Week 2 Assignment: Presentation

.title[
# Week 2 Assignment: Presentation
]
.subtitle[
## MLR of CO2 Emissions for Vehicles
]
.author[
### Alice Xiang & Angelo Saporito
]
.date[
### 2024-02-25
]

---

<h2 align="center"> Table of Contents</h2>

- Research Question

- Full Model
    - Analysis
    - Residual analysis
    - Variance Inflation Factor (VIF)
    - Discussion

- Edited Model
    - Analysis
    - Residual analysis
    - Variance Inflation Factor (VIF)
    - Discussion
]

.pull-right[
- Transformed Model
    - Box-Cox Transform
    - Log Transform
    - Residual Analysis
    - Discussion

- The Bootstrap

- Model Selection
    - Goodness-of-fit
    - Final Model

- Conclusion
]
---
class: inverse center middle

## Introduction to the Dataset

---
.pull-left[
## Introduction

We chose [this dataset](https://www.kaggle.com/datasets/bhuviranga/co2-emissions) on CO2 emissions of different vehicles to do multiple linear regression analysis. 
]

The following are the variables included in the dataset (6 continuous, 2 categorical):

- Engine.Size

- Cylinders

- Fuel.Type

- Fuel.Consumption.City

- Fuel.Consumption.Hwy

- Fuel.Consumption.Combined

- Fuel.Consumption.mpg

- CO2.Emissions 
]
---
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## Research Question: How do different predictor variables relate to the CO2 emissions of the vehicle?

---
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## Full Model + Discussion

---

## The Full Model

Using R, we create the full model.

```r
full.model = lm(CO2.Emissions ~ ., data = emissions)
```

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-47eb2c74e6b0332127a1" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-47eb2c74e6b0332127a1">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["(Intercept)","Engine.Size","Cylinders4","Cylinders5","Cylinders6","Cylinders8","Cylinders10","Cylinders12","Cylinders16","Fuel.TypeE","Fuel.TypeN","Fuel.TypeX","Fuel.TypeZ","Fuel.Consumption.City","Fuel.Consumption.Hwy","Fuel.Consumption.Combined","Fuel.Consumption.mpg"],[93.867,0.822,-0.958,-3.55,-0.09,1.308,6.303,8.606,28.896,-137.706,-111.334,-30.526,-31.112,6.096,5.483,8.195,-0.965],[1.642,0.144,0.523,1.103,0.589,0.738,1.095,0.957,3.068,0.536,4.926,0.384,0.389,0.742,0.612,1.347,0.025],[57.156,5.704,-1.834,-3.218,-0.153,1.773,5.759,8.99,9.42,-256.687,-22.602,-79.414,-80.027,8.211,8.959,6.084,-38.13],[0,0,0.067,0.001,0.878,0.076,0,0,0,0,0,0,0,0,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>Estimate<\/th>\n      <th>Std. Error<\/th>\n      <th>t value<\/th>\n      <th>Pr(>|t|)<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. Error","targets":2},{"name":"t value","targets":3},{"name":"Pr(>|t|)","targets":4}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

## Residual Analysis

---

|                          |    GVIF| Df| GVIF^(1/(2*Df))|
|:-------------------------|-------:|--:|---------------:|
|Engine.Size               |   11.67|  1|            3.42|
|Cylinders                 |   14.40|  7|            1.21|
|Fuel.Type                 |    2.48|  4|            1.12|
|Fuel.Consumption.City     | 2069.97|  1|           45.50|
|Fuel.Consumption.Hwy      |  568.00|  1|           23.83|
|Fuel.Consumption.Combined | 4651.99|  1|           68.21|
|Fuel.Consumption.mpg      |   10.26|  1|            3.20|
]

## Issues We See

- nonconstant variance

- residuals not normal (Q-Q plot)

- multicollinearity between all Fuel Consumption variables

]

---
class: inverse center middle

## Edited Model + Discussion

---

## Edited Model: Removing Predictors due to Multicollinearity

Of the Fuel Consumption variables, we keep only Fuel.Consumption.mpg and create the following model.

