Hw2R -Moneyball

Part 1

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 exercise 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 (and load up mosaic while we’re at it!).

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.

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?

SOLUTION:

xyplot(runs ~ at_bats, data=mlb11)

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

SOLUTION:

cor(runs ~ at_bats, data=mlb11)

## [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 between two quantitative variables, such as runs and at_bats above.

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.

SOLUTION: There seems to be a weak positive linear relationship between the two variables.

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 represents 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)

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. What are residuals?

The most common way to do linear regression is to select the line that minimizes the sum of squared residuals.

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?

YOU CAN SKIP THIS PROBLEM

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 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)
sum(residuals(m1)^2)

With this table, what is the least squares regression line?

SOLUTION:

\[\widehat{runs} = -2789.24 + 0.63 at\_bats\]

fitted runs = -2789.24 + 0.63at_bats

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?

SOLUTION:

xyplot(runs ~ homeruns, data=mlb11)

m2 <- lm(runs ~ homeruns, data=mlb11)
summary(m2)

## 
## 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.

xyplot(runs ~ at_bats, data=mlb11, type=c("p", "r"))

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.

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 5,579 at-bats? Is this an overestimate or an underestimate, and by how much?

SOLUTION:

pred1 <- makeFun(m1)
pred1(at_bat=5579)
filter(mlb11, at_bats==5579)

Calculate the residual:

713 - pred1(at_bat=5579)

They would predict 729 runs. Phillies had 713 runs, and 5579 at_bats, so we overestimated.

Model Diagnostics

To assess whether the linear model is reliable, we need to check for Linearity, Independence, Normal errors, and Equal Variance.

Linearity: You already checked if the relationship between runs and at-bats is linear using a scatterplot.
Equal Variance: We want look at a plot of the residuals against the fitted values. If we see a change in the spread of the residuals for larger values of the fitted values, we would be uncomfortable with using a linear regression model as is.

xyplot(resid(m1) ~ fitted(m1), data=mlb11, type=c("p", "r"))

Based on this, does the equal variance condition appear to be met?

SOLUTION:

Independence: We don’t really have independence here, since each observation in the dataset reflects on how a particular team did when playing against the other teams. We’ll overlook this bit.
Normal errors: To check this condition, we can look at a histogram of the residuals or a normal quantile plot:

histogram(~residuals(m1), width=50)

qqmath(~resid(m1))
ladd(panel.qqmathline(resid(m1)))

Do the residuals appear normally distributed?

SOLUTION: No, they do not appear normally distributed, the quantiles do not line up with that of a normal distribution.

Do there appear to be any outliers?

SOLUTION: There seem to be some outliers away from the rest of the data.

Unusual points such as this can be: detrimental to the strength of the fit.

YOU CAN SKIP THIS PROBLEM

high leverage: the observation is far away from the other points in the x-direction.

set.seed(15)
a <- c(runif(20), 4)
b <- a*5+rnorm(21)
xyplot(b~a, pch=16, type=c("p", "r"))

high influence: the observation, when included, will significantly move the regression line.

b[21] <- b[21] -25
xyplot(b~a, pch=16, type=c("p", "r"))

neither: the observation appears to be really far away from the rest of the observations, but due to its position, is unlikely to move the regression line substantially.

set.seed(15)
a <- sort(runif(20))
b <- a*5 + rnorm(20)
b[10] <- 10
xyplot(b~a, pch=16, type=c("p", "r"))

Is the point that you’ve identified high leverage, high influence, or both?

SOLUTION:

Part 2

Now ,this is your turn (working in groups is a nice solution)

Choose another traditional variable from mlb11 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?

At a glance there seems to be a strong linear relationship.

xyplot(runs ~ hits, data=mlb11)

m_hits <- lm(runs ~ hits, data = mlb11)

summary(m_hits)

## 
## Call:
## lm(formula = runs ~ hits, data = mlb11)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -103.718  -27.179   -5.233   19.322  140.693 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -375.5600   151.1806  -2.484   0.0192 *  
## hits           0.7589     0.1071   7.085 1.04e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 50.23 on 28 degrees of freedom
## Multiple R-squared:  0.6419, Adjusted R-squared:  0.6292 
## F-statistic:  50.2 on 1 and 28 DF,  p-value: 1.043e-07

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?

