Lineare Modelle

# Lineare Modelle
## Kapitel 5

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## Gliederung

</br>

2. [Teil 2: Die Post-Verteilung befragen](#teil-2)

3. [Teil 3: Die PPV befragen](#teil-3)

4. [Hinweise](#hinweise)
]

---

# Teil 1

## Post-Verteilung der Regression

---

## Einfache Regression

- Die (einfache) Regression prüft, inwieweit zwei Variablen, `$Y$` und `$X$` linear zusammenhängen.
- Je mehr sie zusammenhängen, desto besser kann man `$X$` nutzen, um `$Y$` vorherzusagen (und umgekehrt).
- Hängen `$X$` und `$Y$` zusammen, heißt das nicht (unbedingt), dass es einen *kausalen* Zusammenhang zwische `$X$` und `$Y$` gibt.
- Linear bedeutet, der Zusammenhang ist additiv und konstant: wenn `$X$` um eine Einheit steigt, steigt `$Y$` immer um `$b$` Einheiten (nicht kausal, sondern deskriptiv gemeint).

]

]

---

## Statistiken zum !Kung-Datensatz

[Datenquelle](https://raw.githubusercontent.com/sebastiansauer/2021-wise/main/Data/Howell1a.csv), <a name=cite-mcelreath_statistical_2020></a>([McElreath, 2020](#bib-mcelreath_statistical_2020))

```r
library(tidyverse)
library(rstatix)
Kung_path <- "https://tinyurl.com/jr7ckxxj"  # Datenquelle s.o.

d <- read_csv(Kung_path)

d2 <- d %>% filter(age > 18)

get_summary_stats(d2)
```

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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">n</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">min</th>
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    <tr><td class="gt_row gt_left">age</td>
<td class="gt_row gt_right">346.0</td>
<td class="gt_row gt_right">19.0</td>
<td class="gt_row gt_right">88.0</td>
<td class="gt_row gt_right">40.0</td>
<td class="gt_row gt_right">29.0</td>
<td class="gt_row gt_right">51.0</td>
<td class="gt_row gt_right">22.0</td>
<td class="gt_row gt_right">16.3</td>
<td class="gt_row gt_right">41.5</td>
<td class="gt_row gt_right">15.8</td>
<td class="gt_row gt_right">0.8</td>
<td class="gt_row gt_right">1.7</td></tr>
    <tr><td class="gt_row gt_left">height</td>
<td class="gt_row gt_right">346.0</td>
<td class="gt_row gt_right">136.5</td>
<td class="gt_row gt_right">179.1</td>
<td class="gt_row gt_right">154.3</td>
<td class="gt_row gt_right">148.6</td>
<td class="gt_row gt_right">160.7</td>
<td class="gt_row gt_right">12.1</td>
<td class="gt_row gt_right">8.5</td>
<td class="gt_row gt_right">154.6</td>
<td class="gt_row gt_right">7.8</td>
<td class="gt_row gt_right">0.4</td>
<td class="gt_row gt_right">0.8</td></tr>
    <tr><td class="gt_row gt_left">male</td>
<td class="gt_row gt_right">346.0</td>
<td class="gt_row gt_right">0.0</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">0.0</td>
<td class="gt_row gt_right">0.0</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">0.0</td>
<td class="gt_row gt_right">0.5</td>
<td class="gt_row gt_right">0.5</td>
<td class="gt_row gt_right">0.0</td>
<td class="gt_row gt_right">0.1</td></tr>
    <tr><td class="gt_row gt_left">weight</td>
<td class="gt_row gt_right">346.0</td>
<td class="gt_row gt_right">31.5</td>
<td class="gt_row gt_right">63.0</td>
<td class="gt_row gt_right">45.0</td>
<td class="gt_row gt_right">40.3</td>
<td class="gt_row gt_right">49.4</td>
<td class="gt_row gt_right">9.0</td>
<td class="gt_row gt_right">6.7</td>
<td class="gt_row gt_right">45.0</td>
<td class="gt_row gt_right">6.5</td>
<td class="gt_row gt_right">0.3</td>
<td class="gt_row gt_right">0.7</td></tr>
  </tbody>
  
  
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Das mittlere Körpergewicht (`weight`) liegt bei ca. 45kg (sd 7 kg).

