IMSE 440 - logistic regression

class: center, middle

## IMSE 440
## Applied Statistical Models in Engineering
<br>

## Logistic regression

[ISLR book](https://www.statlearning.com): Chapter 4.3

---
# Will a credit card customer default?

```
import pandas as pd
df = pd.read_csv('./data/default.csv')
df.head()
```

.center[![:scale 70%](images/default_dataframe.png)]

---

$$\text{default_binary}=
\begin{cases}
    1, & \text{if } \text{default = Yes,} \\\
    0, & \text{if } \text{default = No.}
\end{cases}
$$

```
plt.scatter(x='balance', y='default_binary', data=df, 
            s=200, alpha=.1)
```

<br>
.center[![:scale 100%](images/default_scatter.png)]

---
# Logistic regression

$$p(x)=\frac{e^{\beta_0+\beta_1x}}{1+e^{\beta_0+\beta_1x}}$$

$p(x)$: probability of default given that the balance is $x$.

<br>
.center[![:scale 100%](images/default_model.png)]

---
# Logistic regression

$$p(x)=\frac{e^{\beta_0+\beta_1x}}{1+e^{\beta_0+\beta_1x}}$$

???

$$
\begin{aligned}
p(x)&=\frac{e^{\beta_0+\beta_1x}}{1+e^{\beta_0+\beta_1x}} \\\
\\\
\frac{p(x)}{1-p(x)}&=e^{\beta_0+\beta_1x} \\\
\\\
\ln{\frac{p(x)}{1-p(x)}}&=\beta_0+\beta_1x \\\
\end{aligned}
$$

The log-odds (or logit) is a linear function of $x$.

---
# Interpretation of the slope parameter $\beta_1$

$$\frac{p(x)}{1-p(x)}=e^{\beta_0+\beta_1x}$$

When we increase $x$ by $1$, the odds

$$
\begin{aligned}
\frac{p(x+1)}{1-p(x+1)}&=e^{\beta_0+\beta_1(x+1)} \\\
&=e^{\beta_1}e^{\beta_0+\beta_1x} \\\
&=e^{\beta_1}\frac{p(x)}{1-p(x)}
\end{aligned}
$$

increase by a ratio of $e^{\beta_1}$.

---
# Interpretation of the slope parameter $\beta_1$

$$\ln{\frac{p(x)}{1-p(x)}}=\beta_0+\beta_1x$$

When we increase $x$ by $1$, the log-odds

$$
\begin{aligned}
\ln{\frac{p(x+1)}{1-p(x+1)}}&=\beta_0+\beta_1(x+1) \\\
&=\beta_0+\beta_1x+\beta_1\\\
&=\ln{\frac{p(x)}{1-p(x)}}+\beta_1
\end{aligned}
$$

increase by $\beta_1$.