Set up

Before beginning the lab, download the data file mlb23.csv from Canvas. I’ll show you how to import it and get you set up to start!

Batter up

The movie Moneyball focuses on the “quest for the secret of success in baseball”. It follows a low-budget team, the Oakland Athletics, who believed that underused statistics, such as a player’s ability to get on base, better predict the ability to score runs than typical statistics like home runs, RBIs (runs batted in), and batting average. Obtaining players who excelled in these underused statistics turned out to be much more affordable for the team.

In this lab we’ll be looking at data from all 30 Major League Baseball teams and examining the linear relationship between runs scored in a season and a number of other player statistics. Our aim will be to summarize these relationships both graphically and numerically in order to find which SINGLE variable, if any, helps us best predict a team’s runs scored in a season. Remember, you can use previous labs to remind you functions for plotting, etc.

The data

Load the data for the 2023 MLB season and take a look at the dataset; it’s called mlb23. You will need to upload the data first into posit cloud.

library(readr)
mlb23 <- read_csv("mlb23.csv")

In addition to runs scored, there are seven traditionally used variables in the data set: at-bats, hits, home runs, batting average, strikeouts, stolen bases, and wins. There are also three newer variables: on-base percentage, slugging percentage, and on-base plus slugging. If you’re not sure what all of these are, google them for a description of the different variables. For the first portion of the analysis we’ll consider the seven traditional variables. At the end of the lab, you’ll work with the newer variables on your own.

  1. What type of plot would you use to display the relationship between runs and one of the other numerical variables? Plot this relationship using the variable at_bats as the predictor. Does the relationship look linear? If you knew a team’s at_bats, would you be comfortable using a linear model to predict the number of runs?

If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient.

mlb23 %>% summarise(cor(runs, at_bats))

Sum of squared residuals

Think back to the way that we described the distribution of a single variable. Recall that we discussed characteristics such as center, spread, and shape. It’s also useful to be able to describe the relationship of two numerical variables, such as runs and at_bats above.

  1. Looking at your plot from the previous exercise, describe the relationship between these two variables. Make sure to discuss the form, direction, and strength of the relationship as well as any unusual observations.

Just as we used the mean and standard deviation to summarize a single variable, we can summarize the relationship between these two variables by finding the line that best follows their association. Use the following interactive function to select the line that you think does the best job of going through the cloud of points.

plot_ss(x = at_bats, y = runs, data=mlb23)

After running this command, you’ll be prompted to click two points on the plot to define a line. Once you’ve done that, the line you specified will be shown in black and the residuals in blue. Note that there are 30 residuals, one for each of the 30 observations. Recall that the residuals are the difference between the observed values and the values predicted by the line:

\[ e_i = y_i - \hat {y}_i \]

The most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, you can rerun the plot command and add the argument showSquares = TRUE.

plot_ss(x = at_bats, y = runs, data = mlb23, showSquares = TRUE)

Note that the output from the plot_ss function provides you with the slope and intercept of your line as well as the sum of squares.

  1. Using plot_ss, choose a line that does a good job of minimizing the sum of squares. Run the function several times. What was the smallest sum of squares that you got? How does it compare to your neighbors? (Note that you should run this code in the console and do not put it in your Markdown document. Only include text as the answer for #3.)

The linear model

It is rather cumbersome to try to get the correct least squares line, i.e. the line that minimizes the sum of squared residuals, through trial and error. Instead we can use the lm function in R to fit the linear model (a.k.a. regression line).

m1 <- lm(runs ~ at_bats, data = mlb23)

The first argument in the function lm is a formula that takes the form y ~ x. Here it can be read that we want to make a linear model of runs as a function of at_bats. The second argument specifies that R should look in the mlb23 data frame to find the runs and at_bats variables.

