Week 2 Assignment: Presentation

.title[
# Week 2 Assignment: Presentation
]
.subtitle[
## MLR of CO2 Emissions for Vehicles
]
.author[
### Alice Xiang & Angelo Saporito
]
.date[
### 2024-02-20
]

---

<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 
]
---
class: inverse center middle

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

---
class: inverse center middle

## 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" id="htmlwidget-6631d9dc57969ac8dfad" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-6631d9dc57969ac8dfad">{"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

---

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> GVIF </th>
   <th style="text-align:right;"> Df </th>
   <th style="text-align:right;"> GVIF^(1/(2*Df)) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Engine.Size </td>
   <td style="text-align:right;"> 11.67 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 3.42 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders </td>
   <td style="text-align:right;"> 14.40 </td>
   <td style="text-align:right;"> 7 </td>
   <td style="text-align:right;"> 1.21 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.Type </td>
   <td style="text-align:right;"> 2.48 </td>
   <td style="text-align:right;"> 4 </td>
   <td style="text-align:right;"> 1.12 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.Consumption.City </td>
   <td style="text-align:right;"> 2069.97 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 45.50 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.Consumption.Hwy </td>
   <td style="text-align:right;"> 568.00 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 23.83 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.Consumption.Combined </td>
   <td style="text-align:right;"> 4651.99 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 68.21 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.Consumption.mpg </td>
   <td style="text-align:right;"> 10.26 </td>
   <td style="text-align:right;"> 1 </td>
   <td style="text-align:right;"> 3.20 </td>
  </tr>
</tbody>
</table>
]

## 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" id="htmlwidget-46d0ecfeabc4fec384f4" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-46d0ecfeabc4fec384f4">{"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,-8.934,-13.777,-5.948,11.741,34.316,44.381,145.252,-77.751,-100.318,-25.916,-29.994,-6.053],[2.777,0.432,1.601,3.383,1.806,2.258,3.334,2.867,9.265,1.466,15.114,1.177,1.193,0.042],[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

## 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 
]
---
class: inverse center middle

## Transformed Model + Discussion

---

## Box-Cox Transformation

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

<img src="Untitled_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" id="htmlwidget-6b73aa2760e6c73a7924" style="width:100%;height:auto;"></div>
<script type="application/json" data-for="htmlwidget-6b73aa2760e6c73a7924">{"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,-0.009,0.002,0.001,0.003,0.007,0.039,-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,-2.359,0.857,0.565,0.699,2.181,3.837,-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="Untitled_files/figure-html/unnamed-chunk-17-1.png" width="70%" height="70%" style="display: block; margin: auto;" />
]

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

.pull-center[
<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> Estimate </th>
   <th style="text-align:left;"> Std. Error </th>
   <th style="text-align:left;"> t value </th>
   <th style="text-align:left;"> Pr(&gt;|t|) </th>
   <th style="text-align:left;"> btc.ci.95 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:left;"> 8.891 </td>
   <td style="text-align:left;"> 0.006 </td>
   <td style="text-align:left;"> 1414.010 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ 8.8792 ,  8.9023 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Engine.Size </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> 0.798 </td>
   <td style="text-align:left;"> 0.425 </td>
   <td style="text-align:left;"> [ -5e-04 ,  0.0013 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders4 </td>
   <td style="text-align:left;"> -0.002 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> -1.388 </td>
   <td style="text-align:left;"> 0.165 </td>
   <td style="text-align:left;"> [ -0.0051 ,  1e-04 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders5 </td>
   <td style="text-align:left;"> -0.009 </td>
   <td style="text-align:left;"> 0.004 </td>
   <td style="text-align:left;"> -2.359 </td>
   <td style="text-align:left;"> 0.018 </td>
   <td style="text-align:left;"> [ -0.014 ,  -0.0037 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders6 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.856 </td>
   <td style="text-align:left;"> 0.392 </td>
   <td style="text-align:left;"> [ -0.0012 ,  0.0045 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders8 </td>
   <td style="text-align:left;"> 0.001 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.564 </td>
   <td style="text-align:left;"> 0.572 </td>
   <td style="text-align:left;"> [ -0.0026 ,  0.0053 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders10 </td>
   <td style="text-align:left;"> 0.003 </td>
   <td style="text-align:left;"> 0.004 </td>
   <td style="text-align:left;"> 0.699 </td>
   <td style="text-align:left;"> 0.485 </td>
   <td style="text-align:left;"> [ -0.004 ,  0.0089 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders12 </td>
   <td style="text-align:left;"> 0.007 </td>
   <td style="text-align:left;"> 0.003 </td>
   <td style="text-align:left;"> 2.181 </td>
   <td style="text-align:left;"> 0.029 </td>
   <td style="text-align:left;"> [ 0.0012 ,  0.0127 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.TypeE </td>
   <td style="text-align:left;"> -0.492 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> -292.677 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.4963 ,  -0.4878 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.TypeX </td>
   <td style="text-align:left;"> -0.141 </td>
   <td style="text-align:left;"> 0.001 </td>
   <td style="text-align:left;"> -108.385 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.1425 ,  -0.1391 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.TypeZ </td>
   <td style="text-align:left;"> -0.142 </td>
   <td style="text-align:left;"> 0.001 </td>
   <td style="text-align:left;"> -108.189 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.1439 ,  -0.1406 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> log(Fuel.Consumption.mpg) </td>
   <td style="text-align:left;"> -0.988 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> -654.666 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.9904 ,  -0.9848 ] </td>
  </tr>
</tbody>
</table>
]
---
<h2 align="center">Bootstrapping Residuals</h2>

