Factors Influencing Golf Earnings

class: center, middle, inverse, title-slide

.title[
# Factors Influencing Golf Earnings
]
.subtitle[
## Linear Regression
]
.author[
### Tyler Battaglini & Ryan Lebo
]
.date[
### 2025-02-16
]

---

<h1 align="center"> Table of Contents</h1>
<BR>

.pull-left[
- Introduction
- Variables
- Research Question
- Exploratory Data Analysis
- Linear Model
- Log Model
- Bootstrapping
- Model Selection
- Conclusion

]

---

<h1 align = "center"> Introduction <font color="orange"></font></h1>
<BR>
.pull-left[
- PGA 2004 data set (196 participants)
- What is the PGA?
- Data set provides (earnings and player stats)

]

---
<h1 align = "center"> Variables <font color="orange"></font></h1>
<BR>

.pull-left[
- Name
- Age
- Avg Drive
- Driving Accuracy
- Greens in Regulation
- Avg Number of Putts

]

.pull-right[
- Save Percentage
- Money Rank
- Number of Events
- Total Winnings
- Average Winnings

]

---
<h1 align = "center"> Research Question <font color="orange"></font></h1>
<BR>

- What variables affect the players winnings of this given season?
- Looking at average drive vs earnings

---
## Exploratory Data Analysis

- Check for high correlation
- Take out missing observations
- Remove some variables

---
<h1 align = "center"> Linear Model <font color="orange"></font></h1>
<BR>

- Non-normal distribution
- Several Outliers
- Non-constant variance

---

<h1 align = "center"> Linear Model Cont. <font color="orange"></font></h1>
<BR>

.pull-left[

- All below 5
- Little to no multicollinearity

]

---
##

<div style="display: flex; justify-content: center; align-items: center; height: 80vh; font-size: 50px;">
  Box-Cox Transformation
</div>

---
## Box-Cox Transformation

- All the lambda values are close to 0
- Proceed with log transformation

---

## Log Transformation Model

- Response variable – log (Average Winnings)
- Explanatory Variables – Average Drive, Greens on Regulation, Save Percentage, Number of Events, and Age Above 30

|                 | Estimate| Std. Error| t value| Pr(>&#124;t&#124;)|
|:----------------|--------:|----------:|-------:|------------------:|
|(Intercept)      |   -5.451|      2.383|  -2.287|              0.023|
|Average_Drive    |    0.006|      0.007|   0.819|              0.414|
|Greens_on_reg    |    0.192|      0.019|   9.875|              0.000|
|Save_Percent     |    0.056|      0.010|   5.562|              0.000|
|Number_events    |   -0.045|      0.012|  -3.857|              0.000|
|Age_Above_30TRUE |    0.033|      0.139|   0.240|              0.811|

---

## Goodness of Fit Measures

- Improvement in constant variance
- Improvement in Normality

---

## Comparison of Models

- Log model outperforms linear model 
- SSE signifies better fit
- R squared and R adjusted better in log model

|             |     SSE|  R.sq| R.adj| Cp|      AIC|     SBC|   PRESS|
|:------------|-------:|-----:|-----:|--:|--------:|-------:|-------:|
|full.model   | 123.038| 0.335| 0.316|  6|  -64.865| -45.510| 135.110|
|log.winnings | 100.246| 0.455| 0.440|  6| -102.970| -83.615| 107.534|

---

## Comparison of Models Cont.

- Vast improvement in Log transformation model
- Normality?

---

## Comparison of Model Cont.

- Residuals vs. Fitted improvement in log transformation
- Can assume constant variance

<img src="Presentation-1-Capstone-_files/figure-html/unnamed-chunk-12-1.png" width="100%" />
---
##

<div style="display: flex; justify-content: center; align-items: center; height: 80vh; font-size: 50px;">
  Bootstrapping
</div>
---
## Bootstrapping

