      Exposure          Age       Tolerance   
##  S0009  : 5   Female:45   Min.   :0.810   Min.   :11   Min.   :1.00  
##  S0045  : 5   Male  :35   1st Qu.:0.922   1st Qu.:12   1st Qu.:1.22  
##  S0268  : 5               Median :1.145   Median :13   Median :1.50  
##  S0314  : 5               Mean   :1.191   Mean   :13   Mean   :1.62  
##  S0442  : 5               3rd Qu.:1.395   3rd Qu.:14   3rd Qu.:1.99  
##  S0514  : 5               Max.   :1.990   Max.   :15   Max.   :3.46  
##  (Other):50
```

![](msmr_lec05-1_IndivDiffs_Ext_files/figure-html/unnamed-chunk-2-1.png)

---
## Research question 1

**Does tolerance increase with age and is it modulated by gender?**

```r
m.base <- lmer(Tolerance ~ Age*Gender + (Age | SubjID),
                 contrasts=list(Gender="contr.sum"), data=DeviantBehavior)
```
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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">effect</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">group</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">term</th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">std.error</th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">df</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">p.value</th>
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  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">(Intercept)</td>
<td class="gt_row gt_right">-0.12563</td>
<td class="gt_row gt_right">0.52444</td>
<td class="gt_row gt_right">-0.2395</td>
<td class="gt_row gt_right">14.04</td>
<td class="gt_row gt_right">0.814143</td></tr>
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<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Age</td>
<td class="gt_row gt_right">0.13488</td>
<td class="gt_row gt_right">0.04403</td>
<td class="gt_row gt_right">3.0634</td>
<td class="gt_row gt_right">14.04</td>
<td class="gt_row gt_right">0.008399</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Gender1</td>
<td class="gt_row gt_right">0.35552</td>
<td class="gt_row gt_right">0.52444</td>
<td class="gt_row gt_right">0.6779</td>
<td class="gt_row gt_right">14.04</td>
<td class="gt_row gt_right">0.508852</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Age:Gender1</td>
<td class="gt_row gt_right">-0.03255</td>
<td class="gt_row gt_right">0.04403</td>
<td class="gt_row gt_right">-0.7392</td>
<td class="gt_row gt_right">14.04</td>
<td class="gt_row gt_right">0.471957</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">SubjID</td>
<td class="gt_row gt_left">sd__(Intercept)</td>
<td class="gt_row gt_right">1.75026</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">SubjID</td>
<td class="gt_row gt_left">cor__(Intercept).Age</td>
<td class="gt_row gt_right">-0.99242</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">SubjID</td>
<td class="gt_row gt_left">sd__Age</td>
<td class="gt_row gt_right">0.15204</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">Residual</td>
<td class="gt_row gt_left">sd__Observation</td>
<td class="gt_row gt_right">0.27235</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
  </tbody>
  
  
</table>
</div>

Significant increase with age, no significant effects of gender

---
## Research question 2

**Is this modulated by exposure to deviant behavior?**

`Exposure` scale is not clear and other effects would be estimated at `Exposure=0`, which is not an attested value.

If we center `Exposure`, its effects and the effects of other predictors will be easier to interpret.

```r
# center Exposure, but do not re-scale it
DeviantBehavior$Exposure.center <- scale(DeviantBehavior$Exposure, scale=FALSE)
```

Fit the full model

```r
m.exp1 <- lmer(Tolerance ~ Age*Gender*Exposure.center + (Age | SubjID), 
               data=DeviantBehavior, contrasts=list(Gender="contr.sum"))
```

---

