Movtivating regression

class: center, middle, inverse, title-slide

# Movtivating regression
### Prof. Maria Tackett

---

## Sales vs. Advertising

- Suppose you are a data scientist on the marketing team and the company wants to improve the sales of their premiere product

- You want to understand the relationship between .vocab[advertising budget] and .vocab[total sales]

- The goal is to advise the marketing team about how to set the advertising budget based on their target sales goals

---

## Advertising vs. Sales

```r
glimpse(advertising)
```

```
## Rows: 200
## Columns: 4
## $ tv <dbl> 230.1, 44.5, 17.2, 151.5, 180.8, 8.7, 57.5, 120.2, 8.6, 199…
## $ radio <dbl> 37.8, 39.3, 45.9, 41.3, 10.8, 48.9, 32.8, 19.6, 2.1, 2.6, 5…
## $ newspaper <dbl> 69.2, 45.1, 69.3, 58.5, 58.4, 75.0, 23.5, 11.6, 1.0, 21.2, …
## $ sales <dbl> 22.1, 10.4, 9.3, 18.5, 12.9, 7.2, 11.8, 13.2, 4.8, 10.6, 8.…
```

- .vocab[Observations]: 200 markets

- .vocab[Variables]:
    - `tv`: Spending on TV ads (in $thousands)
    - `radio`: Spending on radio ads (in $thousands)
    - `newspaper`: Spending on newspaper ads (in $thousands)
    - `sales`: total sales (in $millions)
   
---

## Terminology

- `sales` is the .vocab[response variable]
    - variable whose variation we want to understand / variable we wish to predict
    - also known as *outcome* or *dependent* variable

- `tv`, `radio`, `newspaper` are the .vocab[predictor variables]
    - variables used to account for variation in the outcome
    - also known as *explanatory*, *independent*, or *input* variables

---

## Let's look at the data

Each line represents model we could use to predict `sales` using `tv`, `radio`, or `newspaper`

---

## Let's look at the data

.alert[
`$$\text{sales} = f(\text{tv}, \text{radio}, \text{newspaper}) + \epsilon$$`
]

---

## Model

.alert[
`$$\text{sales} = f(\text{tv}, \text{radio}, \text{newspaper}) + \epsilon$$`
]

- **Goal**: Define `$f$`

- How do we define `$f$`?
  - Make an assumption about the functional form `$f$`
  - Use the data to fit a model based on that form 
    
---

## How to define `$f$`

In general,

1. Choose the functional form of `$f$`, i.e. choose the appropriate model given the data
 - Ex: `$f$` is a linear model
 `$$f(\mathbf{X}) = \beta_0 + \beta_1 X_1 + \dots + \beta_p X_p$$`

2. Use the data to fit (or train) the model, i.e. estimate the model parameters
 - Ex: Find estimates of `$\beta_0, \beta_1, \ldots, \beta_p$`

---

## Why?

.alert[
`$$\hat{\text{sales}} = \hat{\beta}_0 + \hat{\beta}_1 \times \text{tv} + \hat{\beta}_2 \times \text{radio} + \hat{\beta}_3 \times \text{newspaper}$$`
]

.vocab[Prediction:]

What do we expect `sales` to be in a market where there is $100,000 spent on TV ads, $30,000 spent on radio ads, and $10,000 spent on newspaper ads?

.vocab[Inference:]

What is the relationship between spending on TV ads and sales after accounting for spending on radio and newspaper ads?

---

## Course Outline

.pull-left[

.vocab[Unit 1: Quantitative Response Variables]
  - Simple Linear Regression 
  - Multiple Linear Regression

.vocab[Unit 3: Looking Ahead]
  - Weighted least squares
  - Dealing with missing data
  - Modeling in practice

]

.pull-right[

- .vocab[Unit 2: Categorical Response Variable]
  - Logistic Regression 
  - Multinomial Logistic Regression

]