<img src="Untitled_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>

<table>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:left;"> Estimate </th>
   <th style="text-align:left;"> Std. Error </th>
   <th style="text-align:left;"> Pr(&gt;|t|) </th>
   <th style="text-align:left;"> btc.ci.95 </th>
   <th style="text-align:left;"> btr.ci.95 </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> (Intercept) </td>
   <td style="text-align:left;"> 8.891 </td>
   <td style="text-align:left;"> 0.006 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ 8.8792 ,  8.9023 ] </td>
   <td style="text-align:left;"> [ 8.8788 ,  8.9036 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Engine.Size </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> 0.425 </td>
   <td style="text-align:left;"> [ -5e-04 ,  0.0013 ] </td>
   <td style="text-align:left;"> [ -6e-04 ,  0.0013 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders4 </td>
   <td style="text-align:left;"> -0.002 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.165 </td>
   <td style="text-align:left;"> [ -0.0051 ,  1e-04 ] </td>
   <td style="text-align:left;"> [ -0.0056 ,  0.0013 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders5 </td>
   <td style="text-align:left;"> -0.009 </td>
   <td style="text-align:left;"> 0.004 </td>
   <td style="text-align:left;"> 0.018 </td>
   <td style="text-align:left;"> [ -0.014 ,  -0.0037 ] </td>
   <td style="text-align:left;"> [ -0.0158 ,  -0.0016 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders6 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.392 </td>
   <td style="text-align:left;"> [ -0.0012 ,  0.0045 ] </td>
   <td style="text-align:left;"> [ -0.002 ,  0.0058 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders8 </td>
   <td style="text-align:left;"> 0.001 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.572 </td>
   <td style="text-align:left;"> [ -0.0026 ,  0.0053 ] </td>
   <td style="text-align:left;"> [ -0.0033 ,  0.0065 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders10 </td>
   <td style="text-align:left;"> 0.003 </td>
   <td style="text-align:left;"> 0.004 </td>
   <td style="text-align:left;"> 0.485 </td>
   <td style="text-align:left;"> [ -0.004 ,  0.0089 ] </td>
   <td style="text-align:left;"> [ -0.0045 ,  0.0098 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Cylinders12 </td>
   <td style="text-align:left;"> 0.007 </td>
   <td style="text-align:left;"> 0.003 </td>
   <td style="text-align:left;"> 0.029 </td>
   <td style="text-align:left;"> [ 0.0012 ,  0.0127 ] </td>
   <td style="text-align:left;"> [ 9e-04 ,  0.0132 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.TypeE </td>
   <td style="text-align:left;"> -0.492 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.4963 ,  -0.4878 ] </td>
   <td style="text-align:left;"> [ -0.4951 ,  -0.4887 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.TypeX </td>
   <td style="text-align:left;"> -0.141 </td>
   <td style="text-align:left;"> 0.001 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.1425 ,  -0.1391 ] </td>
   <td style="text-align:left;"> [ -0.1433 ,  -0.1381 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Fuel.TypeZ </td>
   <td style="text-align:left;"> -0.142 </td>
   <td style="text-align:left;"> 0.001 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.1439 ,  -0.1406 ] </td>
   <td style="text-align:left;"> [ -0.1448 ,  -0.1397 ] </td>
  </tr>
  <tr>
   <td style="text-align:left;"> log(Fuel.Consumption.mpg) </td>
   <td style="text-align:left;"> -0.988 </td>
   <td style="text-align:left;"> 0.002 </td>
   <td style="text-align:left;"> 0.000 </td>
   <td style="text-align:left;"> [ -0.9904 ,  -0.9848 ] </td>
   <td style="text-align:left;"> [ -0.9908 ,  -0.9847 ] </td>
  </tr>
</tbody>
</table>

---
class: inverse center middle

## Model Selection

---

## Comparison of the Models' Goodness of Fit

<table>
<caption>Goodness-of-fit Measures of Candidate Models</caption>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:right;"> SSE </th>
   <th style="text-align:right;"> R.sq </th>
   <th style="text-align:right;"> R.adj </th>
   <th style="text-align:right;"> Cp </th>
   <th style="text-align:right;"> AIC </th>
   <th style="text-align:right;"> SBC </th>
   <th style="text-align:right;"> PRESS </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Full Model </td>
   <td style="text-align:right;"> 1673191 </td>
   <td style="text-align:right;"> 0.934 </td>
   <td style="text-align:right;"> 0.934 </td>
   <td style="text-align:right;"> 14.015 </td>
   <td style="text-align:right;"> 40077 </td>
   <td style="text-align:right;"> 40174 </td>
   <td style="text-align:right;"> 2293146 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Edited Model </td>
   <td style="text-align:right;"> 1673191 </td>
   <td style="text-align:right;"> 0.934 </td>
   <td style="text-align:right;"> 0.934 </td>
   <td style="text-align:right;"> 14.015 </td>
   <td style="text-align:right;"> 40077 </td>
   <td style="text-align:right;"> 40174 </td>
   <td style="text-align:right;"> 2293146 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Transformed Model </td>
   <td style="text-align:right;"> 2 </td>
   <td style="text-align:right;"> 0.995 </td>
   <td style="text-align:right;"> 0.995 </td>
   <td style="text-align:right;"> Inf </td>
   <td style="text-align:right;"> -60600 </td>
   <td style="text-align:right;"> -60517 </td>
   <td style="text-align:right;"> 2 </td>
  </tr>
</tbody>
</table>

---

## 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)
```

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---
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: First half of the presentation up until log-transformed model

Angelo: Second half (log-transform, bootstrap, model selection)