class: center, middle, inverse, title-slide

# Recap

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layout: true

<div class="my-footer">
<span>
Dr. Lucy D'Agostino McGowan
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</div>

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## Simple linear regression

`\(\Huge y = \beta_0 + \beta_1 x + \epsilon\)`

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## Simple linear regression

`\(\Huge \hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\)`

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## Simple linear regression

.question[
What is the method used to estimate `\(\hat{\beta}_0\)` and `\(\hat{\beta}_1\)`?
]

`\(\Huge \hat{y} = \hat{\beta}_0 + \hat{\beta}_1 x\)`

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## Simple linear regression

.question[
What is the method used to estimate `\(\hat{\beta}_0\)` and `\(\hat{\beta}_1\)`?
]

.center[
![](img/03/least-squares-vis.gif)
]

---

## Simple linear regression

.question[
What is least squares minimizing?
]

.center[
![](img/03/least-squares-vis.gif)
]

---

## Simple linear regression
.question[
What is least squares minimizing?
]

* Sum of square errors (SSE)
 
.center[
![](img/03/least-squares-vis.gif)
]
---

## Simple linear regression
.question[
How do we calculate the SSE?
]

* SSE: `\(\sum (y-\hat{y})^2\)`
 
.center[
![](img/03/least-squares-vis.gif)
]

---
## Simple linear regression
.question[
How do we calculate the SSE?
]

* SSE: `\(\sum (y-\hat{y})^2\)`
 
.center[
![](img/03/least-squares-vis.gif)
]
---

## Simple linear regression
.question[
How do we calculate the a residual?
]

---

## Simple linear regression
.question[
How do we calculate the a residual?
]

* residual: `\(y - \hat{y}\)`
---

## Simple linear regression

.question[
How do we calcualte the regression standard error (residual standard error)?
]
 
---

## Simple linear regression

.question[
How do we calcualte the regression standard error (residual standard error)?
]

* `\(\hat{\sigma}_\epsilon = \sqrt{\frac{SSE}{n-2}}\)`

---

## Simple linear regression

.question[
How do we interpret the regression standard error?
]

* `\(\hat{\sigma}_\epsilon = \sqrt{\frac{SSE}{n-2}}\)`

---

## Simple linear regression

.question[
What are the (six) assumptions for simple linear regression?
]

---
## Simple linear regression

.question[
What are the (six) assumptions for simple linear regression?
]

* Linearity
* Zero Mean
* Constant Variance
* Independence
* Random
* Normality
---
## Simple linear regression

`\(\Huge\epsilon \sim N(0, \sigma_\epsilon)\)`

* Linearity
* Zero Mean
* Constant Variance
* Independence
* Random
* Normality
---

## Simple linear regression

.question[
What are the three parameters we estimate with simple linear regression?
]

---

## Simple linear regression

.question[
What are the three parameters we estimate with simple linear regression?
]

- `\(\Huge\hat{\beta}_0\)`
- `\(\Huge\hat{\beta}_1\)`
- `\(\Huge\hat{\sigma}_\epsilon\)`

Notes for current slide

Notes for next slide

Recap1 / 19

Simple linear regression

$y = β_{0} + β_{1} x + ϵ$

2 / 19

Simple linear regression

$\hat{y} = {\hat{β}}_{0} + {\hat{β}}_{1} x$

3 / 19

Simple linear regression

What is the method used to estimate ${\hat{β}}_{0}$ and ${\hat{β}}_{1}$ ?

$\hat{y} = {\hat{β}}_{0} + {\hat{β}}_{1} x$

4 / 19

Simple linear regression

What is the method used to estimate ${\hat{β}}_{0}$ and ${\hat{β}}_{1}$ ?

5 / 19

Simple linear regression

What is least squares minimizing?

6 / 19

Simple linear regression

What is least squares minimizing?

Sum of square errors (SSE)

7 / 19

Simple linear regression

How do we calculate the SSE?

SSE: $\sum (y - \hat{y})^{2}$

8 / 19

Simple linear regression

How do we calculate the SSE?

SSE: $\sum (y - \hat{y})^{2}$

9 / 19

Simple linear regression

How do we calculate the a residual?

10 / 19

Simple linear regression

How do we calculate the a residual?

residual: $y - \hat{y}$

11 / 19

Simple linear regression

How do we calcualte the regression standard error (residual standard error)?

12 / 19

Simple linear regression

How do we calcualte the regression standard error (residual standard error)?

${\hat{σ}}_{ϵ} = \sqrt{\frac{S S E}{n - 2}}$

13 / 19

Simple linear regression

How do we interpret the regression standard error?

${\hat{σ}}_{ϵ} = \sqrt{\frac{S S E}{n - 2}}$

14 / 19

Simple linear regression

What are the (six) assumptions for simple linear regression?

15 / 19

Simple linear regression

What are the (six) assumptions for simple linear regression?

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

16 / 19

Simple linear regression

$ϵ \sim N (0, σ_{ϵ})$

Linearity
Zero Mean
Constant Variance
Independence
Random
Normality

17 / 19

Simple linear regression

What are the three parameters we estimate with simple linear regression?

18 / 19

Simple linear regression

What are the three parameters we estimate with simple linear regression?

${\hat{β}}_{0}$
${\hat{β}}_{1}$
${\hat{σ}}_{ϵ}$

19 / 19

Simple linear regression

$y = β_{0} + β_{1} x + ϵ$

2 / 19

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