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?

We would use a dot plot, with at_bats as the x-axis and runs as the y-axis.

plot(mlb11$at_bats,mlb11$runs)

The relationship looks like there is some correlation, but not very strong.

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 would be best described as positive, yet weak correlation. There are some outliers in the center/top of the graph as well as bottom/right which throw off the correlation measurements.

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?

The smallest I got was: 166077

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?
plot(mlb11$homeruns, mlb11$runs)

hr1 <- lm(runs ~ homeruns, data = mlb11)
summary(hr1)
## 
## 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?

There isn’t an exact data point for 5,578 at bats, so I went with the next best, 5579. I assume this is what you were looking for, since we wouldn’t be able to tell the direction of the residual using R-squared/etc.

pred <- -2789.2429 + 0.6305*5579

#Predicted valued based on SS line

pred
## [1] 728.3166
actual <- mlb11$runs[mlb11$at_bats == 5579]

#Actual value in dataset:

actual
## [1] 713
residual <- actual - pred

#Difference between pred and actual is the residual:

residual
## [1] -15.3166

The estimate is an overestimate (higher than the actual value).

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 seem to be an apparently pattern in the residual plot. The residuals seem to be randomly scattered in the positive and negative. This is a good indication about a linearity of relationship between runs and at-bats.

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?

It does, both the histogram and normal probability plot look to be roughly normal.

Constant variability:

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

Yes, the variability doesn’t seem to change as at_bats increase or decrease, so the variability is apparently constant.
* * *

On Your Own

“hits” seems like a logical place to start:

hm <- lm(runs ~ hits, data = mlb11)
plot(mlb11$hits, mlb11$runs)
abline(hm)

Looks pretty linear to me!

R-squared for runs ~ at_bats: 0.3729 R-squared for runs ~ hits: 0.6419

hits does seem to predict r-squared better. R-squared is a good indicator of how well the line fits reality, as it explains the % of variability explained by the predictor line. It also happens to be the square of the correlation coefficient, which attempts to give a value to how closely two variable are correlated.

#Calculate R-squared for each of the traditional variables:
cor(mlb11$at_bats, mlb11$runs)^2
## [1] 0.3728654
cor(mlb11$hits, mlb11$runs)^2
## [1] 0.6419388
cor(mlb11$homeruns, mlb11$runs)^2
## [1] 0.6265636
cor(mlb11$bat_avg, mlb11$runs)^2
## [1] 0.6560771
cor(mlb11$strikeouts, mlb11$runs)^2
## [1] 0.1693579
cor(mlb11$stolen_bases, mlb11$runs)^2
## [1] 0.002913993
cor(mlb11$wins, mlb11$runs)^2
## [1] 0.3609712

Looks like batting average has the best relationship based on R-Squared, closely followed by hits. Let’s look graphically:

m2 <- lm(runs ~ bat_avg, data = mlb11)
plot(mlb11$bat_avg, mlb11$runs)
abline(m2)

cor(mlb11$new_onbase, mlb11$runs)^2
## [1] 0.8491053
cor(mlb11$new_slug, mlb11$runs)^2
## [1] 0.8968704
cor(mlb11$new_obs, mlb11$runs)^2
## [1] 0.9349271

On-base plus slugging (OBS) seems to be the best predictor for runs scored, and each of the 3 are better predictors than the best of the traditional variables.

m3 <- lm(runs ~ new_obs, data = mlb11)
plot(mlb11$new_obs, mlb11$runs)
abline(m3)

The line is extremely well fit, as is also indicated by the very high r-squared value.

The result does make sense. When you consider the traditional variables, none of them took account of ‘walks’, which places a player on first base, and can in some cases be viewed as a single.

Furthermore, slugging percentage seeks to weight hits differently based on how many bases the hitter reaches (single = 1, double = 2, triple = 3, homerun = 4). This is a much more accurate indicator of importance rather than simple “hits”, which weighs each hit equally.

Combining those two values seems to be the best predictor of runs scored, and this does intuitively make sense.

Linearity

plot(m3$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)

No apparent pattern in the residual plot.

Nearly Normal

hist(m3$residuals)

qqnorm(m3$residuals)
qqline(m3$residuals)

Very symmetric histogram and tight fit to the QQ plot.

Constant variability

Based on the plot from the linearity section above, we can see that the variance away from the x-axis doesn’t change is any real way, with points seemingly randomly scattered. This indicates a constant variability.

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.