Week 8: The Linear Model (ctd)

class: center, middle, inverse, title-slide

.title[
# <b>Week 8: The Linear Model (ctd)</b>
]
.subtitle[
## Univariate Statistics and Methodology using R
]
.author[
### 
]
.institute[
### Department of Psychology<br/>The University of Edinburgh
]

---

class: inverse, center, middle
# Part 1
## Quick Refresh

---
# Some new data

.pull-left[
![](lecture_8_files/figure-html/frecap2-1.svg)
]
.pull-right[
![](lecture_8_files/img/playmo_police.jpg)

]

---
# Some new data

.pull-left[
![](lecture_8_files/figure-html/frecap22-1.svg)

.tc[
"for every extra 0.01% blood alcohol, reaction time slows down by around 32 ms"
]
]
.pull-right[
![](lecture_8_files/img/playmo_police.jpg)
]

---
# The Model

.pull-left[

```r
mod <- lm(RT~BloodAlc, data=dat)
summary(mod)
```

```
## 
## Call:
## lm(formula = RT ~ BloodAlc, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -115.92  -40.42    1.05   42.93  126.64 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      321         91    3.53  0.00093 ***
## BloodAlc        3228        888    3.64  0.00067 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 55.8 on 48 degrees of freedom
## Multiple R-squared:  0.216,	Adjusted R-squared:   0.2 
## F-statistic: 13.2 on 1 and 48 DF,  p-value: 0.000673
```
]

---
# Another (identical) Model

```r
dat <- dat %>% mutate(BloodAlc100 = BloodAlc*100)
mod2 <- lm(RT~BloodAlc100, data=dat)
summary(mod2)
```

```
## 
## Call:
## lm(formula = RT ~ BloodAlc100, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -115.92  -40.42    1.05   42.93  126.64 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   321.24      91.05    3.53  0.00093 ***
## BloodAlc100    32.28       8.88    3.64  0.00067 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 55.8 on 48 degrees of freedom
## Multiple R-squared:  0.216,	Adjusted R-squared:   0.2 
## F-statistic: 13.2 on 1 and 48 DF,  p-value: 0.000673
```

---
class: inverse, center, middle
# Part 2
## Checking Assumptions

---
# Assumptions of Linear Models

.pull-left[
.br3.bg-gray.white.pa2[
### required
- **linearity** of relationship(!)

- for the _residuals_:
  + **normality**
  + **homogeneity of variance**
  + **independence**
]
]
.pull-right[
.br3.bg-gray.white.pa2[
### desirable

- no 'bad' (overly influential) observations
]]

---
# Residuals

.pull-left[
`$$y_i=b_0+b_1\cdot{}x_i+\epsilon_i$$`
`$$\color{red}{\epsilon\sim{}N(0,\sigma)~\text{independently}}$$`
- normally distributed (mean should be `$\simeq{}$` zero)

- homogeneous (differences from `$\hat{y}$` shouldn't be systematically smaller or larger for different `$x$`)

- independent (residuals shouldn't influence other residuals)
]
.pull-right[
![](lecture_8_files/figure-html/resid-1.svg)
]

---
# At A Glance

```r
summary(mod)
```

```
## 
## Call:
## lm(formula = RT ~ BloodAlc, data = dat)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
*## -115.92  -40.42    1.05   42.93  126.64 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      321         91    3.53  0.00093 ***
## BloodAlc        3228        888    3.64  0.00067 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 55.8 on 48 degrees of freedom
## Multiple R-squared:  0.216,	Adjusted R-squared:   0.2 
## F-statistic: 13.2 on 1 and 48 DF,  p-value: 0.000673
```

---
# In More Detail

.pull-left[
### linearity

```r
plot(mod,which=1)
```
- plots residuals `$\epsilon_i$` against fitted values `$\hat{y}_i$`

- the 'average residual' is roughly zero across `$\hat{y}$`, so relationship is likely to be linear
]
.pull-right[
![](lecture_8_files/figure-html/resid1-1.svg)
]