---

## Visualisierung von `weight` und `height`

Explorative Datenanalyse (keine Inferenz auf Populationswerte, sondern auf die Stichprobe bezogen)

---

## Prädiktor zentrieren 1/2

- Zieht man von jedem Gewichtswert den Mittelwert ab, so bekommt man die Abweichung des Gewichts vom Mittelwert (Prädiktor "zentrieren").
- Wenn man den Prädiktor (`weight`) zentriert hat, ist der Achsenabschnitt, `$\alpha$`, einfacher zu verstehen.
- In einem Modell mit zentriertem Prädiktor (`weight`) gibt der Achsenabschnitt die Größe einer Person mit durchschnittlichem Gewicht an. 
- Würde man `weight` nicht zentrieren, gibt der Achsenabschnitt die Größe einer Person mit `weight=0` an, was nicht wirklich sinnvoll zu interpretieren ist.

<a name=cite-gelman_regression_2021></a>([Gelman, Hill, and Vehtari, 2021](#bib-gelman_regression_2021)Kap. 10.4,  12.2)

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## Prädiktor zentrieren 2/2

```r
d2 <-
  d2 %>% 
  mutate(
    weight_c = weight - 
      mean(weight))
```

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      <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">male</th>
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<td class="gt_row gt_right">48</td>
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<td class="gt_row gt_right">1</td>
<td class="gt_row gt_right">3</td></tr>
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<td class="gt_row gt_right">36</td>
<td class="gt_row gt_right">63</td>
<td class="gt_row gt_right">0</td>
<td class="gt_row gt_right">−9</td></tr>
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<td class="gt_row gt_right">32</td>
<td class="gt_row gt_right">65</td>
<td class="gt_row gt_right">0</td>
<td class="gt_row gt_right">−13</td></tr>
  </tbody>
  
  
</table>
</div>

]

---

## Bei jedem Prädiktorwert eine Post-Verteilung für `$\mu$`

---

## Modelldefinition von `m43`

- Für jede Ausprägung des Prädiktors (`weight`), `$h_i$`, wird eine Post-Verteilung für die abhängige Variable (`height`) berechnet.
- Der Mittelwert `$\mu$` für jede Post-Verteilung ergibt sich aus dem .green[linearen Modell (unserer Regressionsformel)].
- Die Post-Verteilung berechnet sich auf Basis der .blue[Priori-Werte] und des .red[Likelihood] (Bayes-Formel).
- Wir brauchen .blue[Priori-Werte] für die Steigung `$\beta$` und den Achsenabschnitt `$\alpha$` der Regressionsgeraden.
- Außerdem brauchen wir einen .blue[Priori-Wert], der die Streuung `$\sigma$` der Größe (`height`) angibt; dieser Wert wird als exonentialverteilt angenommen.
- Der .green[Likelihood] gibt an, wie wahrscheinlich ein Wert `height` ist, gegeben `$\mu$` und `$\sigma$`.

$$
`\begin{align*}
\color{red}{\text{height}_i} & \color{red}\sim \color{red}{\operatorname{Normal}(\mu_i, \sigma)} && \color{red}{\text{Likelihood}} \\
\color{green}{\mu_i} & \color{green}= \color{green}{\alpha + \beta\cdot \text{weight}_i}  && \color{green}{\text{Lineares Modell} } \\
\color{blue}\alpha & \color{blue}\sim \color{blue}{\operatorname{Normal}(178, 20)} && \color{blue}{\text{Priori}} \\
\color{blue}\beta  & \color{blue}\sim \color{blue}{\operatorname{Normal}(0, 10)}  && \color{blue}{\text{Priori}}\\
\color{blue}\sigma & \color{blue}\sim \color{blue}{\operatorname{Exp}(0.1)}  && \color{blue}{\text{Priori}}
\end{align*}`
$$

---

## Likelihood, `m43`

$$
`\begin{aligned}
\color{red}{\text{height}_i} & \color{red}\sim \color{red}{\operatorname{Normal}(\mu_i, \sigma)} && \color{red}{\text{Likelihood}}
\end{aligned}`
$$

- Der Likelihood von `m43` ist ähnlich zu den vorherigen Modellen (`m41, m42`).
- Nur gibt es jetzt ein kleines "Index-i" am `$\mu$` und am `$h$` (h wie `heights`).
- Es gibt jetzt nicht mehr nur einen Mittelwert `$\mu$`, sondern für jede Beobachtung (Zeile) einen Mittelwert `$\mu_i$`.
- Lies  etwa so:

>    "Die Wahrscheinlichkeit, eine bestimmte Größe bei Person `$i$` zu beobachten, gegeben `$\mu$` und `$\sigma$` ist normalverteilt (mit Mittelwert `$\mu$` und Streuung `$\sigma$`)".