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-affae824d132c4a3fe2a" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-affae824d132c4a3fe2a">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["(Intercept)","Engine.Size","Cylinders4","Cylinders5","Cylinders6","Cylinders8","Cylinders10","Cylinders12","Cylinders16","Fuel.TypeE","Fuel.TypeN","Fuel.TypeX","Fuel.TypeZ","Fuel.Consumption.mpg"],[425.436,7.5,-8.933999999999999,-13.777,-5.948,11.741,34.316,44.381,145.252,-77.751,-100.318,-25.916,-29.994,-6.053],[2.777,0.432,1.601,3.383,1.806,2.258,3.334,2.867,9.265000000000001,1.466,15.114,1.177,1.193,0.042],[153.178,17.35,-5.579,-4.072,-3.294,5.2,10.294,15.478,15.677,-53.043,-6.638,-22.012,-25.142,-145.335],[0,0,0,0,0.001,0,0,0,0,0,0,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>Estimate<\/th>\n      <th>Std. Error<\/th>\n      <th>t value<\/th>\n      <th>Pr(>|t|)<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. Error","targets":2},{"name":"t value","targets":3},{"name":"Pr(>|t|)","targets":4}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

## Residual Plots of Edited Model

---
.pull-left[
## VIF of Edited Model

|                     |  GVIF| Df| GVIF^(1/(2*Df))|
|:--------------------|-----:|--:|---------------:|
|Engine.Size          | 11.15|  1|            3.34|
|Cylinders            | 11.79|  7|            1.19|
|Fuel.Type            |  1.41|  4|            1.04|
|Fuel.Consumption.mpg |  2.95|  1|            1.72|
]

## Edited Model Discussion

We see that the residual plots improved, and the issues with multicollinearity have been resolved.

### Remaining Issues: 
- variances still nonconstant
- assumption of normality still violated 
]
---
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## Transformed Model + Discussion

---

## Box-Cox Transformation

We proceed by performing several box-cox transformations on the data.

<img src="Final-Presentation_files/figure-html/unnamed-chunk-10-1.png" width="100%" />
The plots show that a log transformation of Fuel Consumption impacts lambda.

---

## Log Transformed Model

Using a log transformed mpg, we create the following model with a log of the response variable CO2.Emissions:

```r
log.model = lm(log(CO2.Emissions) ~ Engine.Size + Cylinders + Fuel.Type + log(Fuel.Consumption.mpg), data = emissions.edit)
```

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-6613b72cc40dd855cde9" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-6613b72cc40dd855cde9">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["(Intercept)","Engine.Size","Cylinders4","Cylinders5","Cylinders6","Cylinders8","Cylinders10","Cylinders12","Cylinders16","Fuel.TypeE","Fuel.TypeN","Fuel.TypeX","Fuel.TypeZ","log(Fuel.Consumption.mpg)"],[8.891,0,-0.002,-0.008999999999999999,0.002,0.001,0.003,0.007,0.039,-0.492,-0.48,-0.141,-0.142,-0.988],[0.006,0,0.002,0.004,0.002,0.002,0.004,0.003,0.01,0.002,0.017,0.001,0.001,0.002],[1414.202,0.798,-1.388,-2.359,0.857,0.5649999999999999,0.699,2.181,3.837,-292.716,-28.813,-108.4,-108.204,-654.755],[0,0.425,0.165,0.018,0.392,0.572,0.484,0.029,0,0,0,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>Estimate<\/th>\n      <th>Std. Error<\/th>\n      <th>t value<\/th>\n      <th>Pr(>|t|)<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. Error","targets":2},{"name":"t value","targets":3},{"name":"Pr(>|t|)","targets":4}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---

## Residual Plots of Transformed Model

---

## Transformed Model Discussion

- Significant improvements from earlier models
- Curvature in residual plot greatly improved
- Q-Q plot closest to normal
---

