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<!DOCTYPE html>
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    <title>Simple Linear Regression (SLR)</title>
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class: center, middle, inverse, title-slide

# Simple Linear Regression (SLR)
### STAT 021 with Prof Suzy
### Swarthmore College

---


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## A simple statistical model

`$$Y\mid x = f(x) + \epsilon$$`

- `\(f\)` is a smooth function.

  - In linear regression, we consider functions with linear coefficients. These coefficients are our model parameters. (I.e. `\(f\)` is just the equation for a line.)

- `\(x\)` is a fixed/known covariate.

- `\(\epsilon\)` is some random measurement error. Note that, against convention, even though this is a Greek letter, `\(\epsilon\)` represents a **random variable**!


---
## Simple linear regression


Statistical convention represents a regression line with a intercept, `\(\beta_0\)`, and a slope `\(\beta_1\)` so that we have the following **simple linear regression model**:
`$$Y\mid x = \beta_0 + \beta_1 x + \epsilon.$$`


- `\(Y\)` is the response (output) variable. We assume that there is some random error associated with our observations of `\(Y\)`.
- `\(x\)` is a predictor (explanatory, covariate, input) variable. We assume there is **no** random error associated with `\(x\)`, i.e. that all values of `\(x\)` are fixed, so it's not a random variable.
- The behavior of `\(Y\)` is modeled conditional upon the predictor `\(x\)`.
- `\(\beta_{0}\)`, `\(\beta_{1}\)` are the regression model coefficients (the intercept and slope, respectively).

***

Compare this to the the typical algebraic notation for the equation of a line:
`$$Y = ax + b.$$`


For more information on interpreting negative intercept values &lt;a href="https://statisticsbyjim.com/regression/interpret-constant-y-intercept-regression/"&gt;go here&lt;/a&gt;.

---
## Simple linear regression

`$$Y \mid x = \beta_0 + \beta_1 x + \epsilon.$$`

It's called a linear model because `\(f\)` is linear with respect to the coefficients `\(\beta_{i}\)`, for `\(i=1,2\)`.


**Question:** Which of the following are linear models?

1. `\(Y = \beta_{0} + \beta_{1}x^2 + \epsilon\)`

1. `\(Y = \sqrt{\beta_{0} + \beta_{1}x} + \epsilon\)`

---
## Simple linear regression

`$$Y \mid x = \beta_0 + \beta_1 x + \epsilon.$$`

It's called a linear model because `\(f\)` is linear with respect to the coefficients `\(\beta_{i}\)`, for `\(i=1,2\)`.


**Question:** Which of the following are linear models?

1. `\(Y = \beta_{0} + \beta_{1}x^2 + \epsilon\)` (this is!)

1. `\(Y = \sqrt{\beta_{0} + \beta_{1}x} + \epsilon\)` (not this!)


---
## Simple linear regression

`$$Y \mid x = \beta_0 + \beta_1 x + \epsilon.$$`

For now, we are only going to consider the case where `\(x\)` and `\(Y\)` both represent **quantitative, continuous** random variables.


We will be generalizing this SLR (simple linear regression) model to cases where

  - X is a discrete and quantitative variable;

  - X is a categorical variable (ANOVA);

  - We have more than just one predictor variable (MLR);

  - Y is a binary variable (logistic regression) - time permitting.


---
## Simple linear regression

In SLR, the data we observe are pairs `\((x_{1},y_{1}), \dots, (x_{n},y_{n})\)`, of continuous, quantitative variables.

The model `\(Y \mid x = \beta_0 + \beta_1 x + \epsilon\)` means that we are assuming
`$$y_{i} = \beta_0 + \beta_1 x_{i} + \epsilon_{i},$$`
for each data point we observe where `\(\epsilon_{i}\)` represents an (unobserved) measurement error associated with our response variable.


---
## Simple linear regression

`$$Y \mid x = \beta_0 + \beta_1 x + \epsilon$$`

**Assumptions**

- For estimation: The measurement error has mean `\(E[\epsilon]=0\)` and (unknown) variance `\(Var[\epsilon]=\sigma^2\)` and all measurement errors are independent of each other.

- For inference: If we wish to conduct statistical inference, we must also assume that the measurement error, `\(\epsilon\)`, follows a standard normal distribution.


--
**Question:** What do theses assumptions imply about `\(Y\)`?


--
**Another question:** What if there was no random error in our observations of `\(Y\)`? How do we find the line of best fit in this case?


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