Week 9: Scaling, Contrasts, Interactions

# <b>Week 9: Scaling, Contrasts, Interactions</b>
## Univariate Statistics and Methodology using R
### Martin Corley
### Department of Psychology<br/>The University of Edinburgh

---

---
# Learning to Read

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![](lecture_8_files/img/playmo_teach.jpg)
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<div id="yzrzfegnqd" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">hrs_wk</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">method</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">R_AGE</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.115</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.971</td>
      <td class="gt_row gt_center">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">14.272</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">9.940</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.677</td>
      <td class="gt_row gt_center">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">13.692</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.060</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.619</td>
      <td class="gt_row gt_center">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.353</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">9.269</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.894</td>
      <td class="gt_row gt_center">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">12.744</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.991</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.035</td>
      <td class="gt_row gt_center">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">15.353</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.535</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.272</td>
      <td class="gt_row gt_center">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.798</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.150</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.871</td>
      <td class="gt_row gt_center">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.691</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">7.941</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.053</td>
      <td class="gt_row gt_center">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.988</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.233</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.474</td>
      <td class="gt_row gt_center">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.713</td>
    </tr>
    <tr>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.219</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">4.038</td>
      <td class="gt_row gt_center">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.908</td>
    </tr>
  </tbody>
  
  
</table></div>
]]

---
# Learning to Read

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -2.423      2.472   -0.98    0.332    
## age            0.938      0.206    4.55 0.000038 ***
## hrs_wk         0.964      0.418    2.31    0.025 *  
## ...
```

---
count: false
# Learning to Read

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
*## (Intercept)   -2.423      2.472   -0.98    0.332    
## age            0.938      0.206    4.55 0.000038 ***
## hrs_wk         0.964      0.418    2.31    0.025 *  
## ...
```

- as we noted last week, the _intercept_ for this model is nonsensical

+ "children aged zero who read for zero hours a week have a predicted reading age of -2.423"

- perhaps there's something we can do about this?

---
# One-Predictor Model

```r
# model
mod2 <- lm(R_AGE ~ age,data=reading)

# figure
p <- reading %>% ggplot(aes(x=age,y=R_AGE)) +
  xlab("age") + ylab("reading age") +
  geom_point(size=3) +
  geom_smooth(method="lm")
p
```

]
.pull-right[
![](lecture_8_files/figure-html/ggp-1.svg)
]

.footnote[
<sup>1</sup> we know this model doesn't meet assumptions, but it will work for an illustration
]
???
- animate the x-axis changes? (Unlikely to work, maybe as .svg)
---
# Changing the Intercept

- we can just pick a "useful-looking" value

- for example, we might want the intercept to tell us about students at age 8

+ this is a decision; no magic about it
]

]
---
count: false

# Changing the Intercept

- we can just pick a "useful-looking" value

- for example, we might want the intercept to tell us about students at age 8

+ this is a decision; no magic about it
]

]

---
# A Model With a New Intercept

### original model

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.764      1.753    1.01     0.32    
## age            1.012      0.212    4.76 0.000018 ***
## ...
```

### new model

```r
*mod2b <- lm(R_AGE ~ I(age-8), data=reading)
summary(mod2b)
```

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    9.862      0.383   25.77  < 2e-16 ***
## I(age - 8)     1.012      0.212    4.76 0.000018 ***
## ...
```

---
# Fit Remains Unchanged

### original model

```
## ...
## Multiple R-squared:  0.321,	Adjusted R-squared:  0.307 
## F-statistic: 22.7 on 1 and 48 DF,  p-value: 0.0000179
```

### new model

```
## ...
## Multiple R-squared:  0.321,	Adjusted R-squared:  0.307 
## F-statistic: 22.7 on 1 and 48 DF,  p-value: 0.0000179
```

---
# A Model with a New Slope

- we can just pick a "useful" scale

- for example, we might want to examine the effect per month of age

+ this is a decision; no magic about it
]

---
count: false
# A Model with a New Slope

- we can just pick a "useful" scale

- for example, we might want to examine the effect per month of age

+ this is a decision; no magic about it
]

---
# A Model With a New Slope

### original model

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    1.764      1.753    1.01     0.32    
## age            1.012      0.212    4.76 0.000018 ***
## ...
```

### new model

