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")

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.

sample(mlb11)

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

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?

A scatter plot should show the relationship between numeric variables.

plot(mlb11$at_bats,mlb11$runs, xlab = "homeruns", ylab = "Runs", col = 'darkblue')

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

cor(mlb11$runs, mlb11$homeruns)

## [1] 0.7915577

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.

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.

A fairly strong, positive linear relationship exists between homeruns and runs.

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$homeruns, y = mlb11$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

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$hits, y = mlb11$runs, showSquares = TRUE)

## 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

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?

I got the smallest sum of squares using the variable hits (70,638), homeruns was 73,671 and at_bats was 123,721.

***I'm not sure I'm understanding what is being asked here.***

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.

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

The slope tells us that run is postively related to homeruns.  A team can expect approximately 1.8 runs for every homerun that is hit.  y^=415.2389+1.8345 x 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.

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?

#From summary above 5579 hits produced 713 run (Phillies). Therefore 5578 should produce approximately 712.
at_bats <- 5578
y_hat <- -2789.2429 + 0.6305 * at_bats
y_hat

## [1] 727.6861

728 is greater than 712, so it appears 728 is an overestimate, by approximately 15 to 16 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

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

There does not appear to be a disernable. Residuals appear randomly scattered above and below the line. This indicates the residuals should be near normal.

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

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

  Yes, although there appears to be some skew, the plot indicates near normality.

Constant variability:

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

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$bat_avg,mlb11$runs, xlab = 'Batting Avg', ylab = 'Runs' ,col = 'steelblue3')

Yes, there seems to be a positive linear relationship between batting average and runs.

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(mlb11$runs ~ mlb11$bat_avg)
summary(m3)

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

Batting average is more predictive of runs than at bats. The R-squared for batting average was 0.65 ca 0.37 for at bats. We can tell this by comparing the R-squareds - this tell that batting average can explain more than 60% of the variability while ate bats can explain less than 40%.

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).

Batting average best predicts runs. Support follows:

ba <- lm(mlb11$runs ~  mlb11$bat_avg )
summary(ba)

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

plot(mlb11$bat_avg, mlb11$runs, xlab = 'Batting Ave', ylab= 'Runs', col = 'darkblue')
abline(ba, col = 'blue')

hist(resid(ba))

plot(mlb11$bat_avg, resid(ba))
abline(h = 0, lty = 3)

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?

Onbase plus slugging seems to be the best predictor by far.  The R-squared for new_obs is 93% far above batting average.  This makes intuitives sences because obs seems to combine batting average and home runs (or slugging), this like explains the higher R-squared.   The new stats are all better than the traditional.

obs1 <- lm(mlb11$runs ~  mlb11$new_obs)
summary(obs1)

## 
## Call:
## lm(formula = mlb11$runs ~ mlb11$new_obs)
## 
## 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 ***
## mlb11$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

plot(mlb11$new_obs, mlb11$runs, xlab = 'Batting Ave', ylab= 'Runs', col = 'darkblue')
abline(obs1, col = 'blue')

hist(resid(obs1))

plot(mlb11$new_obs, resid(obs1))
abline(h = 0, lty = 3)

qqnorm(resid(obs1))
qqline(resid(obs1))

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

See above.

This is a product of OpenIntro that is released under a Creative Commons Attribution-ShareAlike 3.0 Unported. This lab was adapted for OpenIntro by Andrew Bray and Mine Çetinkaya-Rundel from a lab written by the faculty and TAs of UCLA Statistics.