The output of lm is an object that contains all of the information we need about the linear model that was just fit. We can access this information using the summary function.

summary(m1)

Run this commands and let’s consider this output piece by piece. First, the formula used to describe the model is shown at the top. After the formula you find the five-number summary of the residuals. The “Coefficients” table shown next is key; its first column displays the linear model’s y-intercept and the coefficient of at_bats. With this table, we can write down the least squares regression line for the linear model:

\[ \hat{y} = -3149.4259 + 0.7111 * \mbox{at_bats} \]

One last piece of information we will discuss from the summary output is the Multiple R-squared, or more simply, \(R^2\). The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 37.91% of the variability in runs is explained by at-bats.

  1. Fit a new model that uses homeruns to predict runs. Call this model m2 so it doesn’t conflict with m1, which is used later in the lab. Using the estimates from the R output, write the equation of the regression line. What does the slope tell us in the context of the relationship between success of a team and its home runs?

Prediction and prediction errors

Let’s create a scatterplot with the least squares line laid on top.

ggplot(data = mlb23, mapping=aes(x=at_bats, y=runs)) +
  geom_point() +
  stat_smooth(method = "lm", se = FALSE)

Here we are literally adding a layer on top of our plot. stat_smooth creates the line by fitting a linear model. It can also show us the standard error se associated with our line, but we’ll suppress that for now.

This line can be used to predict \(y\) at any value of \(x\). When predictions are made for values of \(x\) that are beyond the range of the observed data, it is referred to as extrapolation and is not usually recommended. However, predictions made within the range of the data are more reliable. They’re also used to compute the residuals.

  1. If a team manager saw the least squares regression line and not the actual data, about how many runs would he or she predict for a team with 5,578 at-bats?

Model diagnostics

To assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.

Linearity: You already checked if the relationship between runs and at-bats is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. fitted (predicted) values.

ggplot(data = m1, aes(x = .fitted, y = .resid)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = "dashed") +
  xlab("Fitted values") +
  ylab("Residuals")

Notice here that our model object m1 can also serve as a data set because stored within it are the fitted values \((\hat{y})\) and the residuals. Also note that we’re getting fancy with the code here. After creating the scatterplot on the first layer (first line of code and the geom_point), we overlay a horizontal dashed line at \(y = 0\) (to help us check whether residuals are distributed around 0), and we also adjust the axis labels to be more informative.

  1. Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between runs and at-bats? In other words, if the data are linear, what should the residuals look like?


Nearly normal residuals: To check this condition, we can look at a histogram

ggplot(data = m1, aes(x = .resid)) +
  geom_histogram(bins=8) +
  xlab("Residuals")

or a normal probability plot of the residuals.

ggplot(data = m1, aes(sample = .resid)) +
  geom_qq_line() +
  geom_qq()

Note that the syntax for making a normal probability plot is a bit different than what you’re used to seeing: we set sample equal to the residuals instead of x. Recall that qq, which stands for “quantile-quantile”, is another name commonly used for normal probability plots.

  1. Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?


Constant variability:

The constant variability condition means that the variability doesn’t change based on the values of the variables. You can asses this by looking at the residuals plot, and seeing whether or not the size of the residuals has any relation to the fitted values, or whether it appears to be random.

  1. Based on the residuals vs. fitted plot, does the constant variability condition appear to be met?

On Your Own

In the following problems, the newer variables are onbase percentage, slugging percentage, and onbase pct + slugging. The traditional variables are all the others.

  1. Choose another traditional variable from mlb23 that you think might be a good predictor of runs. Produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?

  2. How does this model compare to the regression between runs and at_bats? Use the R\(^2\) values from the two model summaries to determine whether your variable seems to predict runs better than at_bats.

  3. Now that you can summarize the linear relationship between two variables, investigate the relationship between runs and one of the three newer variables. Support your conclusions using the graphical and numerical methods we’ve discussed, including examining your assumptions and writing your conclusions in paragraph form.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by Mark Hansen of UCLA Statistics and further adapted by your professor.