Introduction to linear regression

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")

mlb11

##                     team runs at_bats hits homeruns bat_avg strikeouts
## 1          Texas Rangers  855    5659 1599      210   0.283        930
## 2         Boston Red Sox  875    5710 1600      203   0.280       1108
## 3         Detroit Tigers  787    5563 1540      169   0.277       1143
## 4     Kansas City Royals  730    5672 1560      129   0.275       1006
## 5    St. Louis Cardinals  762    5532 1513      162   0.273        978
## 6          New York Mets  718    5600 1477      108   0.264       1085
## 7       New York Yankees  867    5518 1452      222   0.263       1138
## 8      Milwaukee Brewers  721    5447 1422      185   0.261       1083
## 9       Colorado Rockies  735    5544 1429      163   0.258       1201
## 10        Houston Astros  615    5598 1442       95   0.258       1164
## 11     Baltimore Orioles  708    5585 1434      191   0.257       1120
## 12   Los Angeles Dodgers  644    5436 1395      117   0.257       1087
## 13          Chicago Cubs  654    5549 1423      148   0.256       1202
## 14       Cincinnati Reds  735    5612 1438      183   0.256       1250
## 15    Los Angeles Angels  667    5513 1394      155   0.253       1086
## 16 Philadelphia Phillies  713    5579 1409      153   0.253       1024
## 17     Chicago White Sox  654    5502 1387      154   0.252        989
## 18     Cleveland Indians  704    5509 1380      154   0.250       1269
## 19  Arizona Diamondbacks  731    5421 1357      172   0.250       1249
## 20     Toronto Blue Jays  743    5559 1384      186   0.249       1184
## 21       Minnesota Twins  619    5487 1357      103   0.247       1048
## 22       Florida Marlins  625    5508 1358      149   0.247       1244
## 23    Pittsburgh Pirates  610    5421 1325      107   0.244       1308
## 24     Oakland Athletics  645    5452 1330      114   0.244       1094
## 25        Tampa Bay Rays  707    5436 1324      172   0.244       1193
## 26        Atlanta Braves  641    5528 1345      173   0.243       1260
## 27  Washington Nationals  624    5441 1319      154   0.242       1323
## 28  San Francisco Giants  570    5486 1327      121   0.242       1122
## 29      San Diego Padres  593    5417 1284       91   0.237       1320
## 30      Seattle Mariners  556    5421 1263      109   0.233       1280
##    stolen_bases wins new_onbase new_slug new_obs
## 1           143   96      0.340    0.460   0.800
## 2           102   90      0.349    0.461   0.810
## 3            49   95      0.340    0.434   0.773
## 4           153   71      0.329    0.415   0.744
## 5            57   90      0.341    0.425   0.766
## 6           130   77      0.335    0.391   0.725
## 7           147   97      0.343    0.444   0.788
## 8            94   96      0.325    0.425   0.750
## 9           118   73      0.329    0.410   0.739
## 10          118   56      0.311    0.374   0.684
## 11           81   69      0.316    0.413   0.729
## 12          126   82      0.322    0.375   0.697
## 13           69   71      0.314    0.401   0.715
## 14           97   79      0.326    0.408   0.734
## 15          135   86      0.313    0.402   0.714
## 16           96  102      0.323    0.395   0.717
## 17           81   79      0.319    0.388   0.706
## 18           89   80      0.317    0.396   0.714
## 19          133   94      0.322    0.413   0.736
## 20          131   81      0.317    0.413   0.730
## 21           92   63      0.306    0.360   0.666
## 22           95   72      0.318    0.388   0.706
## 23          108   72      0.309    0.368   0.676
## 24          117   74      0.311    0.369   0.680
## 25          155   91      0.322    0.402   0.724
## 26           77   89      0.308    0.387   0.695
## 27          106   80      0.309    0.383   0.691
## 28           85   86      0.303    0.368   0.671
## 29          170   71      0.305    0.349   0.653
## 30          125   67      0.292    0.348   0.640

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?

plot(mlb11$at_bats, mlb11$runs)

There is a bit of a relationship, but not strong enough that I would make any major decisions based on this alone.

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

This is about what I would have expected.

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.

2. 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 between the two features is loosly positive and has 1 clear outlier, but 3 that could be considered outliers depending on how tightly you define it.

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.

3. 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?

   142550.7

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.

4. 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(runs ~ homeruns, data = mlb11)

plot(mlb11$homeruns, mlb11$runs)

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

y = 1.83x + 415.24

The slope tells us that there is a positive correlation between the two metrics and that when one increases so will the other. The number of runs as predicted by number of home runs grows at nearly twice the rate as the number of home runs.\

For every homerun hit a player should have scored about 417 runs. It also suggests that the less someone has played the more likely their home run to run ratio should be.

