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.
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.
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
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.
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:
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.
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?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.
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).
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
Constant variability:
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?
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?
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).
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?
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.