---
# In More Detail

.pull-left[
### normality

```r
hist(resid(mod))
```
]
.pull-right[
![](lecture_8_files/figure-html/resid2-1.svg)
]

---
count: false
# In More Detail

.pull-left[
### normality

```r
plot(density(resid(mod)))
```

- check that residuals `$\epsilon$` are approximately normally distributed

- in fact there's a better way of doing this...
]
.pull-right[
![](lecture_8_files/figure-html/resid3-1.svg)
]

---
name: QQ
# In More Detail

.pull-left[
### normality

```r
plot(mod,which=2)
```

- **Q-Q plot** compares the residuals `$\epsilon$` against a known distribution (here, normal)

- observations close to the straight line mean residuals are approximately normal

- numbered observations refer to _row numbers_ in the original data, for checking
]
.pull-right[
![](lecture_8_files/figure-html/resid4-1.svg)
]

---
# Q-Q Plots

.pull-left[
#### y axis

- Our residuals, in terms of "standard deviations from the mean": `$\text{standardized residual} = \frac{\text{residual}-mean(\text{residual})}{sd(\text{residual})}$`

```r
scale(resid(mod))
```
```
[1] -2.10063 -1.51541 -1.46399 -1.33279 -1.30350 
[6] -1.27782 -1.19769 -1.12129 -1.09559 -0.86440
...
...
```
]

.pull-right[
#### x axis

- we have 50 residuals.

- for a normal distribution, what values _should_ 1/50th, 2/50th, 3/50th (etc) of the observations lie below?

- expressed in "standard deviations from the mean"

```r
qnorm(c(1/50,2/50,3/50))
```

```
## [1] -2.054 -1.751 -1.555
```
]

- Q-Q Plot shows these values plotted against each other

---
template: QQ
---
# In More Detail

.pull-left[
### homogeneity of variance

```r
plot(mod,which=3)
```

- the _size_ of the residuals is approximately the same across values of `$\hat{y}$`
]
.pull-right[
![](lecture_8_files/figure-html/resid5-1.svg)
]

---
# Visual vs Other Methods

- statistical ways of checking assumptions are introduced in the reading

- they tend to have limitations (for example, they're susceptible to sample size)

- nothing beats looking at plots like these (and `plot(<model>)` makes it easy)

- however, two things:

.pt2[
1. there are no criteria for deciding exactly when assumptions are sufficiently met

+ it's a matter of experience and judgement
  
1. we need to talk about **independence** of residuals
]

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

---
class: inverse, center, middle
# Part 3
## Independence, Influence

---
# Independence

- no easy way to check independence of residuals

- in part, because it depends on the _source_ of the observations

.pt2[
- one determinant might be a single person being observed multiple times

- e.g., my reaction times might tend to be slower than yours<br/>
  `$\rightarrow$` multivariate statistics
]

---
# Independence

- another determinant might be _time_

- observations in a sequence might be autocorrelated
  
- can be checked using the Durbin-Watson Test from the `car` package

```r
library(car)
dwt(mod)
```

```
##  lag Autocorrelation D-W Statistic p-value
*##    1         -0.1377          2.22    0.42
##  Alternative hypothesis: rho != 0
```

- shows no autocorrelation at lag 1

---
# Influence
.center[
![](lecture_8_files/figure-html/baddata1-1.svg)
]
- even substantial **outliers** may only have small effects on the model

- here, only the intercept is affected

---
# Influence
.center[
![](lecture_8_files/figure-html/baddata2-1.svg)
]
- observations with high **leverage** are inconsistent with other data, but may not be distorting the model

---
# Influence
.center[
![](lecture_8_files/figure-html/baddata3-1.svg)
]
- we care about observations with high **influence** (outliers with high leverage)

---
# Cook's Distance

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

- a standardised measure of "how much the model differs without observation `$i$`"

---
.flex.items-center[.w-10.pa1[![:scale 70%](lecture_8_files/img/danger.svg)] .f1.w-80.pa1[Cook's Distance]]