<li>Have not assumed normality in our QQ plot.</li>  
  
  <li>Uses a nonparametric model for comparison.</li>  
  
  <li>Estimating confidence intervals.</li>

</ul>

</div>

---
## Boostrapping Cont. with Cases

- The red line in the curves is used to show the p-values and uses the estimated regression coefficients and their corresponding standard error in the output of the regression procedure
- The blue curve is used to used to show the bootstrap CI which is based on a non-parametric data-driven estimate of the density of bootstrap sampling distribution
- All appear to be normal

<img src="Presentation-1-Capstone-_files/figure-html/unnamed-chunk-13-1.png" width="100%" />
---
## Bootstrapping Cont. with Residuals

<img src="Presentation-1-Capstone-_files/figure-html/unnamed-chunk-15-1.png" width="100%" />
---
## Bootstrapping Cont. CIs

- CIs match our p-values
- Violation in two variables

Table: Final Combined Inferential Statistics: Coefficients, p-values, and Bootstrap CIs

|Coefficients |95% CI (Bootstrap t)  |95% CI (Bootstrap r)    |p-values |
|:------------|:---------------------|:-----------------------|:--------|
|-5.4513      |[ -10.21 ,  -0.763 ]  |[ -10.3252 ,  -0.6365 ] |0.0233   |
|0.0058       |[ -0.0067 ,  0.0186 ] |[ -0.0075 ,  0.0198 ]   |0.4139   |
|0.0563       |[ 0.148 ,  0.2356 ]   |[ 0.1546 ,  0.2302 ]    |0.0000   |
|-0.0450      |[ 0.0363 ,  0.0748 ]  |[ 0.0368 ,  0.0747 ]    |0.0002   |
|0.0334       |[ -0.068 ,  -0.0206 ] |[ -0.0676 ,  -0.0226 ]  |0.8108   |
|-5.4513      |[ -0.2485 ,  0.3697 ] |[ -0.2338 ,  0.3216 ]   |0.0233   |
---
## Model Selection

<li><u>Log Transformation</u>  
      <br> - Positives: Can assume normality and constant variance, best R-squared and adjusted R-squared values.  
      <br> - Negatives: Age and Average Drive are insignificant.</li>
  
  <li><u>Linear Model</u>  
      <br> - Positives: Only Age is insignificant, and the model is simple.  
      <br> - Negatives: Cannot assume constant variance or normality.</li>

</ul>

</div>

---
##

<div style="display: flex; justify-content: center; align-items: center; height: 80vh; font-size: 50px;">
  Conclusion
</div>

---
## Conclusion

<ul style="margin-top: 15px; margin-bottom: 15px;">
  
  <li>Greens on Regulation is the biggest indicator of increase in Winnings 21.18%.</li>
  
  <li>When holding all other variables constant, an increase of one unit in Average Drive leads to a 0.58% increase in Average Winnings.</li>
  
  <li>Short game (e.g., greens on regulation or save percent) has a larger impact on winnings compared to the long game. <br>
      - Short game impact:5.8% <br>
      - Long game (Average Drive) impact: 0.58%</li>
  
  <li>Number of Events decreases Average Winnings by 4.4%.</li>

</ul>

</div>

---
## Limitations

<li>Average Drive and Age are insignificant values.</li>  
  
  <li>Log transformation is sensitive to outliers and could amplify small values.</li>  
  
  <li>Assumes a linear relationship.</li>  
  
  <li>Factors such as injury, weather, start time, and mental focus are not included in this dataset.</li>

</ul>

</div>

---
##

<div style="display: flex; justify-content: center; align-items: center; height: 80vh; font-size: 50px;">
  Questions?
</div>

---
## Works Cited

<li>Datasets. (n.d.). <a href="https://users.stat.ufl.edu/~winner/datasets.html">https://users.stat.ufl.edu/~winner/datasets.html</a></li>

</ul>

</div>

---
##

<div style="display: flex; justify-content: center; align-items: center; height: 100%; font-size: 50px;">
  Thank You!
</div>

---
## Slide Contributors

<div style="font-size: 40px; line-height: 2;">
  <ul>
    <li>Ryan Lebo did slides from the Introduction to the Linear Regression Model</li>
    <li>Tyler Battaglini did the Box-Cox Transformation to the Conclusion</li>
  </ul>
</div>