IDS 702: Module 1.10

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# IDS 702: Module 1.10
## Bringing the MLR pieces together I (illustration)
### Dr. Olanrewaju Michael Akande

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## Diamonds data

- A diamond's value is often determined using four factors known as the 4Cs: color, clarity, cut (certification) and carat weight.

+ Color: evaluation based on absence of color; how pure the diamond is. .hlight[This is a categorical variable with 6 levels.]
  
--
  
  + Clarity: evaluation based on absence of blemishes. .hlight[This is a categorical variable with 5 levels.]
  
--

+ Certification: how well the diamond is cut; how well a diamond's facets interacts with light. .hlight[This is a categorical variable with 3 levels.]

+ Carats: carat weight measuring how much the diamond weighs. .hlight[This is a continuous variable.]
  
--

- We will use some data to draw inference about how these factors affect a diamond's price .hlight[(continuous)].

- You can read more about the 4Cs [here](https://4cs.gia.edu/en-us/4cs-diamond-quality/).

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## Multiple regression of diamonds data

- A good starting model is
.block[
`$$y_i = \boldsymbol{x}_i\boldsymbol{\beta} + \epsilon_i; \ \ \epsilon_i \sim N(0, \sigma^2).$$`
]

where `$y_i$` is the price for observation `$i$`, and `$\boldsymbol{x}_i$` is the vector containing the corresponding values for Carats, Color, Clarity, and Certification.
  
--

- Alternatively, write
.block[
.midsmall[
$$
`\begin{split}
\text{Price}_i & = \beta_0 + \beta_1 \text{Carats}_i + \sum_{j=2}^6 \beta_{2j} \mathbb{1}[\text{Color}_i = j] + \sum_{j=2}^5 \beta_{3j} \mathbb{1}[\text{Clarity}_i = j] \\
& \ \ \ + \sum_{j=2}^3 \beta_{4j} \mathbb{1}[\text{Certification}_i = j] +  \epsilon_i; \ \ \epsilon_i \sim N(0, \sigma^2).
\end{split}`
$$
]
]

- Can also write
.block[
.midsmall[
$$
`\begin{split}
\widehat{\text{Price}}_i & = \hat{\beta}_0 + \hat{\beta}_1 \text{Carats}_i + \sum_{j=2}^6 \hat{\beta}_{2j} \mathbb{1}[\text{Color}_i = j] + \sum_{j=2}^5 \hat{\beta}_{3j} \mathbb{1}[\text{Clarity}_i = j] \\
& \ \ \ + \sum_{j=2}^3 \hat{\beta}_{4j} \mathbb{1}[\text{Certification}_i = j].
\end{split}`
$$
]
]

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## Multiple regression of diamonds data

- This is just a candidate model.

- We will go through the full (almost!) modeling process and we will see if this model makes sense or if we need to make changes to it.

- We will start by doing EDA, all the way down to model assessment, including investigating multicollinearity.

- We will explore transformations, polynomial forms, interactions, etc.

- The data is in the file `diamonds.csv` on Sakai.

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# Move to the R script [here](https://ids-702-f20.github.io/Course-Website/slides/Diamonds.R).

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# What's next?

### Move on to the readings for the next module!