.pt3[
`$$D_i=\frac{\sum_{j=1}^{n}{(\hat{y}_j-\hat{y}_{j(i)})^2}}{(p+1)\hat{\sigma}^2}$$`
- `$\hat{y}_j$` is the `$j$`th fitted value
- `$\hat{y}_{j(i)}$` is the `$j$`th value from a fit which doesn't include observation `$i$`
- `$p$` is the number of regression coefficients
- `$\hat{\sigma}^2$` is the estimated variance from the fit, i.e., mean squared error
]

---
# Cook's Distance

.pull-left[

```r
plot(mod,which=4)
```

- observations labelled by row

- various rules of thumb, but start looking when Cook's Distance > 0.5
]
.pull-right[
![](lecture_8_files/figure-html/cook2-1.svg)
]

---
class: animated, rollIn
# Learning to Read

.pull-left[
![](lecture_8_files/img/playmo_teach.jpg)
]
.pull-right[
- the Playmo School has been evaluating its reading programmes, using 50 students

- ages of students

- hours per week students spend reading of their own volition

- whether they are taught using phonics or whole-word methods

- **outcome: "reading age"**
]

---
count: false
# Learning to Read

.pull-left[
![](lecture_8_files/img/playmo_teach.jpg)
]
.pull-right[
.center[
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</style>
<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" scope="col">age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">hrs_wk</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col">method</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">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">phonics</td>
<td class="gt_row gt_right">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">phonics</td>
<td class="gt_row gt_right">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">phonics</td>
<td class="gt_row gt_right">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">phonics</td>
<td class="gt_row gt_right">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">phonics</td>
<td class="gt_row gt_right">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">word</td>
<td class="gt_row gt_right">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">word</td>
<td class="gt_row gt_right">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">word</td>
<td class="gt_row gt_right">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">word</td>
<td class="gt_row gt_right">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">word</td>
<td class="gt_row gt_right">5.908</td></tr>
  </tbody>
  
  
</table>
</div>
]]
---
count: false
# Learning to Read

.pull-left[
![](lecture_8_files/img/playmo_teach.jpg)
]
.pull-right[
.center[
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<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" scope="col">age</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">hrs_wk</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_center" rowspan="1" colspan="1" scope="col">method</th>
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    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">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">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">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">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">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">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">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">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">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">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>
]]
---
# Does Practice Affect Reading Age?

.pull-left[

```r
p <- reading %>% 
  ggplot(aes(x=hrs_wk,y=R_AGE)) +
  geom_point(size=3) +
  ylab("reading age") +
  xlab("hours reading/week")
p
```

]
.pull-right[
![](lecture_8_files/figure-html/plot1-1.svg)
]

- hours per week is correlated with reading age: `$r=0.3483$`,
`$p=0.0132$`

- we can use a linear model to say something about the effect size

---
# Does Practice Affect Reading Age?

.pull-left[

```r
p + geom_smooth(method="lm")
```
]
.pull-right[
![](lecture_8_files/figure-html/plot2-1.svg)
]

- each extra hour spent reading a week adds 1.26 years to reading age

---
# A Linear Model

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

```
## 
## Call:
## lm(formula = R_AGE ~ hrs_wk, data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
*## -5.567 -2.991  0.378  2.385  5.351 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)  
*## (Intercept)     3.65       2.47    1.48    0.146  
*## hrs_wk          1.26       0.49    2.57    0.013 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 3.08 on 48 degrees of freedom
*## Multiple R-squared:  0.121,	Adjusted R-squared:  0.103 
*## F-statistic: 6.63 on 1 and 48 DF,  p-value: 0.0132
```

---
class: center, middle, inverse
.f1[but...]