```r
*mod2c <- lm(R_AGE ~ I(age*12), data=reading)
summary(mod2c)
```

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   1.7638     1.7534    1.01     0.32    
## I(age * 12)   0.0844     0.0177    4.76 0.000018 ***
## ...
```

---
# Fit Remains Unchanged

### original model

```
## ...
## Multiple R-squared:  0.321,	Adjusted R-squared:  0.307 
## F-statistic: 22.7 on 1 and 48 DF,  p-value: 0.0000179
```

### new model

```
## ...
## Multiple R-squared:  0.321,	Adjusted R-squared:  0.307 
## F-statistic: 22.7 on 1 and 48 DF,  p-value: 0.0000179
```

---
# We Can Get Fancy About This

```r
mod.mb <- lm(R_AGE ~ I((age-8)*12) + I(hrs_wk-mean(hrs_wk)), data=reading)
summary(mod.mb)
```

```
## 
## Call:
## lm(formula = R_AGE ~ I((age - 8) * 12) + I(hrs_wk - mean(hrs_wk)), 
##     data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -4.385 -2.251  0.326  2.395  3.201 
## 
## Coefficients:
##                          Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                9.8659     0.3665   26.92  < 2e-16 ***
## I((age - 8) * 12)          0.0782     0.0172    4.55 0.000038 ***
## I(hrs_wk - mean(hrs_wk))   0.9636     0.4176    2.31    0.025 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.59 on 47 degrees of freedom
## Multiple R-squared:  0.39,	Adjusted R-squared:  0.364 
## F-statistic:   15 on 2 and 47 DF,  p-value: 0.00000896
```

---
# Which Has a Bigger Effect?

.flex.items-center[
.w-40.pa2[
![](lecture_8_files/img/playmo_age.jpg)
]

.w-60.pa2[
- in our two-predictor model, is age more important than practise?  Or vice-versa?

- hard to tell because the predictors are in different _units_

-  how do we compare effects of a year of age to those of an hour per week of practise?
]]

---
# Standardisation

- _if_ the predictors and outcome are very roughly normally distributed...

.center[
![](lecture_8_files/figure-html/hist-1.svg)
]
- we can calculate `$z$`-scores by subtracting the mean and dividing by the standard deviation

`$$z_i=\frac{x_i-\bar{x}}{\sigma_x}$$`

---
# Standardisation

- in R, the `scale()` function calculates `$z$`-scores

- in R, you don't need to create new columns!
  + also don't need to use `I()` because no ambiguity (though you can use it if you want)

```r
mod.ms <- lm(scale(R_AGE) ~ scale(age) + scale(hrs_wk), data=reading)
```

- the variables are now _all_ in terms of standard deviations from the mean

- at the _intercept_, `age` is the mean of age and `hrs_wk` is the mean of hrs_wk

- _slopes_: "how many standard deviations does `R_AGE` change for a one standard deviation change in the predictor?"

---
# Standardisation

```r
summary(mod.ms)
```

```
## ...
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   -6.86e-17   1.13e-01    0.00    1.000    
## scale(age)     5.25e-01   1.15e-01    4.55 0.000038 ***
## scale(hrs_wk)  2.66e-01   1.15e-01    2.31    0.025 *  
## ...
```

- `R_AGE` changes 0.52 sds for a 1-sd change in `age`, and 0.27 sds for a 1-sd change in `hrs_wk`

- reasonable conclusion might be that age has a greater effect on reading age than does practice

???
- we _expect_ the intercept to be zero because it's the mean of everything

- model fit doesn't change with standardisation

```
## ...
## Multiple R-squared:  0.39,	Adjusted R-squared:  0.364 
## F-statistic:   15 on 2 and 47 DF,  p-value: 0.00000896
```

---
# Standardisation Post-Hoc

- we can convert "raw" model coefficients `$b$` to standardised coefficients `$\beta$` without re-running the regression]

- for predictor `$x$` of outcome `$y$`:

`$$\beta_x=b_x\cdot{}\frac{\sigma_x}{\sigma_y}$$`
- or there's a function to do it for you

```r
library(lsr)
standardCoefs(mod.m)
```

```
##             b   beta
## age    0.9378 0.5250
## hrs_wk 0.9636 0.2662
```

