Week 10: The Generalized Linear Model

class: center, middle, inverse, title-slide

.title[
# <b>Week 10: The Generalized Linear Model</b>
]
.subtitle[
## Univariate Statistics and Methodology using R
]
.author[
### 
]
.institute[
### Department of Psychology<br/>The University of Edinburgh
]

---

class: inverse, center, middle
# Part 1
## Bigger and Better

---
# Learning to Read

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<td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.798</td></tr>
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<td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">8.691</td></tr>
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<td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.474</td>
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<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>
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<td class="gt_row gt_right" style="background-color: rgba(208,217,255,0.8); color: #000000;">5.908</td></tr>
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]]

---
count: false
# Learning to Read

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![](lecture_10_files/img/playmo_teach.jpg)
]
.pull-right[
![](lecture_10_files/figure-html/plotme1-1.svg)

]
---
# Learning to Read

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<table class="gt_table">
  
  <thead class="gt_col_headings">
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      <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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  </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>
]]
---
count: false
# Learning to Read

.pull-left[
![](lecture_10_files/img/playmo_teach.jpg)
]
.pull-right[
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]
---
# Bigger and Better

- easy to build models including more predictors

.br3.pa2.bg-gray.white[
`$$\hat{y}_i=b_0+b_1x_{1i}+b_2x_{2i}+\ldots{}+b_kx_{ki}+\ldots{}+b_mx_{1i}x_{2i}+b_{m+1}x_{2i}x_{3i}+\ldots$$`
]

- for example

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

- NB., order of predictors **can** matter (judgement is important)
    - if we conduct `anova(mod.mm)` we test _incremental addition_ of each predictor

- first question:  is it worth it building such a complex model?

---
# Does Each New Predictor Improve Fit?

```r
anova(mod.mm)
```

```
## Analysis of Variance Table
## 
## Response: R_AGE
##               Df Sum Sq Mean Sq F value  Pr(>F)    
## age            1  166.0   166.0  599.36 < 2e-16 ***
## hrs_wk         1   35.7    35.7  128.99 1.1e-14 ***
## method         1  300.2   300.2 1083.79 < 2e-16 ***
## hrs_wk:method  1    2.8     2.8   10.25  0.0025 ** 
## age:hrs_wk     1    0.1     0.1    0.27  0.6078    
## Residuals     44   12.2     0.3                    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

- adding `age:hrs_wk` doesn't improve the model any further over a model without it

```r
mod.mm <- update(mod.mm, ~ . -age:hrs_wk)
# equivalent
# mod.mm <- lm(R_AGE ~ age + hrs_wk + method + hrs_wk:method, data=reading)
```
---
.center[
![](lecture_10_files/figure-html/assumptions-1.svg)
]
---
# The Model

```
## 
## Call:
## lm(formula = R_AGE ~ age + hrs_wk + method + hrs_wk:method, data = reading)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.3637 -0.2737  0.0288  0.2538  1.0491 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         0.6849     0.6114    1.12   0.2686    
## age                 0.9076     0.0428   21.22  < 2e-16 ***
## hrs_wk              0.8785     0.1177    7.46  2.1e-09 ***
## methodword         -2.1643     0.8724   -2.48   0.0169 *  
## hrs_wk:methodword  -0.5599     0.1734   -3.23   0.0023 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.522 on 45 degrees of freedom
## Multiple R-squared:  0.976,	Adjusted R-squared:  0.974 
## F-statistic:  463 on 4 and 45 DF,  p-value: <2e-16
```
- [coef_as_pred.R](coef_as_pred.R)
---
# The Model

.pull-left[
- not always convenient to draw 3d models!

- graphs can show "interesting" results

- here, `age` doesn't interact with anything

- so show plot for _mean age_ (or some other meaningful value)
]
.pull-right[
![](lecture_10_files/figure-html/pp-1.svg)
.f5[_reading age predicted by practice hours per week for children of average age_]
]
---
class: inverse, center, middle, animated, zoomInRight

# End of Part 1

---
class: inverse, center, middle
# Part 2
## Probability, Odds, Log-Odds

---
# Aliens
.center[
<video width= "70%" controls>
  <source src="lecture_10_files/video/alien.mp4" type="video/mp4">
  video not supported by this browser
</video>
]

---
# A Binary World

.center[
![:scale 50%](lecture_10_files/img/playmo_aliens.jpg)
]
---
count: false
# A Binary World

.center[
![:scale 50%](lecture_10_files/img/playmo_aliens_splat.jpg)
]

---
count: false
# A Binary World

.center[
![:scale 50%](lecture_10_files/img/playmo_aliens_10.jpg)
]
---
# 1,000 Aliens

.pull-left[
<div id="rguolnnkzs" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;">
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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_left" rowspan="1" colspan="1" scope="col">id</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">quality</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">SPLATTED</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">The Great Odorjan of Erpod</td>
<td class="gt_row gt_right">84</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Hapetox Bron</td>
<td class="gt_row gt_right">34</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Loorn Molzeks</td>
<td class="gt_row gt_right">92</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Ba'lite Adrflen</td>
<td class="gt_row gt_right">49</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Tedlambo Garilltet</td>
<td class="gt_row gt_right">93</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Goraveola Grellorm</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Colonel Garqun</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Bosgogo Lurcat</td>
<td class="gt_row gt_right">64</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Osajed Voplily</td>
<td class="gt_row gt_right">45</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Subcommander Edorop</td>
<td class="gt_row gt_right">90</td>
<td class="gt_row gt_right">0</td></tr>
  </tbody>
  
  
</table>
</div>
]
.pull-right[
- `quality` = quality of singing

- `SPLATTED` = whether splatted (1 or 0)
]

---
# 1,000 Aliens

.center[
![](lecture_10_files/figure-html/plotm-1.svg)

]
---
count: false
# 1,000 Aliens

.center[
![](lecture_10_files/figure-html/plotme-1.svg)
]
- using `geom_jitter()`

---
# Binomial Regression, Conceptually

- each alien either gets splatted or doesn't

+ each observation is either a 1 or a 0
  
- underlyingly, there's a **binomial** distribution

- for each value of "quality of singing" there's a _probability_ of getting splatted

.pt2[
- for each alien, the outcome is deterministic

- but it's the _probability_ we are ultimately interested in

- we can approximate it by binning our data...
]

---
# Binned Data

.pull-left[