---
.center[
![](lecture_8_files/figure-html/diagp2-1.svg)
]

---
# Assumptions Not Met!
.pull-left[
- it seems that the assumptions aren't met for this model

- (another demonstration on the right)

- one reason for this can be because there's still systematic structure in the residuals

- i.e., _more than one thing_ which can explain the variance
]
.pull-right[
![](lecture_8_files/figure-html/re-1.svg)

]

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

# End of Part 3

---
class: inverse, center, middle

# Part 4
## Multiple Regression

---
# Adding Age into the Equation

.pull-left[
- so far, have focused on effects of practice

- but presumably older children read better?

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aTexcoord;\n#endif\n  \n#if defined(FIXED_SIZE) && !defined(ROTATING)\n  vec4 pos = prMatrix * mvMatrix * vec4(aPos, 1.);\n  pos = pos/pos.w;\n  gl_Position = pos + vec4(aOfs*textScale, 0.);\n#endif\n  \n#if defined(IS_SPRITES) && !defined(FIXED_SIZE)\n  vec4 pos = mvMatrix * vec4(aPos, 1.);\n  pos = pos/pos.w + vec4(aOfs,  0.);\n  gl_Position = prMatrix*pos;\n#endif\n  \n#ifdef FAT_LINES\n  /* This code was inspired by Matt Deslauriers' code in \n   https://mattdesl.svbtle.com/drawing-lines-is-hard */\n  vec2 aspectVec = vec2(uAspect, 1.0);\n  mat4 projViewModel = prMatrix * mvMatrix;\n  vec4 currentProjected = projViewModel * vec4(aPos, 1.0);\n  currentProjected = currentProjected/currentProjected.w;\n  vec4 nextProjected = projViewModel * vec4(aNext, 1.0);\n  vec2 currentScreen = currentProjected.xy * aspectVec;\n  vec2 nextScreen = (nextProjected.xy / nextProjected.w) * aspectVec;\n  float len = uLwd;\n  vec2 dir = vec2(1.0, 0.0);\n  vPoint = aPoint;\n  vLength = length(nextScreen 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]

---
# Another Model

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

]

---
# Another Model

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

```
## 
## Call:
## lm(formula = R_AGE ~ age, data = reading)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
*## -4.577 -2.509 -0.005  2.390  4.392 
## 
## Coefficients:
##             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 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.7 on 48 degrees of freedom
*## Multiple R-squared:  0.321,	Adjusted R-squared:  0.307 
*## F-statistic: 22.7 on 1 and 48 DF,  p-value: 0.0000179
```

---
.center[
![](lecture_8_files/figure-html/diag2-1.svg)
]

---
# Two Models, No Answers

.pull-left[
- we now have two models that don't map well to assumptions

- each suggests an effect

+ one of `age`
  
  + one of `hrs_wk`
]
.pull-right[
- if we run them independently, the chances of a type 1 error are

+ `$\frac{1}{20}$` (`mod`, including `hrs_wk`)
  
  + `$\frac{1}{20}$` (`mod2`, including `age`)
  
- or ** `$\frac{1}{10}$` ** overall
]

--
.pt1[
&nbsp;
]

.br3.pa2.bg-gray.white.tc[
we need to test multiple predictors in _one_ linear model
]

---
# Model Equations Again

`$$\color{red}{\textrm{outcome}_i}=\color{blue}{(\textrm{model})_i}+\textrm{error}_i$$`

`$$\color{red}{y_i}=\color{blue}{b_0\cdot{}1+b_1\cdot{}x_i}+\epsilon_i$$`

### linear model with two predictors

`$$\color{red}{y_i}=\color{blue}{b_0\cdot{}1+b_1\cdot{}x_{1i}+b_2\cdot{}x_{2i}}+\epsilon_i$$`
`$$\color{red}{\hat{y}_i}=\color{blue}{b_0\cdot{}1+b_1\cdot{}x_{1i}+b_2\cdot{}x_{2i}}$$`

--
.center[
`y ~ 1 + x1 + x2`

`R_AGE ~ 1 + hrs_wk + age` &nbsp;&nbsp; or &nbsp;&nbsp; `R_AGE ~ hrs_wk + age`<sup>1</sup>
]

.footnote[
<sup>1</sup> we'll come back to why order can matter in a bit
]

---
# Running a Multiple Regression

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

```
## 
## Call:
## lm(formula = R_AGE ~ age + 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)   -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 *  
## ---
## 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
```

---
# Running a Multiple Regression

```
## ...
## (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 *  
## ...
```

- there are _independent_ effects of age and practice

+ reading age improves by 0.9378 for each year of age

+ reading age improves by 0.9636 for each weekly hour of practice

- note that the _intercept_ (0 years old, 0 hours/week) is meaningless here

- important question:  is this model _better_ than a model based just on age?

