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.

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.

- 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 would use a scatter plot to display the relationship between the variables.**

```
plot(mlb11$at_bats, mlb11$hits, # plot the variables
xlab="At Bats", # x−axis label
ylab="Hits") # y−axis label
```

**Yes, the relationship looks linear.**

**Yes, If I knew a team’s at_bats, I would be comfortable using a linear model to predict the number of runs.**

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

`## [1] 0.610627`

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.

- 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.

```
plot(mlb11$at_bats, mlb11$runs, # plot the variables
xlab="At Bats", # x−axis label
ylab="Runs") # y−axis label
```

**The relationship seems to be a little sparsed or dispersed, the distribution of the points seems to grow in a positive direction with some leverage and the relationship seems to have a positive correlation.**

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.

- 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 sum of squares found was 123721.9**

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.

- 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?

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 415.239 1.835
##
## Sum of Squares: 73671.99
```

`## [1] 0.7915577`

```
##
## 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
```

**Regrssion line equation is: runs = 1.8345.homeruns + 415.2389**

**In this case, the slope tells us that the more homeruns, the more runs the team will have, therfore increasing the chances of the team winning games.**

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.

- 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?

**Since the regression line is y = 0.6305⋅x − 2789.2429**

```
runs <- function(x = NULL)
{
y <- 0.6305 * x - 2789.2429
y <- round(y,0)
return(y)
}
y <- runs(5578)
y
```

`## [1] 728`

```
## runs
## 16 15
```

**Since the manager didn’t look at the data by doing these calculations, he can get a totals of 728.**

**Since the real nearest point from the table is totaled at 713 from Philadelphia Phillies with at bats of 5579, the difference will be of 15, resulting in an over estimation of 15 runs.**

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.

- Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between runs and at-bats?

**We do notice that the residuals are positioned around zero, indicating a constant linearity of the relationship 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.

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

**Looking at both graphs, we can see that the normal residuals condition seems to be met.**

*Constant variability*:

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

**Based on the plot in (1), the points seem to follow a pattern, making constant variability condition to be met.**

- 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?

**Choosing Hits variable:**

```
m3 <- lm(runs ~ hits, data = mlb11)
plot(mlb11$hits, mlb11$runs, # plot the variables
xlab="Hits", # x−axis label
ylab="Runs") # y−axis label
abline(m3)
```

`## [1] 0.8012108`

**At a glance, there seems to be a linear positive relationship.**

- 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?

```
## 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
```

```
##
## 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
```

**For this model, 64.19% of the variability in runs is explained by Hits. Looking at the results from the at_bats in which only 34.3% was explained by it, we can conclude that this model seem to predict runs better.**

- 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).

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 415.239 1.835
##
## Sum of Squares: 73671.99
```

`## [1] 0.7915577`

**Correlation: 0.7915577**

```
##
## 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
```

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## -642.8 5242.2
##
## Sum of Squares: 67849.52
```

`## [1] 0.8099859`

**Correlation is: 0.8099859**

```
##
## 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
```

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 1054.7342 -0.3141
##
## Sum of Squares: 163870.1
```

`## [1] -0.4115312`

**Correlation is: -0.4115312**

```
##
## Call:
## lm(formula = runs ~ strikeouts, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -132.27 -46.95 -11.92 55.14 169.76
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 1054.7342 151.7890 6.949 1.49e-07 ***
## strikeouts -0.3141 0.1315 -2.389 0.0239 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 76.5 on 28 degrees of freedom
## Multiple R-squared: 0.1694, Adjusted R-squared: 0.1397
## F-statistic: 5.709 on 1 and 28 DF, p-value: 0.02386
```

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 677.3074 0.1491
##
## Sum of Squares: 196706.3
```

`## [1] 0.05398141`

**Correlation: 0.05398141**

```
##
## 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
```

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## 342.121 4.341
##
## Sum of Squares: 126068.4
```

`## [1] 0.6008088`

**Correlation: 0.6008088**

```
##
## Call:
## lm(formula = runs ~ wins, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -145.450 -47.506 -7.482 47.346 142.186
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 342.121 89.223 3.834 0.000654 ***
## wins 4.341 1.092 3.977 0.000447 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 67.1 on 28 degrees of freedom
## Multiple R-squared: 0.361, Adjusted R-squared: 0.3381
## F-statistic: 15.82 on 1 and 28 DF, p-value: 0.0004469
```

```
## r2
## 1 0.6266
## 2 0.6561
## 3 0.1694
## 4 0.002914
## 5 0.361
```

**From the table above, we can see that the best approach is by selecting bat_avg.**

- 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?

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## -1118 5654
##
## Sum of Squares: 29768.7
```

`## [1] 0.9214691`

```
##
## 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
```

```
## Click two points to make a line.
## Call:
## lm(formula = y ~ x, data = pts)
##
## Coefficients:
## (Intercept) x
## -375.8 2681.3
##
## Sum of Squares: 20345.54
```

`## [1] 0.9470324`

```
##
## 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
```