---

## Regressionsformel, `m43`

$$
`\begin{aligned}
\color{green}{\mu_i} & \color{green}= \color{green}{\alpha + \beta\cdot \text{weight}_i}  && \color{green}{\text{Lineares Modell} } \\
\end{aligned}`
$$

- `$\mu$` ist jetzt nicht mehr ein Parameter, der (stochastisch) geschätzt werden muss. `$\mu$` wird jetzt (deterministisch) *berechnet*. Gegeben `$\alpha$` und `$\beta$` ist `$\mu$` ohne Ungewissheit bekannt.
- `$\text{weight}_i$` ist der Prädiktorwert (`weight`) der `$i$`ten Beobachtung, also einer !Kung-Person (Zeile `$i$` im Datensatz).
- Lies  etwa so:

>    "Der Mittelwert `$\mu_i$` der `$i$`ten Person berechnet sich als Summe von `$\alpha$` und `$\beta$` mal  `$\text{weight}_i$`".

- `$\mu_i$` ist eine lineare Funktion von `weight`.
- `$\beta$` gibt den Unterschied in `height` zweier Beobachtung an, die sich um eine Einheit in `weight` unterscheiden (Steigung der Regressionsgeraden).
- `$\alpha$` gibt an, wie groß `$\mu$` ist, wenn `weight` Null ist (Achsenabschnitt, engl. intercept).

---

## Priori-Werte des Modells `m43`

$$
`\begin{align*}
\color{blue}\alpha & \color{blue}\sim \color{blue}{\operatorname{Normal}(178, 20)} && \color{blue}{\text{Priori Achsenabschnitt}} \\
\color{blue}\beta  & \color{blue}\sim \color{blue}{\operatorname{Normal}(0, 10)}  && \color{blue}{\text{Priori Regressionsgewicht}}\\
\color{blue}\sigma & \color{blue}\sim \color{blue}{\operatorname{Exp}(0.1)}  && \color{blue}{\text{Priori Sigma}}
\end{align*}`
$$

- Parameter sind hypothetische Kreaturen: Man kann sie nicht beobachten, sie existieren nicht wirklich. Ihre Verteilungen nennt man Priori-Verteilungen.
- `$\alpha$` wurde in `m41` als `$\mu$` bezeichnet, da wir dort eine "Regression ohne Prädiktoren" berechnet haben.
- `$\sigma$` ist uns schon als Parameter bekannt und behält seine Bedeutung aus dem letzten Kapitel.
- Da `height` nicht zentriert ist, der Mittelwert von `$\alpha$` bei 178 und nicht 0.
- `$\beta$` fasst unser Vorwissen, ob und wie sehr der Zusammenhang zwischen Gewicht und Größe positiv (gleichsinnig) ist.
   - 🤔 Moment. Dieser Prior, `$\beta$` erachtet positive und negative Zusammenhang als gleich wahrscheinlich?!
   - Sind wir wirklich indifferent, ob der Zusammenhang von Gewicht und Größe positiv oder negativ ist? [Nein, sind wir nicht.](https://media.giphy.com/media/daPCSjwus6UR2JxRX1/giphy.gif) 
    
    
---

## Priori-Prädiktiv-Verteilung für `m43`

- Was denkt [wir](https://media.giphy.com/media/Aausss8uUBIe3bZ3d2/giphy.gif) bzw. unser Golem *apriori* über den Zusammenhang von Größe und Gewicht?
- Um diese Frage zu beantworten ziehen wir Stichproben aus den Priori-Verteilungen des Modells, also für `$\alpha$`, `$\beta$` und `$\sigma$`.

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]

Jede Zeile definiert eine Regressionsgerade.

