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.

library(statsr)
## Loading required package: BayesFactor
## Loading required package: coda
## Loading required package: Matrix
## ************
## Welcome to BayesFactor 0.9.12-4.2. If you have questions, please contact Richard Morey (richarddmorey@gmail.com).
## 
## Type BFManual() to open the manual.
## ************
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
library(ggplot2)
library('grid')

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.

download.file("http://www.openintro.org/stat/data/mlb11.RData", destfile = "mlb11.RData")
load("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? **I used a scatter plot to display the relationship between ‘runs’ and ‘at_bats’
ggplot(mlb11, aes(x=at_bats, y=runs)) +
  geom_point() +    
  geom_smooth(method=lm)

Answer: Yes, the relationship looks linear. The correlation coefficient is large enough to predict ‘runs’ scored by player from his ‘at_bats’ number.

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.

Answer:
It looks like there is a positive relationship between ‘runs’ and ‘at_bats’
There is a positive correlation coeffidient of 0.610627 with a positive direction.
They are some outliers, but not significant.

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?

Answer: Smallest Sum of Square that I got was: 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?
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

\(\hat{y} = 415.239 + 1.8345 * homeruns\)

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?
    Answer:
    \(\hat{y} = -2789.2429+0.6305 * at-bat\) \(\hat{y} = -2789.2429+0.6305 * 5578\) \(\hat{y} = 727.6861\)
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

Based on the data, we see that at 5579 at_bat, runs are 713. We would expect 0.6305 fewer runs for 5578 \(713-0.6305=712.3695\)
That means the team manager would over estimate the number of runs with the 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 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?
    Answer: There is a linear relationship with some high variation.

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?

Answer: This Normal QQ plot shows that the relationship between the theoretical percentile and sample percentile is approximately linear. Therefore, the normal probability plot of the residuals suggests that the residuals are indeed nearly normal.

Constant variability:

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

Answer: Yes, the constant variability condition appear to be met.

On Your Own

plot_ss(y = mlb11$runs, x = mlb11$hits)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##   -375.5600       0.7589  
## 
## Sum of Squares:  70638.75
ggplot(mlb11, aes(x=hits, y=runs)) +
  geom_point() +    
  geom_smooth(method=lm)

Answer: The relationship does looks linear.

Answer: They both have a similar type of relationship but with less outliers at hits than in at_bats.

m3 <- lm(runs ~ hits, data = mlb11)
summary(m3)$r.squared
## [1] 0.6419388

Answer: \(R^2\) for hits = 0.6419 and \(R^2\) for at_bats 0.3729. Higher \(R^2\) for hits indicates that its a good predictor for runs compared to at_bats.

#R-squared for strikeouts
m4 <- lm(runs ~ strikeouts, data = mlb11)
summary(m4)$r.squared
## [1] 0.1693579
#R-squared for home runs
m5 <- lm(runs ~ homeruns, data = mlb11)
summary(m5)$r.squared
## [1] 0.6265636
#R-squared for stolen_bases
m6 <- lm(runs ~ stolen_bases, data = mlb11)
summary(m6)$r.squared
## [1] 0.002913993
#R-squared for new_onbase  
m7 <- lm(runs ~ new_onbase, data = mlb11)
summary(m7)$r.squared
## [1] 0.8491053
#R-squared for new_slug 
m8 <- lm(runs ~ new_slug, data = mlb11)
summary(m8)$r.squared
## [1] 0.8968704
#R-squared for new_obs
m9 <- lm(runs ~ new_obs, data = mlb11)
summary(m9)$r.squared
## [1] 0.9349271

Answer: By far with the highest \(R^2\) of 0.9349271, new_obs is the best predictor for runs in this model. Lets take a closer look at the top 3 \(R^2\) for runs, which are new_onbase, new_slug and new_obs for a better understanding.

#new_obs vs. runs
ggplot(mlb11, aes(x= new_obs, y=runs)) +
  geom_point() +    
  geom_smooth(method=lm)

#new_onbase vs. runs  
ggplot(mlb11, aes(x= new_onbase, y=runs)) +
  geom_point() +    
  geom_smooth(method=lm)

#new_slug vs. runs
ggplot(mlb11, aes(x= new_slug, y=runs)) +
  geom_point() +    
  geom_smooth(method=lm)

Linearity:

plot(m9$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 0

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

hist(m9$residuals)

qqnorm(m9$residuals)
qqline(m9$residuals)  # adds diagonal line to the normal prob plot