```r
singers <- singers %>% 
  mutate(bin=cut_interval(quality,10))

dat <- singers %>% group_by(bin) %>%
  summarise(prop=mean(SPLATTED))

dat %>% ggplot(aes(x=bin,y=prop)) +
  xlab("bin") + ylab("prop splatted") +
  geom_point(size=3) +
  scale_x_discrete(label=1:10)
```
]
.pull-right[
![](lecture_10_files/figure-html/cut-1.svg)
]

---
# Best Fit Lines

.pull-left[
- we can fit our data using a standard linear model

- but there's something very wrong...
]

.pull-right[
![](lecture_10_files/figure-html/with_line-1.svg)

]
---
# The Problem with Probability

.center[
![](lecture_10_files/figure-html/bplot-1.svg)
]
---
count: false
# The Problem with Probability

.center[
![](lecture_10_files/figure-html/bplot2-1.svg)
]

- a _linear_ model predicts impossible values because probability isn't linear; it's **asymptotic**

---
# The Problem with Probability

.center[
![](lecture_10_files/figure-html/bplot3-1.svg)
]

- variance _necessarily_ covaries with probability

---
# Assumptions
.center[
![](lecture_10_files/figure-html/ass-1.svg)

]
---
# Probability and Odds

.br3.pa2.bg-gray.white.flex.items-center[
.w-50.pa2[
`$$\textrm{odds}(y)=\frac{p(y)}{1-p(y)}$$`
]
.w-50.pa2[
`$$0<p<1$$`
`$$0<\textrm{odds}<\infty$$`
]]

--
.center[
.flex.flex-column[
.flex.items-center[
.w-40[
&nbsp;
]
.w-20[
`$p(y)$`
]
.w-20[
`$\textrm{odds}(y)$`
]
]
.flex.items-center[
.w-40[
throw heads
]
.w-20[
`$\frac{1}{2}$`
]
.w-20[
`$\frac{1}{1}$`
]
]
.flex.items-center[
.w-40[
throw 8 from two dice
]
.w-20[
`$\frac{5}{36}$`
]
.w-20[
`$\frac{5}{31}$`
]
]
.flex.items-center[
.w-40[
get splatted
]
.w-20[
`$\frac{99}{100}$`
]
.w-20[
`$\frac{99}{1}$`
]
]
]
]

---
# Probability and Log-Odds

- `$\log(0)=-\infty$`; `$\log(\infty)=+\infty$`
- `$\log(1)=0$` where odds of 1 are exactly 50:50 ( `$p=0.5$` )

.center[
![](lecture_10_files/figure-html/threeg-1.svg)
]

- if log-odds are _less than zero_, the odds go down (multiply by <1)
- if log-odds are _more than zero_, the odds go up (multiply by >1)
- high odds = high probability

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

---
class: inverse, center, middle

# Part 3
## The Generalized Linear Model

---
name: glm1
# The Generalized Linear Model

- generalises the linear model using mapping functions

- coefficients are in **logit** (log-odds) units

.pt2[

- fit using **maximum likelihood**

- coefficients use **Wald's `$z$` ** instead of `$t$`

]

---
# Likelihood

.flex.items-center[

.w-20.pa2[
![:scale 90%](lecture_10_files/img/50p.jpg)
]
.w-80[
.center[
![](lecture_10_files/figure-html/likely-1.svg)
]]]
- extent to which a sample provides support for a model (![:scale 3%](lecture_10_files/img/danger.svg) &nbsp; [MLE_bend_in_the_road.R](MLE_bend_in_the_road.R)).

---
template: glm1

- but actually it's all quite straightforward...

---
# Alien Singer Splat Probability

.pull-left[
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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" scope="col">id</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">quality</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1" scope="col">SPLATTED</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_left">The Great Odorjan of Erpod</td>
<td class="gt_row gt_right">84</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Hapetox Bron</td>
<td class="gt_row gt_right">34</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Loorn Molzeks</td>
<td class="gt_row gt_right">92</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Ba'lite Adrflen</td>
<td class="gt_row gt_right">49</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Tedlambo Garilltet</td>
<td class="gt_row gt_right">93</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Goraveola Grellorm</td>
<td class="gt_row gt_right">5</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Colonel Garqun</td>
<td class="gt_row gt_right">55</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Bosgogo Lurcat</td>
<td class="gt_row gt_right">64</td>
<td class="gt_row gt_right">1</td></tr>
    <tr><td class="gt_row gt_left">Osajed Voplily</td>
<td class="gt_row gt_right">45</td>
<td class="gt_row gt_right">0</td></tr>
    <tr><td class="gt_row gt_left">Subcommander Edorop</td>
<td class="gt_row gt_right">90</td>
<td class="gt_row gt_right">0</td></tr>
  </tbody>
  
  
</table>
</div>
]
.pull-right[
- use `glm()` `$\text{ }^{*}$` instead of `lm()`

- specify **link function** with `family = binomial` `$\text{ }^{**}$`

.pt2[