## The Bootstrap
---
<h2 align="center">Boostrapping Coefficients</h2>

.pull-center[
<img src="Final-Presentation_files/figure-html/unnamed-chunk-17-1.png" width="70%" height="70%" style="display: block; margin: auto;" />
]

---
<h2 align="center">Confidence Intervals</h2>

|                          |Estimate |Std. Error |t value  |Pr(>&#124;t&#124;) |btc.ci.95              |
|:-------------------------|:--------|:----------|:--------|:------------------|:----------------------|
|(Intercept)               |8.891    |0.006      |1414.010 |0.000              |[ 8.8791 ,  8.9015 ]   |
|Engine.Size               |0.000    |0.000      |0.798    |0.425              |[ -4e-04 ,  0.0013 ]   |
|Cylinders4                |-0.002   |0.002      |-1.388   |0.165              |[ -0.0051 ,  4e-04 ]   |
|Cylinders5                |-0.009   |0.004      |-2.359   |0.018              |[ -0.0143 ,  -0.0036 ] |
|Cylinders6                |0.002    |0.002      |0.856    |0.392              |[ -0.0013 ,  0.005 ]   |
|Cylinders8                |0.001    |0.002      |0.564    |0.572              |[ -0.0028 ,  0.0054 ]  |
|Cylinders10               |0.003    |0.004      |0.699    |0.485              |[ -0.0038 ,  0.009 ]   |
|Cylinders12               |0.007    |0.003      |2.181    |0.029              |[ 0.0012 ,  0.0128 ]   |
|Fuel.TypeE                |-0.492   |0.002      |-292.677 |0.000              |[ -0.4963 ,  -0.4875 ] |
|Fuel.TypeX                |-0.141   |0.001      |-108.385 |0.000              |[ -0.1426 ,  -0.139 ]  |
|Fuel.TypeZ                |-0.142   |0.001      |-108.189 |0.000              |[ -0.1441 ,  -0.1404 ] |
|log(Fuel.Consumption.mpg) |-0.988   |0.002      |-654.666 |0.000              |[ -0.9903 ,  -0.9848 ] |
]
---
<h2 align="center">Bootstrapping Residuals</h2>

<img src="Final-Presentation_files/figure-html/unnamed-chunk-19-1.png" width="60%" height="60%" style="display: block; margin: auto;" />
- Residuals are largely symmetric
- Presence of at least one outlier and some slight right skew

---
<h2 align="center">Bootstrapping Residuals cont.</h2>

---
<h2 align="center">Bootstrapped Coefficients & Residuals</h2>

|                          |Estimate |Std. Error |Pr(>&#124;t&#124;) |btc.ci.95              |btr.ci.95              |
|:-------------------------|:--------|:----------|:------------------|:----------------------|:----------------------|
|(Intercept)               |8.891    |0.006      |0.000              |[ 8.8791 ,  8.9015 ]   |[ 8.8781 ,  8.9035 ]   |
|Engine.Size               |0.000    |0.000      |0.425              |[ -4e-04 ,  0.0013 ]   |[ -6e-04 ,  0.0014 ]   |
|Cylinders4                |-0.002   |0.002      |0.165              |[ -0.0051 ,  4e-04 ]   |[ -0.0058 ,  0.0014 ]  |
|Cylinders5                |-0.009   |0.004      |0.018              |[ -0.0143 ,  -0.0036 ] |[ -0.0156 ,  -0.0015 ] |
|Cylinders6                |0.002    |0.002      |0.392              |[ -0.0013 ,  0.005 ]   |[ -0.0021 ,  0.0058 ]  |
|Cylinders8                |0.001    |0.002      |0.572              |[ -0.0028 ,  0.0054 ]  |[ -0.0034 ,  0.0064 ]  |
|Cylinders10               |0.003    |0.004      |0.485              |[ -0.0038 ,  0.009 ]   |[ -0.0042 ,  0.0101 ]  |
|Cylinders12               |0.007    |0.003      |0.029              |[ 0.0012 ,  0.0128 ]   |[ 8e-04 ,  0.0135 ]    |
|Fuel.TypeE                |-0.492   |0.002      |0.000              |[ -0.4963 ,  -0.4875 ] |[ -0.4951 ,  -0.4883 ] |
|Fuel.TypeX                |-0.141   |0.001      |0.000              |[ -0.1426 ,  -0.139 ]  |[ -0.1433 ,  -0.1382 ] |
|Fuel.TypeZ                |-0.142   |0.001      |0.000              |[ -0.1441 ,  -0.1404 ] |[ -0.1449 ,  -0.1396 ] |
|log(Fuel.Consumption.mpg) |-0.988   |0.002      |0.000              |[ -0.9903 ,  -0.9848 ] |[ -0.9907 ,  -0.9844 ] |