---
class: inverse, center, middle, animated, rotateInDownLeft

# End of Part 1

---
class: inverse, center, middle

# Part 2
## Categorical Predictors

---
# Playmobil vs. SuperZings

.pull-left[
![](lecture_8_files/img/playmo_2.jpg)
]
.pull-right[
- some important pretesting went into these lectures

- every individual figure rated for "usefulness" in explaining stats

- how do we decide which to use?
]

---
count: false
#Playmobil vs. SuperZings

.pull-left[
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<div id="azvvkhqjvb" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <thead class="gt_col_headings">
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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">type</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">UTILITY</th>
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  </thead>
  <tbody class="gt_table_body">
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">3.8</td>
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    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">3.8</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">7.5</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">3.7</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">6.1</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">0.3</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">5.4</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">3.7</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">3.6</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">2.6</td>
    </tr>
  </tbody>
  
  
</table></div>

]
.pull-right[
- some important pretesting went into these lectures

- every individual figure rated for "usefulness" in explaining stats

- how do we decide which to use?

]
---
# Playmobil vs. SuperZings

.pull-left[
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  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">type</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">UTILITY</th>
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  </thead>
  <tbody class="gt_table_body">
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">3.8</td>
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    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">3.8</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">7.5</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">3.7</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">6.1</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">0.3</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">5.4</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">3.7</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(255,0,0,0.3); color: #FFFFFF;">playmo</td>
      <td class="gt_row gt_right">3.6</td>
    </tr>
    <tr>
      <td class="gt_row gt_center" style="background-color: rgba(0,0,255,0.3); color: #FFFFFF;">zing</td>
      <td class="gt_row gt_right">2.6</td>
    </tr>
  </tbody>
  
  
</table></div>
]

- we already know one way to answer this

```r
t.test(UTILITY~type, data=toys,
*      var.equal=TRUE)
```

```
## 
## 	Two Sample t-test
## 
## data:  UTILITY by type
*## t = 2.5, df = 8, p-value = 0.04
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  0.1809 4.7391
## sample estimates:
## mean in group playmo   mean in group zing 
##                 5.28                 2.82
```
]

---
# The Only Equation You'll Ever Need

- which toys are the most useful?

.br3.pa2.bg-gray[
`$$\color{red}{\textrm{outcome}_i}\color{white}=\color{blue}{(\textrm{model})_i}\color{white}{+\textrm{error}_i}$$`

`$$\color{red}{\textrm{utility}_i}\color{white}{=}\color{blue}{(\textrm{some function of type})_i}\color{white}{+\epsilon_i}$$`
]

- we need to represent `type` as a number

- the simplest way of doing this is to use 0 or 1
---
# Quantifying a Nominal Predictor

```r
toys <- toys %>% 
  mutate(is_playmo = 
           ifelse(type=="playmo",1,0))
toys
```

```
## # A tibble: 10 x 3
##    type   UTILITY is_playmo
##    <fct>    <dbl>     <dbl>
##  1 playmo     3.8         1
##  2 zing       3.8         0
##  3 playmo     7.5         1
##  4 zing       3.7         0
##  5 playmo     6.1         1
##  6 zing       0.3         0
##  7 playmo     5.4         1
##  8 zing       3.7         0
##  9 playmo     3.6         1
## 10 zing       2.6         0
```
]

`$$\color{red}{\textrm{utility}_i}=\color{blue}{b_0+b_1\cdot{}\textrm{is_playmo}}+\epsilon_i$$`

- `$\overline{\textrm{utility}}$` for SuperZings is **intercept**

- "change due to playmo-ness" is **slope**

]

---
# Linear Model Using `is_playmo`

```r
mod1 <- lm(UTILITY~is_playmo,data=toys)
summary(mod1)
```

```
## 
## Call:
## lm(formula = UTILITY ~ is_playmo, data = toys)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##  -2.52  -1.17   0.47   0.88   2.22 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)   
## (Intercept)    2.820      0.699    4.04   0.0038 **
*## is_playmo      2.460      0.988    2.49   0.0376 * 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.56 on 8 degrees of freedom
## Multiple R-squared:  0.436,	Adjusted R-squared:  0.366 
## F-statistic:  6.2 on 1 and 8 DF,  p-value: 0.0376
```
???
- note that the `$t$` value (and the `$p$` value) are exactly what we got from our initial `$t$`-test

