Learning Stat Final Project

Analyzing and Interpreting Marathon Results A Deeper Jog

Camille Dominguez, Christina Bui, Youqi Qi

May 2nd, 2024

1. Preliminary Work

Boston Marathon Finishers (2017 data)

• kaggle.com
• "Finishers Boston Marathon 2015, 2016 & 2017"
• This data has the names, times and general demographics of the finishers

• Focus on 2017 data (most relevant)
• Why the Boston Marathon?
• Oldest marathon in US
• You have to qualify to participate!
• Predictors: Age group, countries, gender

'Your content is liked, but it engages much more if it is interactive'

3. Goals

Description of Data

Goals of Analysis

• Develop a model to predict finishing times based on demographic information and past running records

• Cateogrize certain groups and conduct analysis on specific factors

• Use model to predict current or future groups and their finishing times

• 26,410 observations, 8 variables
• Age, gender, City, Country, Half time, 40k time, Pace, Official Time
• Methodologies:
• Data Preprocessing
• Explanatory Data Analysis
• Splitting Data
• Model Building & Diagnostics
• Variable Selection
• Cross-Validation
• Model Evaluation

2. Graphs

Explanatory Data Analysis: Plots and Histograms

• Code: hist(marathon$Age, main="Histogram of Ages")

• Skewed right, center approx. ~45 years

3. Graphs Cont.

Histogram Half Time

• Code: hist(marathon$`40K`, main="Histogram of 40K Times", xlab="40K Time (seconds)", col="red")
• Skewed right, center approx ~ 12500 seconds

Histogram 40K Times

• Code: hist(marathon$Half, main="Histogram of Half Marathon Times", xlab="Half Marathon Time (seconds)", col="green")
• Skewed right, center approx ~ 6250

4. Graphs cont.

Histogram of Pace

• Code: hist(marathon$Pace, main="Histogram of Pace", xlab="Pace (seconds per km)", col="purple")
• Skewed right, center approx ~ 500 seconds per km

Histogram of Official Times

Code: hist(marathon$Official, main="Histogram of Official Times", xlab="Official Time (seconds)", col="orange") Skewed right, center approx ~ 13500 seconds

Pairwise Scatter Plot

• Correlations between finishers' running times
• Age vs. Offical
• Half vs. 40k
• Pace vs. Official Time

Scatter Plot of Marathon Performance by Gender

• Visualization of data
• Seeing the Male (blue) line being positioned slightly higher than the Female (pink) line.
• Overlap and Spread
• Significant overlap
• Blue points being more distributed
• Linear Correlation
• Faster Half times corresponding well to fast Offical times
• Interpretation
• Althetic preparation, biologies, training differences

Splitting Data & Model Building

• Split the data into train and test set
• Used random sampling seed method
• Sample 80% of the dataset for the training set, 20% to the test set
• Performed SLR (fitted using training data)
• Interpretation of results:
• Age, Gender, Location vs '40K' & 'Pace'
• Model fit: Adj. R-squared & R-Sq of 1
• Indicators of overfitting

Model Building Cont.

• Data Preparation:
• Prepare model matrices (x.train and x.test) and response variable (y.train) for training and test data.
• Ridge Regression:
• Fit a Ridge regression model (glmnet) using a sequence of lambda values.
• Perform cross-validation (cv.glmnet) to select the best lambda (lambda.min).
• Refit the Ridge model with the selected lambda (final.ridge).
• Use the final Ridge model to predict on the test data (ridge.predict).
• Lasso Regression:
• Fit a Lasso regression model (glmnet) using a sequence of lambda values.
• Perform cross-validation (cv.glmnet) to select the best lambda (lambda.min).
• Refit the Lasso model with the selected lambda (final.lasso).
• Use the final Lasso model to predict on the test data (lasso.predict)

Model Dianostics

• Ridge Regression
• Utilized optimal lambda value (best.lambda.ridge).
• Achieved a minimum test error of 41070.41.
• Lasso Regression
• Utilized optimal lambda value (best.lambda.lasso).
• Achieved a significantly lower minimum test error of 5687.40.

• Key Findings
• Ridge: MSE= 41070.41
• Lasso: MSE = 5687.40
• Interpretation
• Lasso outperformed Ridge with a notably lower test error.
• Lasso's feature selection likely contributed to enhanced model precision.

Backward Elimination

Code/Stats

We explored three variable selection methods: Backward Elimination, Ridge Regression, and Lasso Regression Method: Starting with a full model containing all predictors, we repeatedly removed the least significant variables until the models performance improved or stabilized The final model retained the following predictors: Gender (M) City (Top City) 40K Pace

```{r, include = TRUE} # Backward Elimination for the linear regression model # Fit the full model with all predictors full.model = lm(Official ~ ., data = train) # Perform backward elimination reduced.model = step(full.model, direction = "backward") # Check the summary of the reduced model to see which variables were kept summary(reduced.model) F Stat: 5.912e+08. p-value: 2.2e-16

Lasso Regression

Ridge Regression

Method: Ridge Regression adds a penalty term to the coefficients to prevent overfitting by shrinking them towards zero Coefficients for Ridge: Age: 1.5672 GenderM: 40.9327 CityTopCity: -13.437 CountryUSA: 16.5415 Half: 0.4066 40K: 0.4057 Pace: 10.708

Method: Lasso Regression adds an L1 penalty term, creating sparcity in the coefficients and effectively selecting variables Coefficients for Lasso: 40K: 0.109 Pace: 22.741

Interpretation of Results

Effect of Predictors:

• Backward Elimination (Linear Regression):
• The reduced model includes variables such as GenderM, CityTopCity, 40K, and Pace indicating that these factors have an impact on the predicted finishing times
• Ridge Regression:
• Notably the coefficients for GenderM, CityTopCity, CountryUSA, 40K, and Pace demonstrate their influence on predicted marathon finishing times
• Lasso Regression:
• The coefficients retained by Lasso, like 40K and Pace suggest their significant impact on the predicted marathon finishing times

Model Selection

• The basic linear regression model provides a straightforward understanding of predictor effects but can suffer from overfitiing if too many variables are included
• Ridge regression generally outperforms the basic linear model by penalizing large coefficients, which helps prevent overfitting and improves generalization to new data
• Lasso regression uses variable selection to enhance prediction accuracy and reduce model complexity by selecting the most relevant predictors while setting less important coefficients to zero
• Therefore, both ridge and lasso regression techniques offer improvements over the basic linear model in terms of balancing complexity and accuracy.

Possible Modifications and Additional Steps

• Future work could focus on refining the predictive models by incorporating additional features such as weather conditions and course elevation

• Weather plays a significant role in outdoor events like races

• Oragnizers can collect historical weather data for the race location and time of year and can include this information as features in their predictive models
• By analyzing the elevation data along the race route, organizers can obtain metrics like total elevation gain, and the distribution of uphill and downhill sections
• Including these features in the model allows for better understanding of how course elevation influences finishing times.

Thank you!