```r
mod.b <- glm(SPLATTED ~ quality,
             family = binomial, 
             data=singers)
```

]

.pt5[
`$\text{ }^*$` can take a 2-level factor DV  
`$\text{ }^{**}$` `family="binomial"` and  
`family=binomial(link="logit")` also work  
]]

---
# Evaluating the Model

.pull-left[
- NB., no statistical test done by default

- **deviance** compares the likelihood of the new model to that of the previous model

+ a generalisation of sums of squares
  
  + _lower_ "residual deviance" is good  
  (*a bit like Residual Sums of Squares*)

]
.pull-right[

```r
summary(mod.b)
```

```
## 
## Call:
## glm(formula = SPLATTED ~ quality, family = binomial, data = singers)
## 
## Deviance Residuals: 
##    Min      1Q  Median      3Q     Max  
## -2.987  -0.374  -0.113   0.333   3.279  
## ...
## ...
## ...
##     Null deviance: 1377.06  on 999  degrees of freedom
## Residual deviance:  577.29  on 998  degrees of freedom
```
]

---
![:scale 5%](lecture_10_files/img/danger.svg) &nbsp;&nbsp; .f1[Evaluating the Model]

```
##     Null deviance: 1377.06  on 999  degrees of freedom
## Residual deviance:  577.29  on 998  degrees of freedom
```

- deviance is `$-2\times$` the **log-likelihood ratio** of the reduced compared to the full model
- _higher_ "deviance" is good (*a bit like F*)

```r
mod.n <- glm(SPLATTED~1,family=binomial,data=singers)
```

.pull-left[

```r
logLik(mod.n)
```

```
## 'log Lik.' -688.5 (df=1)
```

```r
logLik(mod.b)
```

```
## 'log Lik.' -288.6 (df=2)
```
]
.pull-right[

```r
-2*logLik(mod.n)
```

```
## 'log Lik.' 1377 (df=1)
```

```r
-2*logLik(mod.b)
```

```
## 'log Lik.' 577.3 (df=2)
```
]

```r
-2 * (logLik(mod.n)-logLik(mod.b))
```

```
*## 'log Lik.' 799.8 (df=1)
```
---
# Evaluating the Model
- model deviance maps to the `$\chi^2$` distribution

- can specify a `$\chi^2$` test to statistically evaluate model in a similar way to `$F$` ratio

```r
anova(mod.b, test="Chisq")
```

```
## Analysis of Deviance Table
## 
## Model: binomial, link: logit
## 
## Response: SPLATTED
## 
## Terms added sequentially (first to last)
## 
## 
##         Df Deviance Resid. Df Resid. Dev Pr(>Chi)    
## NULL                      999       1377             
*## quality  1      800       998        577   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