---
# Model Fit: `$R^2$`

```
## ...
## 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
```

- in multiple regression, `$R^2$` measures the fit of the entire model
  + sum of individual `$R^2$`s _if predictors not correlated_

- `$R^2=0.3902$` looks better than the `$R^2$` for `mod2` (`age` as a predictor) of `$0.3211$`

- but _any_ predictor will improve `$R^2$` (chance associations guarantee this)

```r
mod2 <- lm(R_AGE ~ age, data=reading)
mod.2r <- update(mod2, ~ . + runif(50))
summary(mod.2r)
```

```
## ...
## Multiple R-squared:  0.361,	Adjusted R-squared:  0.334 
## ...
```

---
# Model Fit: F

```
## ...
## 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
```

- in multiple regression, `$F$` tests the whether the model overall explains more variance than we would expect by chance.  
- can be phrased as a model comparison:

```r
null_mod <- lm(R_AGE ~ 1, data = reading)
mod.m <- lm(R_AGE ~ age + hrs_wk, data=reading)
anova(null_mod, mod.m)
```

```
## Analysis of Variance Table
## 
## Model 1: R_AGE ~ 1
## Model 2: R_AGE ~ age + hrs_wk
##   Res.Df RSS Df Sum of Sq  F Pr(>F)    
## 1     49 517                           
*## 2     47 315  2       202 15  9e-06 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
# Comparing Models

- We can also use `$F$` ratios to compare models in terms of variance explained by each model:

- Models must be "nested" - predictors of one model are a subset of predictors in the other.  
  
  - Models must be fitted to the same data.

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

```
## Analysis of Variance Table
## 
## Model 1: R_AGE ~ age
## Model 2: R_AGE ~ age + hrs_wk
##   Res.Df RSS Df Sum of Sq    F Pr(>F)  
## 1     48 351                           
## 2     47 315  1      35.7 5.33  0.025 *
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
# Partitioning Variance

- Take one model, and examine variance explained by each predictor:

```r
mod.f <- lm(R_AGE ~ age + hrs_wk, data=reading)
anova(mod.m)
```

```
## Analysis of Variance Table
## 
## Response: R_AGE
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## age        1  166.0   166.0   24.75 0.0000092 ***
## hrs_wk     1   35.7    35.7    5.33     0.025 *  
## Residuals 47  315.3     6.7                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
# Order Matters!

- `age` then `hrs_wk`

```
## Analysis of Variance Table
## ...
## Response: R_AGE
##           Df Sum Sq Mean Sq F value    Pr(>F)    
## age        1  166.0   166.0   24.75 0.0000092 ***
## hrs_wk     1   35.7    35.7    5.33     0.025 *  
## Residuals 47  315.3     6.7                      
## ---
```
- `hrs_wk` then `age`

