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 variable, if any, helps us best predict a team’s runs scored in a season.

Getting Started

The data

Let’s load up the data for the 2011 season.

use "mlb11.dta"

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. 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 three 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.

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. Look at the following plots. Which of the plots do you think does the best job of going through the cloud of points?

Plot A

Plot B

Plot C

In each plot, the line specified is 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, look at the plot below:

  1. Look at the three plots A-C above. Which do you think will give the smallest sum of squared residuals?

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 regress function in Stata to fit the linear model (a.k.a. regression line).

regress runs at_bats

The first argument in the function regress is the response variable and the second is the predictor variable.

The output of regress contains all of the information we need about the linear model that was just fit.

Let’s consider this output piece by piece. The box in the top left shows the analysis of variance table with the sum of squares. We will focus on the second table, the “Coefficients” table. Its first column, labelled “Coef.” displays the linear model’s coefficient of at_bats and the y-intercept (_cons). With this table, we can write down the least squares regression line for the linear model:

\[ \hat{y} = -2789.243 + 0.631 \times at\_bats \]

One last piece of information we will discuss from the summary output is the R-squared, or more simply, \(R^2\). The \(R^2\) is shown in the table on the right and its value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 37.3% of the variability in runs is explained by at-bats.

  1. Fit a new model that uses homeruns to predict runs. Using the estimates from the Stata 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 for the regression of runs on at_bats laid on top.

twoway scatter runs at_bats || lfit runs at_bats

Notice that the legend now includes information about the fitted values from the linear regression model, shown as a red line. The blue points still represent the observed runs.

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, how many runs would he or she predict for a team with 5,579 at-bats? Is this an overestimate or an underestimate, and by how much? In other words, what is the residual for this prediction?

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.

regress runs at_bats
rvfplot, yline(0)

After running the regression model in the first line, we use the command rvfplot to plot the residuals against the fitted values. The option yline(0) adds a horizontal dashed line at \(y = 0\) (to help us check whether residuals are distributed around 0).

  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?


Nearly normal residuals: To check this condition, we can look at a histogram or a normal probability plot of the residuals. First, we need to compute the residuals. We can do this with the command predict following the regression of runs on at_bats:

regress runs at_bats
predict runresid, residual

Now, we have a new variable runresid which are the residuals. We can use this new variable to display a histogram of the residuals:

histogram runresid

or a normal probability plot of the residuals.

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


Constant variability:

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

More Practice

  1. Choose another one of the seven traditional variables from mlb11 besides at_bats 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 relationship compare to the relationship between runs and at_bats? Use the \(R^2\) values from the two model summaries to compare. Does your variable seem to predict runs better than at_bats? How can you tell?

  3. Now that you can summarize the linear relationship between two variables, investigate the relationships between runs and each of the other five traditional variables. Which variable best predicts runs? Support your conclusion using the graphical and numerical methods we’ve discussed (for the sake of conciseness, only include output for the best variable, not all five).

  4. Now examine the three newer variables. These are the statistics used by the central character in Moneyball to predict a team’s success. In general, are they more or less effective at predicting runs that the old variables? Explain using appropriate graphical and numerical evidence. Of all ten variables we’ve analyzed, which seems to be the best predictor of runs? Using the limited (or not so limited) information you know about these baseball statistics, does your result make sense?

  5. Check the model diagnostics for the regression model with the variable you decided was the best predictor for runs.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for Stata by Jenna R Krall and adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by the faculty and TAs of UCLA Statistics.