]

---

## Prior-Prädiktiv-Simulation für `m43` mit `stan_glm()`

```r
m43_prior_pred <-
    stan_glm(height ~ weight_c, 
             prior = normal(0, 10),  # beta
             prior_intercept = normal(178, 20),  # alpha
             prior_aux = exponential(0.1),  # sigma
             refresh = FALSE, 
             prior_PD = TRUE,  # DIESER Schalter macht's
             data = d2)

m43_prior_pred_draws <- 
  m43_prior_pred %>% 
  as_tibble() %>% 
  rename(a = `(Intercept)`,
         b = weight_c) %>% 
  slice_sample(n = 50)
```

---

## Visualisieren der Prior-Prädiktiv-Verteilung

```r
d2 %>% ggplot() +
  geom_point(aes(x = weight_c, y = height)) + 
  geom_abline(data = m43_prior_pred_draws,
aes(intercept = a, slope = b), color = "skyblue", size = 0.2) +
  scale_y_continuous(limits = c(0, 500)) +
  geom_hline(yintercept = 272, size = .5) +
  geom_hline(yintercept = 0, linetype = "dashed")
```

🤯 Einige dieser Regressionsgeraden sind unsinnig!

.tiny[Die durchgezogene horizontale Linie gibt die Größe des [größten Menschens, Robert Pershing Wadlow](https://en.wikipedia.org/wiki/Robert_Wadlow), an.]

---

## Ein positiver Wert für `$\beta$` ist plausibler

Eine Normalverteilung mit viel Streuung:

👎 `$\beta=-20$` wäre mit diesem Prior gut möglich: Pro kg Gewicht sind Menschen im Schnitt 20cm kleiner, laut dem Modell. Quatsch.
]

Wir bräuchten eher so eine Verteilung, mit mehr Masse auf der positiven Seite (x>0):

👍 Vermutlich besser: Ein Großteil der Wahrscheinlichkeitsmasse ist `$X>0$`. Allerdings gibt's keine Gewähr, dass unser Prior "richtig" ist.
]

---

## Priori-Prädiktiv-Simulation, 2. Versuch

```r
m43a_prior_pred <-
    stan_glm(
      height ~ weight_c, 
      prior = normal(2, 2),  # Regressionsgewicht
      prior_intercept = normal(178, 20),  # mu
      prior_aux = exponential(0.1),  # sigma
      refresh = FALSE, 
      # Schalter für Prior-Pred-Verteilung:
      prior_PD = TRUE, 
      data = d2)

m43a_prior_pred_draws <- 
  m43a_prior_pred %>% 
  as_tibble() %>% 
  # Spaltennamen kürzen: 
  rename(a = `(Intercept)`) %>%  
  rename(b = weight_c,
         s = sigma)
```

]

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</br>

.footnote[Das Argument `prior_PD = TRUE` sorgt dafür, dass keine Posteriori-Verteilung, sondern eine Prior-Prädiktiv-Verteilung berechnet wird.]

---

## Visualisieren der Prior-Prädiktiv-Verteilung, `m43a`

Unsere Priori-Werte scheinen einigermaßen vernünftige Vorhersagen zu tätigen. Allerdings erwartet unser Golem einige Riesen.

.tiny[Die durchgezogene horizontale Linie gibt die Größe des [größten Menschens, Robert Pershing Wadlow](https://en.wikipedia.org/wiki/Robert_Wadlow), an.]

---

## Moment, kann hier jeder machen, was er will?

</br>

</br>
</br>

>    This is a mistake. There is no more a uniquely correct prior than there is a uniquely correct likelihood. Statistical models are machines for inference. Many machines will work, but some work better than others. Priors can be wrong, but only in the same sense that a kind of hammer can be wrong for building a table.  [McElreath (2020)](#bib-mcelreath_statistical_2020), p. 96.

---

## Hier ist unser Modell, `m43a`

$$
`\begin{align}
\text{height}_i &\sim \operatorname{Normal}(\mu_i, \sigma) \\
\mu_i &= \alpha + \beta \cdot  \text{weight}_i\\
\alpha &\sim \operatorname{Normal}(178, 20)\\
\beta &\sim \operatorname{Normal}(5,3)\\
\sigma &\sim \operatorname{Exp}(0.1)
\end{align}`
$$

```r
# Zufallszahlen festlegen:
set.seed(42)  
# Posteriori-Vert. berechnen:
m43a <-
  stan_glm(
    height ~ weight_c,  # Regressionsformel
    prior = normal(5, 3),  # beta
    prior_intercept = normal(178, 20),  # mu
    prior_aux = exponential(0.1),  # sigma
    refresh = 0,  # zeig mir keine Details
    data = d2)
```

