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, betterpredict 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.

The data

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

load("more/mlb11.RData")

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 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?

    The relationship appears to be positive and linear. It looks like that the number of at bats is a predictor of the number of runs scored. I would feel decently confident using a linear model for this. However, it appears that we may have some high residuals as sum of the data appears that it would produce high residuals.

    library(ggplot2)
gginit <- ggplot(mlb11,aes(x=at_bats,y=runs))
ggtype <- geom_point()
gginit + ggtype + xlab("At Bats") + ylab("Runs")

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

cor(mlb11$runs, mlb11$at_bats)
## [1] 0.610627

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.

    The relationship appears to be linear. Below I have added a linear regression line through the data. It appears that there are roughly the same amount of points above and below the line. This is a positive relationship since the number of runs increases with number of at bats. I would say it is decently strong despite the scatter.

    library(ggplot2)
gginit <- ggplot(mlb11,aes(x=at_bats,y=runs))
ggtype <- geom_point()
line <- geom_smooth(method='lm', se=FALSE)
gginit + ggtype + xlab("At Bats") + ylab("Runs") + line

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 = mlb11$at_bats, y = mlb11$runs)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##  -2789.2429       0.6305  
## 
## Sum of Squares:  123721.9

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 = mlb11$at_bats, y = mlb11$runs, showSquares = TRUE)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##  -2789.2429       0.6305  
## 
## Sum of Squares:  123721.9

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?

    I have ran this code several times, but it won’t let me click anywhere to make a line. Regardless, I do not think I would ever get smaller than the sum of squares since this optimized for all of the data in the set.

    plot_ss(x = mlb11$at_bats, y = mlb11$runs, showSquares = TRUE)

    ## Click two points to make a line.

## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##  -2789.2429       0.6305  
## 
## Sum of Squares:  123721.9

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 = mlb11)

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 mlb11 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)
## 
## Call:
## lm(formula = runs ~ at_bats, data = mlb11)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -125.58  -47.05  -16.59   54.40  176.87 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -2789.2429   853.6957  -3.267 0.002871 ** 
## at_bats         0.6305     0.1545   4.080 0.000339 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 66.47 on 28 degrees of freedom
## Multiple R-squared:  0.3729, Adjusted R-squared:  0.3505 
## F-statistic: 16.65 on 1 and 28 DF,  p-value: 0.0003388

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:

\[ \hat{y} = -2789.2429 + 0.6305 * atbats \]

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, 37.3% 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 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?

    The R-code below gives us the intercept and slope. The equation is:

    \(\hat{y} = 415.2389 + 1.8345x\)

    The slope provides the increase in \(y\) for every unit increase of \(x\). Therefore, the number of runs is increased by an average of 1.8345 for every homerun that is hit.

    r_hr <- lm(runs ~ homeruns, data=mlb11)
summary(r_hr)
    
Call:
lm(formula = runs ~ homeruns, data = mlb11)

Residuals:
    Min      1Q  Median      3Q     Max 
-91.615 -33.410   3.231  24.292 104.631 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 415.2389    41.6779   9.963 1.04e-10 ***
homeruns      1.8345     0.2677   6.854 1.90e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 51.29 on 28 degrees of freedom
Multiple R-squared:  0.6266,    Adjusted R-squared:  0.6132 
F-statistic: 46.98 on 1 and 28 DF,  p-value: 1.9e-07

Prediction and prediction errors

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

plot(mlb11$runs ~ mlb11$at_bats)
abline(m1)

The function abline plots a line based on its slope and intercept. Here, we used a shortcut by providing the model m1, which contains both parameter estimates. 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 5,578 at-bats? Is this an overestimate or an underestimate, and by how much? In other words, what is the residual for this prediction?

    The equation for the linear regression line is:

    \(\hat{y} = -2789.2429 + 0.603\times \text{at-bats}\)

    If we plug 5,578 for at bats, we will get the expected number or runs. Then, we can retrieve the real value from the data and subtract the prediction from the actual value to get the residual.

    b1 <- 0.6305
b0 <- -2789.2429
atbats <- 5579
yhat <- b0 + b1*atbats
yhat
    [1] 728.3166
    head(mlb11[order(-mlb11$at_bats),],10)
                        team runs at_bats hits homeruns bat_avg strikeouts
2         Boston Red Sox  875    5710 1600      203   0.280       1108
4     Kansas City Royals  730    5672 1560      129   0.275       1006
1          Texas Rangers  855    5659 1599      210   0.283        930
14       Cincinnati Reds  735    5612 1438      183   0.256       1250
6          New York Mets  718    5600 1477      108   0.264       1085
10        Houston Astros  615    5598 1442       95   0.258       1164
11     Baltimore Orioles  708    5585 1434      191   0.257       1120
16 Philadelphia Phillies  713    5579 1409      153   0.253       1024
3         Detroit Tigers  787    5563 1540      169   0.277       1143
20     Toronto Blue Jays  743    5559 1384      186   0.249       1184
   stolen_bases wins new_onbase new_slug new_obs
2           102   90      0.349    0.461   0.810
4           153   71      0.329    0.415   0.744
1           143   96      0.340    0.460   0.800
14           97   79      0.326    0.408   0.734
6           130   77      0.335    0.391   0.725
10          118   56      0.311    0.374   0.684
11           81   69      0.316    0.413   0.729
16           96  102      0.323    0.395   0.717
3            49   95      0.340    0.434   0.773
20          131   81      0.317    0.413   0.730
    y <- mlb11[mlb11$at_bats==atbats,c("runs")]
e <- y - yhat
e
    [1] -15.3166

    There was nothing for 5,578 at bats so I picked the closest one at 5,579 and compared them.The prediction for 5,579 at bats is 728.3166 runs. The actual was 713 runs, which makes the residual -15.3166. Therefore, this was an overestimate by 15.3166 runs.

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. at-bats. Recall that any code following a # is intended to be a comment that helps understand the code but is ignored by R.

plot(m1$residuals ~ mlb11$at_bats)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 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?

    There doesn’t appear to be any pattern. It seems that there is an equal amount of points above and below the residual line. Therefore, we can assume a linear relationship.

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

hist(m1$residuals)

or a normal probability plot of the residuals.

qqnorm(m1$residuals)
qqline(m1$residuals)  # adds diagonal line to the normal prob plot

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

    Yes. This histogram appears to be pretty symmetric and unimodal, and the normal plot appears to have the data along the line.

Constant variability:

  1. Based on the plot in (1), does the constant variability condition appear to be met?

    The residuals appear to be evenly spread above and below the line. Therefore, we can assume constant variance.

On Your Own

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 the faculty and TAs of UCLA Statistics.