Multiple Regression 1

class: center, middle, title-slide

# Multiple Regression 1
## AU STAT-615
### Emil Hvitfeldt
### 2021-03-03

---

`$$\require{color}\definecolor{orange}{rgb}{1, 0.603921568627451, 0.301960784313725}$$`
`$$\require{color}\definecolor{blue}{rgb}{0.301960784313725, 0.580392156862745, 1}$$`
`$$\require{color}\definecolor{pink}{rgb}{0.976470588235294, 0.301960784313725, 1}$$`

# Example 1

First order model with two predictor variables

When there are two variables `$X_1$` and `$X_2$`, the regression model is

`$$Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \varepsilon_i$$`

Assuming `$E\{\varepsilon_i\} = 0$` we have

`$$E\{Y_i\} = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2}$$`

The response function is a plane

---

# Example 1

`$$E\{Y_i\} = \blue{\beta_0} + \orange{\beta_1} X_{i1} + \pink{\beta_2} X_{i2}$$`

- `$\blue{\beta_0}$`, y-intercept. If `$X_1 = X_2 = 0$` then `$\beta_0$` represents the mean response `$E\{Y\}$`
- `$\orange{\beta_1}$` Indicates change in mean per unit increase in `$X_1$` when `$X_2$` is constant
- `$\pink{\beta_2}$` Indicates change in mean per unit increase in `$X_2$` when `$X_1$` is constant

---

# Example 2

First order model with more than two predictor variables

For `$p-1$` predictor variables

`$$Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \cdots +  \beta_{p-1} X_{i,p-1} + \varepsilon_i$$`

`$$Y_i = \beta_0 + \sum_{k = 1}^{p-1} \beta_k X_{l1} + \varepsilon_i$$`

---

# Example 2

Assuming `$E\{\varepsilon_i\} = 0$` we obtain

`$$E\{Y_i\} = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \cdots +  \beta_{p-1} X_{i,p-1}$$`

Here the response function is a hyperplane

---

# Qualitative predictor variables

For the model

`$$Y_i = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{i2} + \cdots +  \beta_{p-1} X_{i,p-1} + \varepsilon_i$$`

This model encompasses not only quantitative predictor variables but also qualitative ones such as sex or disability status

For example, let

$$
`\begin{align}
X_1 &= \text{Age of patients}\\
X_2 & = \begin{cases}
    1, & \text{patient female}\\
    0, & \text{patient male}
\end{cases}\\
Y &= \text{Length of hospital stay}
\end{align}`
$$

---

# Qualitative predictor variables

We have

`$$E\{Y\} = \beta_0 + \beta_1 X_{i1} + \beta_2 X_{2}$$`

and for male patients

`$$E\{Y\} = \beta_0 + \beta_1 X_{1}$$`

and for female patients
 
`$$E\{Y\} = \beta_0 + \beta_1 X_{1} + \beta_2 = (\beta_0 + \beta_2)+ \beta_1 X_{1}$$`

These two response functions are straight lines that are parallel with each other

---

# Polynomial Regression

Special case of general linear regression model

`$$Y_i = \beta_0 + \beta_1 X_{i} + \beta_2 X_{i}^2 + \varepsilon_i$$`