Candidate Transformations

1. Load mlb_payrolls_train.csv as mlb_train. Report the ranges of season and relative_payroll, then plot relative_payroll against winning_percentage.

mlb_train <- read_csv("data/mlb_payrolls_train.csv")

2. Fit three models of winning_percentage using relative_payroll. Save them with the names below.

   • lin_payroll_model: relative_payroll
   • log_payroll_model: log(relative_payroll)
   • quad_payroll_model: relative_payroll and relative_payroll^2

   Use I() around the squared term.

lin_payroll_model <- lm(winning_percentage ~ relative_payroll, data = mlb_train)
log_payroll_model <- lm(winning_percentage ~ log(relative_payroll), data = mlb_train)
quad_payroll_model <- lm(winning_percentage ~ ..., data = mlb_train)

3. Create payroll_grid, a tibble containing 150 evenly spaced relative_payroll values from the observed minimum to the observed maximum. Use each model to add a column of fitted winning percentages. Reshape the three columns of fitted values into a long table called payroll_grid_long.

4. Plot the observed training data and add all three curves from payroll_grid_long. Map model to color so that each curve has a different color and the plot includes a legend.

Training Fit

5. Add fitted values from the three models to mlb_train, using the same fitted value column names as in payroll_grid.

6. Define rmse(), a function with predicted and actual arguments that returns RMSE. Use it to calculate the training RMSE for all three models and put the results in a long table called payroll_train_scores.

7. Run the supplied code to add a column of random values, include it in the regression with linear payroll, and add the fitted values as lin_payroll_random. Use rmse() and reframe() to create random_training_scores, a long table comparing the training RMSE of lin_payroll and lin_payroll_random.

set.seed(2026)

mlb_train <- mlb_train %>%
  mutate(random_value = rnorm(n()))

lin_payroll_random_model <- lm(
  winning_percentage ~ relative_payroll + random_value,
  data = mlb_train
)

mlb_train <- mlb_train %>%
  mutate(lin_payroll_random = predict(lin_payroll_random_model))

random_value contains no baseball information. Its model still has slightly lower training RMSE because lm() has another coefficient to fit. A lower training RMSE alone does not identify the better model. The random predictor model is not a candidate for testing.

Adding Previous Season Winning Percentage

8. Calculate the residuals from lin_payroll_model as observed winning percentage minus lin_payroll. Plot the residuals against previous_wp, add a horizontal line at zero, and add a fitted straight line with geom_smooth(). The fitted line should slope upward.

9. Fit three multiple regression models by adding the untransformed previous_wp predictor to each payroll specification. Save them as lin_payroll_prev_wp_model, log_payroll_prev_wp_model, and quad_payroll_prev_wp_model. Add their fitted values to mlb_train as lin_payroll_prev_wp, log_payroll_prev_wp, and quad_payroll_prev_wp.

10. Display the coefficients of log_payroll_prev_wp_model. Calculate the fitted difference in current winning percentage associated with a 0.100 increase in previous_wp, holding relative payroll constant.

11. Calculate the training RMSE for the three multiple regression models. Combine these results with payroll_train_scores in a table called training_scores. Do not select a model or load the testing data.

Testing on Later Seasons

The later seasons measure whether relationships fitted through 2018 carry forward in time.

12. Load mlb_payrolls_test.csv as mlb_test. Do not refit any model. Add predictions from all six fitted models.

13. Calculate testing RMSE for all six models and combine those scores with training_scores. Make a bar chart with the training and testing bars next to each other for each model, and label every bar with its RMSE. Create best_payroll_model containing the model using only payroll with the lowest testing RMSE and best_overall_model containing the model with the lowest testing RMSE overall.