---
class: inverse center middle

## Model Selection

---

## Comparison of the Models' Goodness of Fit

Table: Goodness-of-fit Measures of Candidate Models

|                  |     SSE|  R.sq| R.adj|     Cp|
|:-----------------|-------:|-----:|-----:|------:|
|Full Model        | 1673191| 0.934| 0.934| 14.015|
|Edited Model      | 1673191| 0.934| 0.934| 14.015|
|Transformed Model |       2| 0.995| 0.995|    Inf|

---

## Model Selection

Log Transformed Model selected
- highest adjusted R squared 
- fewest violations to assumptions

## Variable Selection

- Values of Cylinder have large p-values
- Engine.Size p-value also large

We remove Engine.Size from the model

---

## Final Model

```r
log.model = lm(log(CO2.Emissions) ~ Cylinders + Fuel.Type + log(Fuel.Consumption.mpg), data = emissions.edit)
```

<div class="datatables html-widget html-fill-item-overflow-hidden html-fill-item" id="htmlwidget-fbb0cd47497a0d2d445f" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-fbb0cd47497a0d2d445f">{"x":{"filter":"none","vertical":false,"fillContainer":false,"data":[["(Intercept)","Cylinders4","Cylinders5","Cylinders6","Cylinders8","Cylinders10","Cylinders12","Fuel.TypeE","Fuel.TypeX","Fuel.TypeZ","log(Fuel.Consumption.mpg)"],[8.893000000000001,-0.002,-0.008,0.002,0.003,0.004,0.008999999999999999,-0.492,-0.141,-0.142,-0.988],[0.006,0.002,0.004,0.002,0.002,0.003,0.002,0.002,0.001,0.001,0.001],[1545.392,-1.298,-2.291,1.258,1.337,1.218,3.491,-293.233,-108.394,-108.397,-696.1950000000001],[0,0.194,0.022,0.208,0.181,0.223,0,0,0,0,0]],"container":"<table class=\"display\">\n  <thead>\n    <tr>\n      <th> <\/th>\n      <th>Estimate<\/th>\n      <th>Std. Error<\/th>\n      <th>t value<\/th>\n      <th>Pr(>|t|)<\/th>\n    <\/tr>\n  <\/thead>\n<\/table>","options":{"pageLength":5,"columnDefs":[{"className":"dt-right","targets":[1,2,3,4]},{"orderable":false,"targets":0},{"name":" ","targets":0},{"name":"Estimate","targets":1},{"name":"Std. Error","targets":2},{"name":"t value","targets":3},{"name":"Pr(>|t|)","targets":4}],"order":[],"autoWidth":false,"orderClasses":false,"lengthMenu":[5,10,25,50,100]}},"evals":[],"jsHooks":[]}</script>

---
class: inverse center middle

## Conclusions

---

## Conclusions

- Log transformed model chosen as best model due to residual analysis and goodness of fit

- Still shows violations to assumptions

- variation in residuals
  - assumption of normality
  
- Includes outliers

Further analysis can be done through bootstrapping to eliminate some of these issues

---
class: inverse center middle

## Contributions:

Alice: Beginning - Edited Model

Angelo: Transformed Model - Conclusion