I felt like the plot as natural really didn't show the data very well, given the plot. Here's one with the axes set from 0-1000. It does a much better job at showing the upwards trajectory of the line.

plot(mlb11$homeruns, mlb11$runs, xlim=c(0,1000), ylim=c(0,1000))

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.

5. 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?

   Prediction would be around 725

   There is an actual at 5,579 at bats with 713 runs so the model overestimated by 12 runs.

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

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

   No, the residuals are scattered somewhat randomly. It shols that there is not constant variability, therefor there is not a strong linear relationship.

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

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

   The histogram is somewhat fairly normal (if you squint and turn your head a little).
   The probability plot does not show normality; the markers only land on the plot line 11 times (generously).

Constant variability:

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

   No.

On Your Own

• 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?

plot(mlb11$strikeouts, mlb11$runs)

No, there does not seem to be a linear relationship between runs and strikeouts.

• 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?

m3 <- lm(runs ~ strikeouts, data = mlb11)

summary(m1)$r.squared

## [1] 0.3728654

summary(m3)

## 
## 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

R$^2$ for at bats predicting runs is 0.37.\
R$^2$ for strikeouts predicting runs is 0.17.\
At bats is a much stronger 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).

m1 <- lm(runs ~ at_bats, data = mlb11)
m2 <- lm(runs ~ homeruns, data = mlb11)
m3 <- lm(runs ~ strikeouts, data = mlb11)
m4 <- lm(runs ~ hits, data = mlb11)
m5 <- lm(runs ~ bat_avg, data = mlb11)
m6 <- lm(runs ~ stolen_bases, data = mlb11)
m7 <- lm(runs ~ wins, data = mlb11)

at_bats_r2 <- summary(m1)$r.squared
homeruns_r2 <- summary(m2)$r.squared
strikeouts_r2 <- summary(m3)$r.squared
hits_r2 <- summary(m4)$r.squared
bat_avg_r2 <- summary(m5)$r.squared
stolen_bases_r2 <- summary(m6)$r.squared
wins_r2 <- summary(m7)$r.squared

rsq_sum <- data.frame("R_Squared" = c(at_bats_r2, homeruns_r2, strikeouts_r2, hits_r2, bat_avg_r2, stolen_bases_r2, wins_r2))
rsq_sum$metric <- c("AtBats", "HRs", "StrikeOuts", "Hits", "BatAvg", "StolenBases", "Wins")

rsq_sum[order(rsq_sum$R_Squared),]

##     R_Squared      metric
## 6 0.002913993 StolenBases
## 3 0.169357932  StrikeOuts
## 7 0.360971179        Wins
## 1 0.372865390      AtBats
## 2 0.626563570         HRs
## 4 0.641938767        Hits
## 5 0.656077135      BatAvg

Batting average is the best predictor, though hits and home runs aren’t far behind.

par(mfrow = c(2, 2))

plot(mlb11$runs ~ mlb11$bat_avg)
abline(m5)

plot(m5$residuals ~ mlb11$bat_avg)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 0

hist(m5$residuals)

qqnorm(m5$residuals)
qqline(m5$residuals)  # adds diagonal line to the normal prob plot

summary(m5)

## 
## 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

The residusals are more normal than they had been for previous metrics, though still not enough that I would call them fairly normal. Variability is also all over the place. 

• 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?

m8 <- lm(runs ~ new_onbase, data = mlb11)
m9 <- lm(runs ~ new_slug, data = mlb11)
m10 <- lm(runs ~ new_obs, data = mlb11)

onbase_r2 <- summary(m8)$r.squared
slug_r2 <- summary(m9)$r.squared
obs_r2 <- summary(m10)$r.squared

rsq_sum2 <- data.frame("R_Squared" = c(onbase_r2, slug_r2, obs_r2))
rsq_sum2$metric <- c("OnBase", "Slug", "OBS")

rsq_sum2[order(rsq_sum2$R_Squared),]

##   R_Squared metric
## 1 0.8491053 OnBase
## 2 0.8968704   Slug
## 3 0.9349271    OBS

OBS (On Base Something?) had a nearly perfect correlation with an $R^2$ of .93.

par(mfrow = c(2, 2))

plot(mlb11$runs ~ mlb11$new_obs)
abline(m10)

plot(m1$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 0

hist(m10$residuals)

qqnorm(m10$residuals)
qqline(m10$residuals)  # adds diagonal line to the normal prob plot

summary(m10)

## 
## Call:
## lm(formula = runs ~ new_obs, data = mlb11)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -43.456 -13.690   1.165  13.935  41.156 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  -686.61      68.93  -9.962 1.05e-10 ***
## new_obs      1919.36      95.70  20.057  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 21.41 on 28 degrees of freedom
## Multiple R-squared:  0.9349, Adjusted R-squared:  0.9326 
## F-statistic: 402.3 on 1 and 28 DF,  p-value: < 2.2e-16

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

  While the data is fairly normal the residuals are still inconsistent.