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 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?

A scatter plot is the best visualization for runs vs. at bats.

plot(mlb11$at_bats,mlb11$runs, xlab = "At Bats", ylab = "Runs", col = 'darkblue')

The relationship does look linear, and I would be comfortable using at bats in a linear model to predict number of runs since a hit requires an at bat to happen, and hits result in more runs than walks and hit-by-pitches.

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 relationships appears to be linear; you can draw a straight line through the data with equal numbers of data points above an below the line and no trend or pattern to the deviations from the line. The direction is positive, as at bats increases so does runs for the most part. The relationship is moderately strong there is scatter, but there is a definite slope to the trend 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?
plot_ss(mlb11$at_bats,mlb11$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

124275.9 was my best attempt further adjustment to the left or right only increased the sum of squares.

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?
m2 <- lm(mlb11$runs ~ mlb11$homeruns)
summary(m2)
## 
## Call:
## lm(formula = mlb11$runs ~ mlb11$homeruns)
## 
## 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 ***
## mlb11$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

\[ \hat{y} = 415.2389 + 1.8345*homeruns \]

The slope tells us that the more home runs a team has the more runs they have. Runs are the point system in baseball so the more runs the more successful the team.

Prediction and prediction errors

Let’s create a scatter plot 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?

\[ \hat{y} = -2789.2429 + 0.6305 * atbats \\ \hat{y} = -2789.2429 + 0.6305 * 5578 \\ \hat{y} = 727.6861 \]

runs_df <- data.frame(mlb11$runs,mlb11$at_bats)
runs_df
##    mlb11.runs mlb11.at_bats
## 1         855          5659
## 2         875          5710
## 3         787          5563
## 4         730          5672
## 5         762          5532
## 6         718          5600
## 7         867          5518
## 8         721          5447
## 9         735          5544
## 10        615          5598
## 11        708          5585
## 12        644          5436
## 13        654          5549
## 14        735          5612
## 15        667          5513
## 16        713          5579
## 17        654          5502
## 18        704          5509
## 19        731          5421
## 20        743          5559
## 21        619          5487
## 22        625          5508
## 23        610          5421
## 24        645          5452
## 25        707          5436
## 26        641          5528
## 27        624          5441
## 28        570          5486
## 29        593          5417
## 30        556          5421

From the data we see that 5579 at bats resulted in 713 runs. We would expect 0.6305 fewer runs for 5578 for 712.3695. That means our model over estimated with a residual of

\[ \hat{e} = y_i - \hat{y} \\ \hat{e} = 712.3695 - 727.6861 \\ \hat{e} =-15.3166 \]

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 scatter plot. 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?

No, I do not see a pattern in the residuals. The data seems fairly evenly distributed above and below the trend line at 12 above 18 below. This can come about by chance:

dbinom(18,30,0.5)
## [1] 0.08055309

This does not seem to significantly indicate that something other than chance is responsible for the split, and that the data is linear.

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?

There does seem to be a slight right skew to the data, but not enough to invalidate the model by itself. The QQ Normal plot seems to follow the theoretical values well enough. I’d say that the normal residual condition have been met.

Constant variability:

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

When the residuals are plotted they randomly vary form -100 to 100 with the exception of 1 point at 150. This seems like fairly constant variance.

On Your Own

Stolen Bases. Putting runners in a better position may increase runs scored.

plot(mlb11$stolen_bases,mlb11$runs, xlab = 'Stolen Bases', ylab = 'Runs' ,col = 'steelblue3')

There does not seem to be a linear relationship.

m3 <- lm(mlb11$runs ~ mlb11$stolen_bases)
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
summary(m3)
## 
## Call:
## lm(formula = mlb11$runs ~ mlb11$stolen_bases)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -139.94  -62.87   10.01   38.54  182.49 
## 
## Coefficients:
##                    Estimate Std. Error t value Pr(>|t|)    
## (Intercept)        677.3074    58.9751  11.485 4.17e-12 ***
## mlb11$stolen_bases   0.1491     0.5211   0.286    0.777    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 83.82 on 28 degrees of freedom
## Multiple R-squared:  0.002914,   Adjusted R-squared:  -0.0327 
## F-statistic: 0.08183 on 1 and 28 DF,  p-value: 0.7769

The R\(^2\) for at bats is 0.3729. Meaning approximately 37% of the trend is caused by the predicting variable. The R\(^2\) for stolen_bases is 0.002914. Meaning approximately 0.3% of the trend is caused by the predicting variable. At bats is a better predictor of runs than stolen bases.

m4 <- lm(mlb11$runs ~  mlb11$bat_avg )
summary(m4)
## 
## Call:
## lm(formula = mlb11$runs ~ mlb11$bat_avg)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -94.676 -26.303  -5.496  28.482 131.113 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     -642.8      183.1  -3.511  0.00153 ** 
## mlb11$bat_avg   5242.2      717.3   7.308 5.88e-08 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 49.23 on 28 degrees of freedom
## Multiple R-squared:  0.6561, Adjusted R-squared:  0.6438 
## F-statistic: 53.41 on 1 and 28 DF,  p-value: 5.877e-08
plot(mlb11$bat_avg, mlb11$runs, xlab = 'Batting Ave', ylab= 'Runs', col = 'darkblue')
abline(m4, col = 'red')

hist(resid(m4))

plot(mlb11$bat_avg, resid(m4))
abline(h = 0, lty = 3)

qqnorm(resid(m4))
qqline(resid(m4))

Batting average, the number of hits / at bats is the best predictor with R\(^2\) of 0.6561. The conditions for validity also seem to be met.

m5 <- lm(mlb11$runs ~  mlb11$new_obs)
summary(m5)
## 
## Call:
## lm(formula = mlb11$runs ~ mlb11$new_obs)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -43.456 -13.690   1.165  13.935  41.156 
## 
## Coefficients:
##               Estimate Std. Error t value Pr(>|t|)    
## (Intercept)    -686.61      68.93  -9.962 1.05e-10 ***
## mlb11$new_obs  1919.36      95.70  20.057  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 21.41 on 28 degrees of freedom
## Multiple R-squared:  0.9349, Adjusted R-squared:  0.9326 
## F-statistic: 402.3 on 1 and 28 DF,  p-value: < 2.2e-16
plot(mlb11$new_obs, mlb11$runs, xlab = 'On Base + Slugging', ylab= 'Runs', col = 'darkblue')
abline(m5, col = 'red')

On Base + Slugging, a combination of on base percentage and slugging average is the best predictor of Runs with an impressive R\(^2\) of 0.9349. Note that slugging average had R\(^2\) of 0.8969 and on base percentage had R\(^2\) of 0.8491.

All three new stats are better than the traditional stats which had R\(^2\) ~ 0.6.

hist(resid(m5))

plot(mlb11$new_obs, resid(m5))
abline(h = 0, lty = 3)

qqnorm(resid(m5))
qqline(resid(m5))

The residuals are randomly scattered about 0, suggesting constant variance and linearity and normally distributed. On base % + slugging % is a valid predictor of Runs.

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.