- the `$t$`-test is just a special case of the linear model
---
# Let R Do the Work

```r
contrasts(toys$type)
```

```
##        zing
## playmo    0
## zing      1
```

- already built-in to factors

- NB the first value will be the default intercept (because `$b_n=0$` for that value)

+ can change this using the `relevel()` function (or tidyverse `fct_relevel()`)
  
- as long as we have a _factor_, can just use lm() with that column

---
# Linear Model Using `type`

```r
mod2 <- lm(UTILITY~type, data=toys)
summary(mod2)
```

```
## 
## Call:
## lm(formula = UTILITY ~ type, data = toys)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##  -2.52  -1.17   0.47   0.88   2.22 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    5.280      0.699    7.56 0.000066 ***
*## typezing      -2.460      0.988   -2.49    0.038 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.56 on 8 degrees of freedom
## Multiple R-squared:  0.436,	Adjusted R-squared:  0.366 
## F-statistic:  6.2 on 1 and 8 DF,  p-value: 0.0376
```

???
- point out why sign is reversed!

- point out `typezing` label
---
# Graphically

- shows "what the model is doing", but isn't a very good presentation

- the line suggests you can make predictions for types between _playmo_ and _zing_
---
# Graphically

- error bars represent one standard error of the mean

---
# What About Lego Figures?

- we now have three groups

- can't label them `c(0, 1, 2)` because that would express a linear relationship

![](lecture_8_files/figure-html/minig-1.svg)

]

---
# Independent Effects

- "change due to lego-ness" is _independent_ of change due to anything else

- solution: add another predictor

```r
toys <- toys %>% mutate(
  is_playmo = ifelse(type=="playmo",1,0),
  is_lego   = ifelse(type=="lego",1,0)
)
head(toys)
```

```
## # A tibble: 6 x 4
##   type   UTILITY is_playmo is_lego
##   <fct>    <dbl>     <dbl>   <dbl>
## 1 zing       3.8         0       0
## 2 playmo     3.8         1       0
## 3 lego       1.8         0       1
## 4 zing       3.7         0       0
## 5 playmo     7.5         1       0
## 6 lego       3.6         0       1
```

---
# Three-Level Predictor: Two Coefficients

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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">UTILITY</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">is_playmo</th>
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  <tbody class="gt_table_body">
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      <td class="gt_row gt_center" style="background-color: #A2A2FF;">zing</td>
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      <td class="gt_row gt_right">1</td>
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</table></div>
]

.w-60.pa2[

`$$\textrm{UTILITY}_i=\color{blue}{b_0}\color{gray}{+b_1\cdot\textrm{is_playmo}_i+b_2\cdot\textrm{is_lego}_i}+\epsilon_i$$`

`$$\textrm{UTILITY}_i=\color{blue}{b_0}\color{gray}{+b_1\cdot0+b_2\cdot0}+\epsilon_i$$`
.tc.pt3[
"utility of a zing"
]
]]

---
count: false
# Three-Level Predictor: Two Coefficients

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      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">type</th>
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">is_playmo</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">is_lego</th>
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      <td class="gt_row gt_center">lego</td>
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      <td class="gt_row gt_right">0</td>
      <td class="gt_row gt_right">1</td>
    </tr>
  </tbody>
  
  
</table></div>
]

.w-60.pa2[

`$$\textrm{UTILITY}_i=\color{blue}{b_0}\color{red}{+b_1\cdot\textrm{is_playmo}_i}+\color{gray}{b_2\cdot\textrm{is_lego}_i}+\epsilon_i$$`
`$$\textrm{UTILITY}_i=\color{blue}{b_0}\color{red}{+b_1\cdot1}+\color{gray}{b_2\cdot0}+\epsilon_i$$`
.tc.pt3[
"change in utility from a zing due to being a playmo"
]
]]

---
count: false
# Three-Level Predictor: Two Coefficients

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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">is_playmo</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">is_lego</th>
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      <td class="gt_row gt_right">0</td>
    </tr>
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      <td class="gt_row gt_right">1</td>
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      <td class="gt_row gt_center" style="background-color: #FFA2A2;">lego</td>
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</table></div>
]