---

## Eine Zusammenfassung der Posteriori-Verteilung für `m43a`

```
## stan_glm
##  family:       gaussian [identity]
##  formula:      height ~ weight_c
##  observations: 346
##  predictors:   2
## ------
##             Median MAD_SD
## (Intercept) 154.6    0.3 
## weight_c      0.9    0.0 
## 
## Auxiliary parameter(s):
##       Median MAD_SD
## sigma 5.1    0.2   
## 
## ------
## * For help interpreting the printed output see ?print.stanreg
## * For info on the priors used see ?prior_summary.stanreg
```

---

# Teil 2

## Die Post-Verteilung befragen

---

## Mittelwerte von `$\alpha$` und `$\beta$` aus der Post-Verteilung

```r
post_m43a <-
  as_tibble(m43a)
```

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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">id</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">(Intercept)</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">weight_c</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">sigma</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1</td>
<td class="gt_row gt_right">154.8</td>
<td class="gt_row gt_right">0.9</td>
<td class="gt_row gt_right">4.9</td></tr>
    <tr><td class="gt_row gt_right">2</td>
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<td class="gt_row gt_right">4.8</td></tr>
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<td class="gt_row gt_right">1.0</td>
<td class="gt_row gt_right">5.1</td></tr>
  </tbody>
  
  
</table>
</div>

]

```r
names(post_m43a) <- 
  c("a", "b", "sigma")

post_m43a_summary <-
  post_m43a %>% 
  summarise(
    a_mean = mean(a),
    b_mean = mean(b),
    s_mean = mean(sigma))
```

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  <thead class="gt_col_headings">
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<td class="gt_row gt_right">0.9</td>
<td class="gt_row gt_right">5.1</td></tr>
  </tbody>
  
  
</table>
</div>

]

---

## Visualisieren der "mittleren" Regressiongeraden

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  <thead class="gt_col_headings">
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</div>

]

```r
d2 %>% 
  ggplot() +
  aes(x = weight_c, y = height) +
  geom_point() +
  geom_abline(
    slope = 0.9,
    intercept = 154,
    color = "blue")
```

]

---

## Zentrale Statistiken zu den Parametern

In diesem Modell gibt es drei Parameter: `$\mu, \beta, \sigma$`.

.pull-left[
### Mittelwerte
- Mittlere Größe?
- Schätzwert für den Zusammenhang von Gewicht und Größe?
- Schätzwert für Ungewissheit in der Schätzung der Größe?

```r
post_m43a_summary
```

```
## # A tibble: 1 × 3
##   a_mean b_mean s_mean
##    <dbl>  <dbl>  <dbl>
## 1   155.  0.908   5.14
```

]

```r
post_m43a_summary2 <-
  post_m43a %>% 
  summarise(
    a_sd = sd(a),
    b_sd = sd(b),
    s_sd = sd(sigma))
```

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]

---

## Ungewissheit von `$\alpha$` und `$\beta$` aus der Post-Verteilung

```r
d2 %>% 
  ggplot(aes(x = weight_c, 
             y = height)) +
  geom_point() +
  geom_abline(
    data = post_m43a %>% 
      slice_head(n=10),
    aes(slope = b,
        intercept = a),
    alpha = .3)
```

]

```r
d2 %>% 
  ggplot(aes(x = weight_c, 
             y = height)) +
  geom_point() +
  geom_abline(
    data = post_m43a %>% 
      slice_head(n=100),
    aes(slope = b,
        intercept = a),
    alpha = .02)
```

]

---

## Fragen zu Quantilen des Achsenabschnitts

Bei einem zentrierten Prädiktor misst der Achsenabschnitt die mittlere Größe.

.pull-left[
- Welche mittlere Größe mit zu 50%, 90% Wskt. nicht überschritten?
- Welche mittlere Größe mit zu 95% Wskt. nicht unterschritten?
- Von wo bis wo reicht der innere 50%-Schätzbereich der mittleren Größe?

```
## # A tibble: 1 × 3
##    q_50  q_90  q_05
##   <dbl> <dbl> <dbl>
## 1  155.  155.  154.
```

```
## # A tibble: 2 × 1
##   pi_50
##   <dbl>
## 1  154.
## 2  155.
```

]

```r
post_m43a %>% 
  summarise(
    q_50 = 
      quantile(a, prob = .5),
    q_90 = 
      quantile(a, prob = .9),
    q_05 = 
      quantile(a, prob = .05))

post_m43a %>% 
  summarise(          
    pi_50 = 
    quantile(a, 
             prob = c(.25, .75)))
```

