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.

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?

Answer: In this case, scatter plot can be used to display the relationship between runs and other numerical variable. The plot of runs vs at_bats as shown below.Yes, the relationship looks linear; thus, I would be comfortable using a linear model to predict the number of runs given that we could see a linear trend in relationship.

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

## [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: The relationship between the two variables appears to be linear that a straight line best follows their association can be drawn through the data points. The direction of this linear relationship is positive where as at_bats increases, runs will likely to increase as well. The strength of the linear relationship is moderately strong. There are some unusual obervations which deviation distance from the best fit straight line are much bigger than others.

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.

## 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.

## 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: 125196.6 was the smallest sum of squares that I got after running 30 times. I have tried to compare this number with another classmate and my number is smaller than him.

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).

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.

## 
## 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?

Answer: From the output below, the equation of the regression line is runs = 415.2389 + 1.8345 * homeruns. The slope here means that for every 1 homerun increase, there are 1.8345 runs increase.

## 
## 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.

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: The regression line equation is runs = -2789.2429 + 0.6305 * at_bats. So,, if a team with 5,578 at_bats, then the number of runs he or she can predict is runs = -2789.2429 + 0.6305 * 5578 = 727.6861. To figure out whether the predicted value is overestimated or underestimated, we need to find out the observed value where at_bates is 5,578. Using the function ’filter, we can find the observed value. Because at_bats has no 5,578 in the dataset, so I pick the closer one which is 5,579 and its observed value is 713. Therefore, the residual for the prediction is 713 - 727.6861 = -14.6861.

## [1] 713

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.

  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 no any apparent pattern in the residuals plot. You can see that the points in the residual plot are randomly dispersed around the horizontal dashed line at y = 0 and does not show any specific pattern such as U-shaped and inverted-U which better fit for a non-linear model. Therefore, we can say the relationship is linear between runs and at-bats.

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

or a normal probability plot of the residuals.

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

Answer: The histogram of the data appears to be roughly unimodal, symmetric, and without outliers. Also, the normal probabiliy plot shows that the data points are lying fairly close to the straight diagonal line with minimal deviation. Therefore, it seems reasonable to conclude that nearly normal residuals condition is met.

Constant variability:

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

Answer: The variation of residuals along the horizontal dashed line at y = 0 appear to be reasonably constant. If you draw another two horizontal dash lines, one at y =100 and another at y = -100, you will clearly see that most of the residuals variation are constantly bounding between these two lines with y = 0 as a center. Therefore, it seems reasonable to conclude that the constant variability condition appear to have been met.

On Your Own

  1. 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?

Answer: Let’s choose ‘bat_avg’ variable. Below is the scatter plot of ‘runs’ and ‘bat_avg’. Yes, there seem to be a linear relationship betweetn these two variables.

  1. 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?

Answer: It seems to be stronger than the relationship between ‘runs’ and ‘at_bats’. The R\(^2\) value for variable at_bats is 0.3728654 and the R\(^2\) value for variable bat_avg is 0.6560771. Since the R\(^2\) value of bat_avg is significantly higher than R\(^2\) value of at_bats, we can say that bat_avg is a better predictor than at_bats.

## [1] 0.3728654
## [1] 0.6560771
  1. 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).

Answer: So far we have investigated relationship for at_bats and batting average variables. Next, I will investigate relationship for other five traditional variables: home runs, hits, strikeouts, stolen bases, and wins. From my computation result below, the bat_avg variable appears to show the best positive relationship with runs. The R\(^2\) value of bat_avg is higher than R\(^2\) value of other 5 traditional variables. Also, from the scatter plot with the least squares line laid on top in below, you can see that most of the data points which deviation distance from the least squares line is very short. Therefore, we can say that bat_avg is the variable that best predicts ‘runs’.

## [1] 0.6560771

  1. Now examine the three newer variables. These are the statistics used by the author of Moneyball to predict a teams 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?

Answer: From below computation with graphical and numerical evidence, you can see that the three new variables (new_onbase, new_slug, and new_obs) have higher R\(^2\) value to predict a team’s success than the 7 traditional variables. Also, the variable ‘new_obs’ has the highest R\(^2\) value among the 3 new variables. Furthermore, from the 3 scatter plots, you can see that the ‘new_obs’ scatter plot most of its data points which deviation distance from the least squares line is shorter than the other two scatter plots. Therefore, we can say that the new variable ‘new_obs’ is the best predictor of runs.

## [1] 0.3728654
## [1] 0.6265636
## [1] 0.6560771
## [1] 0.6419388
## [1] 0.1693579
## [1] 0.002913993
## [1] 0.3609712
## [1] 0.8491053
## [1] 0.8968704
## [1] 0.9349271

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

Answer: (1) From the residuals plot, we can see that the points in the residuals plot are randomly dispersed around the horizontal dashed line at y = 0 and does not show any specific pattern such as U-shaped and inverted-U which better fit for a non-linear model. Therefore, we can say the relationship is linear between runs and new_obs. (2) The histogram of the data appears to be nearly unimodal, symmetric, and without outliers. Also, the normal probabiliy plot shows that the data points are lying fairly close to the straight diagonal line with minimal deviation. Therefore, it seems reasonable to conclude that nearly normal residuals condition is met. (3) The variation of residuals along the horizontal dashed line at y = 0 appear to be reasonably constant. Most of the residuals variation are constantly bounding between y = 40 and y = -40 with y = 0 as a center. Therefore, it seems reasonable to conclude that the constant variability condition appear to have been met.