---
# Model Coefficients

```
## 
## Call:
## glm(formula = SPLATTED ~ quality, family = binomial, data = singers)
## 
## Deviance Residuals: 
##    Min      1Q  Median      3Q     Max  
## -2.987  -0.374  -0.113   0.333   3.279  
## 
## Coefficients:
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  5.08191    0.33410    15.2   <2e-16 ***
## quality     -0.10557    0.00642   -16.5   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 1377.06  on 999  degrees of freedom
## Residual deviance:  577.29  on 998  degrees of freedom
## AIC: 581.3
## 
## Number of Fisher Scoring iterations: 6
```
---
# Model Coefficients

.flex.items-center[
.w-70.pa2[
#### coefficients are in **logits** (= log-odds)

```
## ...
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  5.08191    0.33410    15.2   <2e-16 ***
## quality     -0.10557    0.00642   -16.5   <2e-16 ***
## ...
```
- zero = "50/50" (odds of 1)

- value below zero: probability of being splatted _decreases_ as quality increases
]
.w-30.pa2[
![](lecture_10_files/figure-html/p3-1.svg)
]]
---
# Log-Odds, Odds, and Probability

```
## ...
##             Estimate Std. Error z value Pr(>|z|)    
## (Intercept)  5.08191    0.33410    15.2   <2e-16 ***
## quality     -0.10557    0.00642   -16.5   <2e-16 ***
## ...
```
.pull-left[
**quality** = 50

- _log-odds_:  `$5.08+-0.11\cdot50=\color{red}{-0.42}$`

- _odds_: 
`$e^{-0.42}=\color{red}{0.657}$`

- _probability_:
`$\frac{0.657}{1+0.657}=\color{red}{0.3965}$`

]
.pull-right[
<br/>
`$$\hat{y}_i=b_0+b_1x_i$$`
`$$\textrm{odds}=e^{\hat{y}_i}$$`
`$$p=\frac{\textrm{odds}}{1+\textrm{odds}}$$`

]

---
# A Useful Function

- intuitive to think in probability

- useful to write a function which takes a value in logits `l` and converts it to a probability `p`

```r
l2p <- function(logits) {
  odds = exp(logits)
  prob = odds/(1+odds)
  return(prob)
}
```
.pull-left[
- singing qualities 50 and 51

```r
l2p(5.08+-0.11*50)
```

```
## [1] 0.3965
```
```r
l2p(5.08+-0.11*51)
```

```
## [1] 0.3705
```
]
.pull-right[
- singing qualities 10 and 11

```r
l2p(5.08+-0.11*10)
```

```
## [1] 0.9817
```
```r
l2p(5.08+-0.11*11)
```

```
## [1] 0.9796
```
]

---
# Representing the Model Graphically

```r
singers %>% ggplot(aes(x=quality,y=SPLATTED)) +
  ylab("p(SPLATTED)") +
  geom_jitter(size=3,width=0,height=.2,alpha=.1) +
  geom_smooth(method="glm",method.args=list(family=binomial)) +
* scale_y_continuous(breaks=seq(0,1,by=.2))
```
.center[
![](lecture_10_files/figure-html/g-1.svg)
]

---
# One Last Trick

- so far we've looked at

+ model _deviance_ and `$\chi^2$` (similar to sums of squares and `$F$`)
  
  + model _coefficients_ and how to map them to probability
  
- what about "explained variance" (similar to `$R^2$`)?

- no really good way of doing this, many proposals

- SPSS uses something called "accuracy" (how well does the model predict actual data?)

- not very informative, but good for learning R

---
# Accuracy

- first, what does the model predict (in logit units)?

```r
guess <- predict(mod.b) # in logit units
```
- if the chance of being splatted is more than .5 (logit > 0) call it a "splat"

```r
guess <- ifelse(guess>0,1,0)
```
- how well do predicted splats match actual splats?

```r
hits <- sum(guess == singers$SPLATTED)
hits/length(singers$SPLATTED)
```

```
## [1] 0.879
```

- present model "correctly predicts" 87.9% of the observations

---
# Other Types of Data

- logit regression is _one type_ of GLM

- others make use of different **link functions** (through `family=...`)

.pt2[

- **poisson**: number of events in a time period

- **inverse gaussian**: time to reach some criterion

- ...
]

---
# GLMs

.flex[
.w-50.br3.mr2.pa2.bg-gray.white[
### Predictor Variables
- linear

- convertible to linear (use `log()` etc.)

- non-convertible (use `contrasts()` etc. to map)

- **don't affect the choice of model**

]
.w-50.br3.mr2.pa2.bg-gray.white[
### Dependent Variables
- linear

- convertible to linear (use `log()` etc.)

- non-convertible (use `glm()` with `family=...`)

- **directly affect the choice of model**

]]

---
class: inverse, center, middle, animated, zoomInRight
# End

---
# Acknowledgements

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