It looks like the relationship between runs and at_bats is weaker than that of hits.

xyplot(runs ~ at_bats, data=mlb11)

m_hits <- lm(runs ~ hits, data = mlb11)

summary(m_hits) #.6419

## 
## Call:
## lm(formula = runs ~ hits, data = mlb11)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -103.718  -27.179   -5.233   19.322  140.693 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -375.5600   151.1806  -2.484   0.0192 *  
## hits           0.7589     0.1071   7.085 1.04e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 50.23 on 28 degrees of freedom
## Multiple R-squared:  0.6419, Adjusted R-squared:  0.6292 
## F-statistic:  50.2 on 1 and 28 DF,  p-value: 1.043e-07

m_atb <- lm(runs ~ at_bats, data = mlb11)

summary(m_atb) #.3729

## 
## 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

The R-square (measure of how much variance in the dependent variable that can be explained by the independent variable) is stronger in hits at .6419 than in at_bats at .3729.

This means that hits is a better predictor.

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).

The R-Square is the highest with the bat_avg variable.

#m_homeruns <- lm(runs ~ homeruns, data = mlb11)

#summary(m_homeruns)

m_batavg <- lm(runs ~ bat_avg, data = mlb11)

summary(m_batavg)

## 
## Call:
## lm(formula = runs ~ bat_avg, data = mlb11)
## 
## 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 ** 
## 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

xyplot(runs ~ bat_avg, data=mlb11)

#m_strikeouts <- lm(runs ~ strikeouts, data = mlb11)

#summary(m_strikeouts)

#m_stolenbases <- lm(runs ~ stolen_bases, data = mlb11)

#summary(m_stolenbases)

#m_wins <- lm(runs ~ wins, data = mlb11)

#summary(m_wins)

Now examine the three newer variables. These are the statistics used by the author of 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?

m_onbase <- lm(runs ~ new_onbase, data = mlb11)

summary(m_onbase)

## 
## Call:
## lm(formula = runs ~ new_onbase, data = mlb11)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -58.270 -18.335   3.249  19.520  69.002 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -1118.4      144.5  -7.741 1.97e-08 ***
## new_onbase    5654.3      450.5  12.552 5.12e-13 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 32.61 on 28 degrees of freedom
## Multiple R-squared:  0.8491, Adjusted R-squared:  0.8437 
## F-statistic: 157.6 on 1 and 28 DF,  p-value: 5.116e-13

xyplot(runs ~ new_onbase, data=mlb11)

m_newslug <- lm(runs ~ new_slug, data = mlb11)

summary(m_newslug)

## 
## Call:
## lm(formula = runs ~ new_slug, data = mlb11)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -45.41 -18.66  -0.91  16.29  52.29 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -375.80      68.71   -5.47 7.70e-06 ***
## new_slug     2681.33     171.83   15.61 2.42e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 26.96 on 28 degrees of freedom
## Multiple R-squared:  0.8969, Adjusted R-squared:  0.8932 
## F-statistic: 243.5 on 1 and 28 DF,  p-value: 2.42e-15

xyplot(runs ~ new_slug, data=mlb11)

m_stolenbases <- lm(runs ~ stolen_bases, data = mlb11)

summary(m_stolenbases)

## 
## Call:
## lm(formula = runs ~ stolen_bases, data = mlb11)
## 
## 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 ***
## 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

xyplot(runs ~ stolen_bases, data=mlb11)

These variables (especially new_onbase and new_slug) are extremely correlated with the target variable. They have even more predictive power than the original 7. This result makes sense because you cannot score without being on base and stealing bases leads to more runs.

Check the model diagnostics for the regression model with the variable you decided was the best predictor for runs.

m_newslug <- lm(runs ~ new_slug, data = mlb11)

summary(m_newslug)

## 
## Call:
## lm(formula = runs ~ new_slug, data = mlb11)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -45.41 -18.66  -0.91  16.29  52.29 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -375.80      68.71   -5.47 7.70e-06 ***
## new_slug     2681.33     171.83   15.61 2.42e-15 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 26.96 on 28 degrees of freedom
## Multiple R-squared:  0.8969, Adjusted R-squared:  0.8932 
## F-statistic: 243.5 on 1 and 28 DF,  p-value: 2.42e-15

xyplot(runs ~ new_slug, data=mlb11)

#Call:
#lm(formula = runs ~ new_slug, data = mlb11)

#Residuals:
#   Min     1Q Median     3Q    Max 
#-45.41 -18.66  -0.91  16.29  52.29 

#Coefficients:
#            Estimate Std. Error t value Pr(>|t|)    
#(Intercept)  -375.80      68.71   -5.47 7.70e-06 ***
#new_slug     2681.33     171.83   15.61 2.42e-15 ***
#---
#Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Residual standard error: 26.96 on 28 degrees of freedom
#Multiple R-squared:  0.8969,   Adjusted R-squared:  0.8932 
#F-statistic: 243.5 on 1 and 28 DF,  p-value: 2.42e-15