]

---
## Fragen zu Wahrscheinlichkeitsmassen des Achsenabschnitts

Bei einem zentrierten Prädiktor misst der Achsenabschnitt die mittlere Größe.

```r
post_m43a %>% 
  count(gross = a >= 155) %>% 
  mutate(prop = n / sum(n))
```

```
## # A tibble: 2 × 3
##   gross     n  prop
##   <lgl> <int> <dbl>
## 1 FALSE  3574 0.894
## 2 TRUE    426 0.106
```

Die Wahrscheinlichkeit beträgt 0.11.
]

- Wie wahrscheinlich ist es, dass die mittlere Größe höchstens 154.5 cm beträgt?

```r
post_m43a %>% 
  count(klein = (a <= 154.5)) %>% 
  mutate(prop = n / sum(n))
```

```
## # A tibble: 2 × 3
##   klein     n  prop
##   <lgl> <int> <dbl>
## 1 FALSE  2810 0.702
## 2 TRUE   1190 0.298
```

Die Wahrscheinlichkeit beträgt 0.3.
]

---

## Ungewissheit von Achsenabschnitt und Steigung

... als Histogramme visualisiert

### Achsenabschnitt

```r
post_m43a %>% 
  ggplot(aes(x = a)) +
  geom_density()
```

]

### Regressionsgewicht (Steigung)

```r
post_m43a %>% 
  ggplot(aes(x = b)) +
  geom_density()
```

]

---

## Ungewissheit für `$\mu|\text{weight}=45, 50$`

```r
mu_at_45_50 <-
  post_m43a %>% 
  mutate(mu_at_45 = a,
         mu_at_50 = a + b * 5)
```

```r
mu_at_45_50 %>% 
  ggplot(aes(x = mu_at_45)) +
  geom_density()
```

```r
mu_at_45_50 %>% 
  ggplot(aes(x = mu_at_50)) +
  geom_density()
```

]

]

---

##  Post-Verteilung bedingt auf einen Prädiktorwert

Was ist das 90% PI für `$\mu|w=50$` ?

```r
mu_at_45_50 %>% 
  summarise(pi = 
              quantile(mu_at_50, 
                       prob = c(0.05, .95)))
```

```
## # A tibble: 2 × 1
##      pi
##   <dbl>
## 1  159.
## 2  160.
```
Die mittlere Größe - gegeben `$w=50$` - liegt mit 90% Wahrscheinlichkeit zwischen den beiden Werten.

]

Welche mittlere Größe wird mit 95% Wahrscheinlichkeit nicht überschritten, wenn die Person 45kg wiegt?

```r
mu_at_45_50 %>% 
  summarise(q_95 = 
              quantile(mu_at_45, 
                       prob = .95))
```

```
## # A tibble: 1 × 1
##    q_95
##   <dbl>
## 1  155.
```
]

---

# Teil 3

## Die PPV befragen

---

## Perzentil-Intervalle für verschiedenen Prädiktor-Werte

Wir erstellen uns eine Sequenz an Prädiktorwerten, die uns interessieren:

```r
weight_df <- tibble(weight_c = seq(-20,20, by = 5))
```

Für diese Werte lassen wir uns dann die Perzentil-Intervalle (PI) ausgeben:

```r
mus <- 
  predictive_interval(
    m43a, 
    newdata = weight_df) %>% 
  as_tibble() %>% 
  bind_cols(weight_df)
```

Um die Perzentilintervalle zu erstellen, wird für jeden Prädiktorwert eine Posteriori-Verteilung erstellt und das 5%- sowie 95%-Quantil berechnet.