```
## Analysis of Variance Table
## ...
## Response: R_AGE
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## hrs_wk     1   62.7    62.7    9.35   0.0037 ** 
## age        1  139.0   139.0   20.72 0.000038 ***
## Residuals 47  315.3     6.7                     
## ---
```

---
# Type 1 vs. Type 3 SS

- order matters because R, by default, uses **Type 1** sums of squares for `anova()`
  + calculate each predictor's improvement to the model _in turn_
  
- compare to **Type 3** sums of squares
  + calculate each predictor's improvement to the model _taking all other predictors into account_
  
- huge debate about which is "better" (nobody likes Type 2)

- if using Type 1, _predictors should be entered into the model in a theoretically-motivated order_

---
# Type 1 vs. Type 3 SS
.pull-left[
### type 1
.center[
![:scale 70%](lecture_8_files/t13/jk/type0-0.svg)
]]
.pull-right[
### type 3
.center[
![:scale 70%](lecture_8_files/t13/jk/type0-0.svg)
]]

---

count: false
# Type 1 vs. Type 3 SS
.pull-left[
### type 1
.center[
![:scale 70%](lecture_8_files/t13/jk/type1-1.svg)
]]
.pull-right[
### type 3
.center[
![:scale 70%](lecture_8_files/t13/jk/type0-0.svg)
]]

---
count: false
# Type 1 vs. Type 3 SS
.pull-left[
### type 1
.center[
![:scale 70%](lecture_8_files/t13/jk/type1-2.svg)
]]
.pull-right[
### type 3
.center[
![:scale 70%](lecture_8_files/t13/jk/type0-0.svg)
]]

---
count: false
# Type 1 vs. Type 3 SS
.pull-left[
### type 1
.center[
![:scale 70%](lecture_8_files/t13/jk/type1-3.svg)
]]
.pull-right[
### type 3
.center[
![:scale 70%](lecture_8_files/t13/jk/type0-0.svg)
]]

---

count: false
# Type 1 vs. Type 3 SS
.pull-left[
### type 1
.center[
![:scale 70%](lecture_8_files/t13/jk/type1-3.svg)
]]
.pull-right[
### type 3
.center[
![:scale 70%](lecture_8_files/t13/jk/type3-1.svg)
]]

---

# Type 1 vs. Type 3 SS

.pull-left[
__Type 1 - "Incremental" (Order Matters)__

```r
# age then hrs_wk:   `
anova(lm(R_AGE~age+hrs_wk,data=reading))
```

```r
# hrs_wk then age: 
anova(lm(R_AGE~hrs_wk+age,data=reading))
```

```
## Analysis of Variance Table
## 
## Response: R_AGE
##           Df Sum Sq Mean Sq F value   Pr(>F)    
## hrs_wk     1   62.7    62.7    9.35   0.0037 ** 
## age        1  139.0   139.0   20.72 0.000038 ***
## Residuals 47  315.3     6.7                     
## ---
```
]
.pull-right[
__Type 3 - "Last one in"__

```r
mod.m <- lm(R_AGE~hrs_wk+age,data=reading)
drop1(mod.m, test="F")
```

```
## Single term deletions
## 
## Model:
## R_AGE ~ hrs_wk + age
##        Df Sum of Sq RSS   AIC F value   Pr(>F)    
## <none>              315  98.1                     
## hrs_wk  1      35.7 351 101.4    5.33    0.025 *  
## age     1     139.0 454 114.3   20.72 0.000038 ***
## ---
```
]

---
# So far..

What can we do with multiple regressions?

- Examine proportion of variance explained - `$R^2$`

- Test whether the model improves over chance - `$F$` test at the bottom of `summary(model)`

- Conduct comparisons between _nested_ models
    - e.g. `lm(y ~ x1)` vs `lm(y ~ x1 + x2 + x3 + x4)`  
    - using `anova(model1, model2)`

- Test the variance explained by each predictor in the model, either... 
    - incrementally (in the order inputted into the model)  
      `anova(model)`
    - after accounting for all other predictors  
      `drop1(model, test = "F")`

---
# Two Subtly Different Questions

.pull-left[

> After accounting for `age`, *does* `hrs_wk` influence `R_AGE`?

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

]
.pull-right[

> After accounting for `age`, *__how__ does* `hrs_wk` influence `R_AGE`?

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

```
## ...
##             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 *  
## ---
## ...
```

]

---
# The Two-Predictor Model

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

---

# The Two-Predictor Model

.flex.items-center[
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&nbsp;
]
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.w-25[
&nbsp;
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---
class: inverse, center, middle, animated, flip

# End

---
# Acknowledgements

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