```r
gt(tidy(m.exp1))
```

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<table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">effect</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">group</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">term</th>
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  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">(Intercept)</td>
<td class="gt_row gt_right">-0.38880</td>
<td class="gt_row gt_right">0.43008</td>
<td class="gt_row gt_right">-0.904</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.3837728</td></tr>
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<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Age</td>
<td class="gt_row gt_right">0.15603</td>
<td class="gt_row gt_right">0.03492</td>
<td class="gt_row gt_right">4.468</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.0007678</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Gender1</td>
<td class="gt_row gt_right">0.65192</td>
<td class="gt_row gt_right">0.43008</td>
<td class="gt_row gt_right">1.516</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.1554571</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Exposure.center</td>
<td class="gt_row gt_right">-4.22668</td>
<td class="gt_row gt_right">1.47540</td>
<td class="gt_row gt_right">-2.865</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.0142277</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Age:Gender1</td>
<td class="gt_row gt_right">-0.05921</td>
<td class="gt_row gt_right">0.03492</td>
<td class="gt_row gt_right">-1.696</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.1156925</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Age:Exposure.center</td>
<td class="gt_row gt_right">0.38532</td>
<td class="gt_row gt_right">0.11979</td>
<td class="gt_row gt_right">3.217</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.0073996</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Gender1:Exposure.center</td>
<td class="gt_row gt_right">3.62705</td>
<td class="gt_row gt_right">1.47540</td>
<td class="gt_row gt_right">2.458</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.0301300</td></tr>
    <tr><td class="gt_row gt_left">fixed</td>
<td class="gt_row gt_left">NA</td>
<td class="gt_row gt_left">Age:Gender1:Exposure.center</td>
<td class="gt_row gt_right">-0.28576</td>
<td class="gt_row gt_right">0.11979</td>
<td class="gt_row gt_right">-2.386</td>
<td class="gt_row gt_right">12</td>
<td class="gt_row gt_right">0.0344162</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">SubjID</td>
<td class="gt_row gt_left">sd__(Intercept)</td>
<td class="gt_row gt_right">1.22120</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">SubjID</td>
<td class="gt_row gt_left">cor__(Intercept).Age</td>
<td class="gt_row gt_right">-0.98960</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">SubjID</td>
<td class="gt_row gt_left">sd__Age</td>
<td class="gt_row gt_right">0.10379</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
    <tr><td class="gt_row gt_left">ran_pars</td>
<td class="gt_row gt_left">Residual</td>
<td class="gt_row gt_left">sd__Observation</td>
<td class="gt_row gt_right">0.27224</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td>
<td class="gt_row gt_right">NA</td></tr>
  </tbody>
  
  
</table>
</div>

* Significant main effects: **Age** and **Exposure**
* Significant interactions: **Age-by-Exposure**, **Gender-by-Exposure**, **Age-by-Gender-by-Exposure**

---
# How to plot the three-way interaction?

The three-way Age-by-Gender-by-Exposure interaction is a relationship among four variables (Tolerance for deviant behaviour, Exposure to deviant behaviour, Age, and Gender), three of which are continuous variables. **This is hard to visualize.**

To make it easier to visualize, can split the `Exposure` into levels

Option 1: Median split

```r
DeviantBehavior$ExposureMedSpl <- ifelse(
  DeviantBehavior$Exposure >= median(DeviantBehavior$Exposure), "High", "Low")
```

---

```r
ggplot(DeviantBehavior, aes(Age, Tolerance, color=ExposureMedSpl)) + 
  facet_wrap(~ Gender) + 
  stat_summary(fun=mean, geom="point", size=3) +
  stat_summary(fun.data=mean_se, geom="errorbar", width=0.2) + 
  stat_summary(aes(y=fitted(m.exp1)), fun=mean, geom="line") +
  labs(x="Age", y="Tolerance for deviant behavior", 
       color="Exposure\n(Median Split)") + 
  theme_bw() 
```

![](msmr_lec05-1_IndivDiffs_Ext_files/figure-html/unnamed-chunk-9-1.png)

Adolescents with higher Exposure to deviant behaviour tend to have increased Tolerance for deviant behaviour as they get older, and more so for males than females.

---
# How to plot the three-way interaction?

Option 2: tertile split provides a little more info than median split

```r
# define break points
b <- quantile(DeviantBehavior$Exposure, probs=seq(0, 1, by=1/3)) 
# split continuous predictor and provide level labels
DeviantBehavior$Exposure3 <- cut(DeviantBehavior$Exposure, 
                                 breaks=b, include.lowest=T, 
                                 labels=c("Low", "Medium", "High"))
```

---

```r
ggplot(DeviantBehavior, aes(Age, Tolerance, color=Exposure3)) + 
  facet_wrap(~ Gender) + 
  stat_summary(fun=mean, geom="point", size=3) + 
  stat_summary(fun.data=mean_se, geom="errorbar", width=0.2) + 
  stat_summary(aes(y=fitted(m.exp1)), fun=mean, geom="line") +
  labs(x="Age", y="Tolerance for deviant behavior", 
       color="Exposure\n(Tertiles)") + 
  theme_bw() 
```

![](msmr_lec05-1_IndivDiffs_Ext_files/figure-html/unnamed-chunk-11-1.png)

**Warning**: reviewers and readers may get confused about whether your *model* used continuous or categorical predictors, so you will need to be extra clear about this in your write-up.

---
# Key points

* Individual differences provide deeper insights into group-level phenomena
* Can assess them by adding them to the multilevel model
* Dividing a continuous variable into levels can help to visualise complex interactions