.w-60.pa2[

`$$\textrm{UTILITY}_i=\color{blue}{b_0}\color{gray}{+b_1\cdot\textrm{is_playmo}_i}+\color{red}{b_2\cdot\textrm{is_lego}_i}+\epsilon_i$$`
`$$\textrm{UTILITY}_i=\color{blue}{b_0}\color{gray}{+b_1\cdot0}+\color{red}{b_2\cdot1}+\epsilon_i$$`
.tc.pt3[
"change in utility from a zing due to being a lego"
]
]]

---
# R Has Our Backs

- known as **treatment** (or **dummy**) contrasts

```r
contrasts(toys$type)
```

```
##        playmo lego
## zing        0    0
## playmo      1    0
## lego        0    1
```
]

```r
# core R: alphabetical
contrasts(as.factor(toys$type))
```

```
##        playmo zing
## lego        0    0
## playmo      1    0
## zing        0    1
```

```r
# tidyverse: order of mention
contrasts(as_factor(toys$type))
```

```
##        playmo lego
## zing        0    0
## playmo      1    0
## lego        0    1
```

]
---
# A Linear Model

```r
mod <- lm(UTILITY ~ type, data=toys)
summary(mod)
```

```
## 
## Call:
## lm(formula = UTILITY ~ type, data = toys)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##  -2.52  -0.70   0.12   0.88   2.22 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    2.820      0.612    4.61  0.00061 ***
## typeplaymo     2.460      0.866    2.84  0.01487 *  
## typelego      -0.020      0.866   -0.02  0.98195    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.37 on 12 degrees of freedom
## Multiple R-squared:  0.475,	Adjusted R-squared:  0.387 
## F-statistic: 5.42 on 2 and 12 DF,  p-value: 0.021
```
???
- this is like a form of multiple regression

+ 3 levels -> 2 predictors

- so we can say that playmo are better than zing (we can see that zing is the intercept because it's omitted from the table)

- we can say that lego are not better than zing

- _playmo and lego are not directly compared_

- to do this, we would need a different contrast coding scheme

---
![:scale 5%](lecture_8_files/img/danger.svg) &nbsp;&nbsp; .f1[Graphically]

```r
gd <- toys %>% group_by(type) %>%
  summarise(mean_se(UTILITY))
gd
```

```
## # A tibble: 3 x 4
##   type       y  ymin  ymax
##   <fct>  <dbl> <dbl> <dbl>
## 1 zing    2.82  2.15  3.49
## 2 playmo  5.28  4.55  6.01
## 3 lego    2.8   2.42  3.18
```

```r
gd %>% ggplot(aes(x=type,y=y,
                  ymin=ymin,ymax=ymax)) +
  geom_bar(stat="identity") +
  geom_errorbar(width=.2) +
  ylab("UTILITY")
```

]
.pull-right[
![](lecture_8_files/figure-html/ggpB2-1.svg)
]
---
# Contrast Coding

- there may be a different contrast coding which better suits our research interests

- for a predictor with `$g$` levels (or "groups") there are `$g-1$` possible contrasts

- these can be anything you like (values don't have to be zero or one):  there are a few built in to R

- usefulness depends on your research question

- contrasts act like "tests of differences of interest" once your model has been fit

- model fit is not affected by the choice of contrasts<sup>1</sup>

---
class: inverse, center, middle, animated, rotateInDownLeft

# End of Part 2
---
class: inverse, center, middle

# Part 3
## Interactions
---
# Back to Reading

.pull-left[
![](lecture_8_files/img/playmo_teach.jpg)
]
.pull-right[
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<div id="lwnkvpxysq" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">hrs_wk</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1">method</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">R_AGE</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr>
      <td class="gt_row gt_right">10.115</td>
      <td class="gt_row gt_right">4.971</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">14.272</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">9.940</td>
      <td class="gt_row gt_right">4.677</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">13.692</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">6.060</td>
      <td class="gt_row gt_right">4.619</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">10.353</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">9.269</td>
      <td class="gt_row gt_right">4.894</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">12.744</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">10.991</td>
      <td class="gt_row gt_right">5.035</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">phonics</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">15.353</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">6.535</td>
      <td class="gt_row gt_right">5.272</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.798</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">8.150</td>
      <td class="gt_row gt_right">6.871</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.691</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">7.941</td>
      <td class="gt_row gt_right">4.053</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">6.988</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">8.233</td>
      <td class="gt_row gt_right">5.474</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.713</td>
    </tr>
    <tr>
      <td class="gt_row gt_right">6.219</td>
      <td class="gt_row gt_right">4.038</td>
      <td class="gt_row gt_center" style="background-color: rgba(208,217,255,0.8); color: #000000;">word</td>
      <td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.908</td>
    </tr>
  </tbody>
  