]

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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">weight_c</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">5%</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">95%</th>
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    <tr><td class="gt_row gt_right">−20.0</td>
<td class="gt_row gt_right">128.0</td>
<td class="gt_row gt_right">144.8</td></tr>
    <tr><td class="gt_row gt_right">−15.0</td>
<td class="gt_row gt_right">132.4</td>
<td class="gt_row gt_right">149.5</td></tr>
    <tr><td class="gt_row gt_right">−10.0</td>
<td class="gt_row gt_right">137.0</td>
<td class="gt_row gt_right">154.2</td></tr>
    <tr><td class="gt_row gt_right">−5.0</td>
<td class="gt_row gt_right">141.6</td>
<td class="gt_row gt_right">158.6</td></tr>
    <tr><td class="gt_row gt_right">0.0</td>
<td class="gt_row gt_right">146.3</td>
<td class="gt_row gt_right">163.2</td></tr>
    <tr><td class="gt_row gt_right">5.0</td>
<td class="gt_row gt_right">150.6</td>
<td class="gt_row gt_right">167.9</td></tr>
    <tr><td class="gt_row gt_right">10.0</td>
<td class="gt_row gt_right">155.6</td>
<td class="gt_row gt_right">172.4</td></tr>
    <tr><td class="gt_row gt_right">15.0</td>
<td class="gt_row gt_right">159.7</td>
<td class="gt_row gt_right">176.9</td></tr>
    <tr><td class="gt_row gt_right">20.0</td>
<td class="gt_row gt_right">164.5</td>
<td class="gt_row gt_right">181.6</td></tr>
  </tbody>
  
  
</table>
</div>

]

---

## Perzentilintervalle für verschiedenen Prädiktorwerte visualisiert

<img src="Kapitel_5_chunk-img/unnamed-chunk-30-1.png" width="100%" style="display: block; margin: auto;" />
]

Je dicker die Violine, desto wahrscheinlicher `$\mu|\text{weight_c}_i$`.

]

---

## Die PPV visualisiert

.pull-left[
Vergleichen wir die echten Werte für `height`, `$h$`, mit den von der PPV simulierten Werten für `height`, `$h_{sim}$`.

```r
library(bayesplot)
h <- d2$height
h_sim <- 
  posterior_predict(m43a, 
                    draws = 50)
ppc_dens_overlay(
  h, h_sim)
```

`?posterior_predict` zeigt Hilfe für diese Funktion. 
Die Funktion zeigt die Vorhersagen für die AV laut der Posteriori-Verteilung.

]

Die zwei Gipfel hat unser Modell nicht mitgekriegt, ansonsten decken sich die Vorhersagen der PPV gut mit den echten Daten.

]

---

## PPV plotten, von Hand

```r
set.seed(42)
ppv_m43a <- posterior_predict(
  m43a,
  newdata = weight_df,
  draws = 100) %>% 
  as_tibble() %>% 
  pivot_longer(
    cols = everything(),
    names_to = "weight_condition",
    values_to = "height")
```

```r
ppv_m43a %>% 
  ggplot(aes(x = height)) +
  geom_density()
```

---

## Fragen an die PPV

.pull-left[
- Wie groß sind die !Kung im Schnitt?
- Welche Größe wird von 90% der Personen nicht überschritten?
- Wie groß sind die 10% kleinsten?

```r
ppv_m43a %>% 
  summarise(
    q_10 = quantile(
      height, prob = .1),
    height_mean = mean(height),
    q_50 = quantile(
      height, prob = .5),
    q_90 = quantile(
      height, prob = .9)
  )
```

```
## # A tibble: 1 × 4
##    q_10 height_mean  q_50  q_90
##   <dbl>       <dbl> <dbl> <dbl>
## 1  138.        154.  154.  172.
```

]

```r
ppv_m43a %>% 
  summarise(
    pi_50 = quantile(
      height, 
      prob = c(.25, .75))
  )
```

```
## # A tibble: 2 × 1
##   pi_50
##   <dbl>
## 1  144.
## 2  165.
```

]

---

# Hinweise

---

## Zu diesem Skript

- Dieses Skript bezieht sich auf folgende [Lehrbücher](#literatur): 
  - Rethink, Kap. 4.4, ROS, Kap. 9.2

- Dieses Skript wurde erstellt am 2021-11-22 12:48:18

- Lizenz: [CC-BY](https://creativecommons.org/licenses/by/4.0/)

- Autor ist Sebastian Sauer.

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---

## Literatur

<a name=bib-gelman_regression_2021></a>[Gelman, A., J.
Hill, and A. Vehtari](#cite-gelman_regression_2021) (2021).
_Regression and other stories_. Analytical methods for
social research. Cambridge University Press.

<a name=bib-mcelreath_statistical_2020></a>[McElreath,
R.](#cite-mcelreath_statistical_2020) (2020). _Statistical
rethinking: a Bayesian course with examples in R and Stan_.
2nd ed. CRC texts in statistical science. Taylor and
Francis, CRC Press.