  
</table></div>
]]

---
# We Know Enough to Fit a Linear Model

```r
mod3 <- lm(R_AGE~method,data=reading)
summary(mod3)
```

```
## 
## Call:
## lm(formula = R_AGE ~ method, data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
##  -3.96  -1.44   0.36   1.39   3.49 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   12.485      0.395   31.59  < 2e-16 ***
## methodword    -5.135      0.559   -9.19  3.8e-12 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.98 on 48 degrees of freedom
## Multiple R-squared:  0.637,	Adjusted R-squared:  0.63 
## F-statistic: 84.4 on 1 and 48 DF,  p-value: 3.76e-12
```

---
# We Know Enough to Draw a Graph

- we also know enough to run model diagnostics
---
.center[
![](lecture_8_files/figure-html/diag-1.svg)
]
---
# Adding Predictors

- we also know that `hrs_wk` affects reading age

- we can build a multiple regression, and inspect the coefficients

```r
mod.m2 <- lm(R_AGE ~ hrs_wk + method,data=reading)
summary(mod.m2)
```

```
## ...
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    7.843      1.524    5.15  5.1e-06 ***
## hrs_wk         0.914      0.291    3.14   0.0029 ** 
## methodword    -4.932      0.518   -9.53  1.5e-12 ***
## ...
```

???
- we're sticking to the one predictor for simplicity here 
---
# Graphically

- note that according to this model the lines are parallel

- an hour of practice has _exactly the same_ effect, however you're taught

---
# Different Effects for Different Methods

```r
mod.m3 <- lm(R_AGE ~ hrs_wk + method + hrs_wk:method,data=reading)
summary(mod.m3)
```

```
## 
## Call:
## lm(formula = R_AGE ~ hrs_wk + method + hrs_wk:method, data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -3.449 -1.377 -0.092  1.428  2.936 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)          4.528      1.916    2.36  0.02238 *  
## hrs_wk               1.567      0.371    4.22  0.00011 ***
## methodword           2.228      2.780    0.80  0.42697    
## hrs_wk:methodword   -1.445      0.552   -2.62  0.01199 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.71 on 46 degrees of freedom
## Multiple R-squared:  0.739,	Adjusted R-squared:  0.722 
## F-statistic: 43.4 on 3 and 46 DF,  p-value: 1.81e-13
```

---
# Different Effects for Different Methods

---
# Interaction is Just Multiplication

.br3.pa2.bg-gray.white[
`$$\hat{y}_i=b_0+b_1x_{1i}+b_2x_{2i}+\color{red}{b_3x_{1i}x_{2i}}$$`
]

`$$\widehat{\textrm{R_AGE}}=b_0+b_1\cdot\textrm{hrs_wk}+b_2\cdot\textrm{word}+b_3\cdot\textrm{hrs_wk}\cdot\textrm{word}$$`
- when `$\textrm{word}=0$`:

`$$\widehat{\textrm{R_AGE}}=b_0+b_1\cdot\textrm{hrs_wk}+\color{gray}{b_2\cdot0+b_3\cdot\textrm{hrs_wk}\cdot0}$$`
- when `$\textrm{word}=1$`:

`$$\widehat{\textrm{R_AGE}}=b_0+b_1\cdot\textrm{hrs_wk}+b_2\cdot1+b_3\cdot\textrm{hrs_wk}\cdot1$$`

---
count: false
# Interaction is Just Multiplication

.br3.pa2.bg-gray.white[
`$$\hat{y}_i=b_0+b_1x_{1i}+b_2x_{2i}+\color{red}{b_3x_{1i}x_{2i}}$$`
]

`$$\widehat{\textrm{R_AGE}}=\color{blue}{4.53}+\color{blue}{1.57}\cdot\textrm{hrs_wk}+\color{blue}{2.23}\cdot\textrm{word}+\color{blue}{-1.44}\cdot\textrm{hrs_wk}\cdot\textrm{word}$$`
- when `$\textrm{word}=0$`:

`$$\widehat{\textrm{R_AGE}}=\color{blue}{4.53}+\color{blue}{1.57}\cdot\textrm{hrs_wk}+\color{gray}{2.23\cdot0+-1.44\cdot\textrm{hrs_wk}\cdot0}$$`
- when `$\textrm{word}=1$`:

`$$\widehat{\textrm{R_AGE}}=\color{blue}{4.53}+\color{blue}{1.57}\cdot\textrm{hrs_wk}+\color{blue}{2.23}\cdot1+\color{blue}{-1.44}\cdot\textrm{hrs_wk}\cdot1$$`

---
# A Nice Graph

```r
reading %>% ggplot(
  aes(x=hrs_wk,y=R_AGE,colour=method)) +
  xlab("practice") +
  ylab("reading age") +
  geom_point(size=3) +
  geom_smooth(method="lm")
```
]

---
# Interaction Really _Is_ Just Multiplication

- in our dataset it's also possible that age and practice interact

.br3.pa2.bg-gray.white.tc[
"effect of practice is not constant across ages"

`$$\hat{y}_i=b_0+b_1x_{1i}+b_2x_{2i}+\color{red}{b_3x_{1i}x_{2i}}$$`
]

```r
mod.m4 <- lm(R_AGE ~ age + hrs_wk + age:hrs_wk, data=reading)
```

.tc.pt2[
`a + b + a:b` can also be written `a * b`
]

```r
mod.m4 <- lm(R_AGE ~ age * hrs_wk, data=reading)
```

---
# Interaction of Age and Practice

```r
summary(mod.m4)
```

```
## 
## Call:
## lm(formula = R_AGE ~ age * hrs_wk, data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -5.445 -1.967 -0.336  2.302  3.925 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   16.596      9.714    1.71    0.094 .
## age           -1.449      1.198   -1.21    0.233  
## hrs_wk        -3.079      2.041   -1.51    0.138  
*## age:hrs_wk     0.504      0.249    2.02    0.049 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.51 on 46 degrees of freedom
## Multiple R-squared:  0.44,	Adjusted R-squared:  0.403 
## F-statistic:   12 on 3 and 46 DF,  p-value: 0.00000607
```

???
- note that the effects of age and of practice are not reliably different from zero
---
# Interaction Effect

```
## ...
##             Estimate Std. Error t value Pr(>|t|)  
## (Intercept)   16.596      9.714    1.71    0.094 .
## age           -1.449      1.198   -1.21    0.233  
## hrs_wk        -3.079      2.041   -1.51    0.138  
## age:hrs_wk     0.504      0.249    2.02    0.049 *
## ...
```

.br3.pa2.bg-gray.white.tc[
`$$\widehat{\textrm{R_AGE}}_i=b_0+b_1\cdot\textrm{age}_i+b_2\cdot\textrm{hrs_wk}_i+b_3\cdot\textrm{age}_i\cdot\textrm{hrs_wk}_i$$`
]

`$$16.6+-1.45\cdot7+-3.08\cdot5+0.5\cdot7\cdot5$$`
`$$\color{red}{=8.55}$$`
]

`$$16.6+-1.45\cdot12+-3.08\cdot6+0.5\cdot12\cdot6$$`
`$$\color{red}{=16.72}$$`
]
---
# Significance

```
## ...
##             Estimate Std. Error t value Pr(>|t|)  
*## (Intercept)   16.596      9.714    1.71    0.094 .
*## age           -1.449      1.198   -1.21    0.233  
*## hrs_wk        -3.079      2.041   -1.51    0.138  
## age:hrs_wk     0.504      0.249    2.02    0.049 *
## ...
```

- note that not all of the effects are significant

- the model's best guess at the data ( `$\widehat{\textrm{R_AGE}}$` ) is expressed by the coefficients

- but we're not confident that the highlighted effects would reliably differ from zero less than 5% of the time if we repeatedly sampled from the same population

- so the _predictions_ of the model are as above (and below) but our _conclusion_ is only that practise is more beneficial the older a child is

---
# Graphical Model

.center[
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]

---

# End

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# Acknowledgements

- icons by Diego Lavecchia from the [Noun Project](https://thenounproject.com/)