Open Intro Statistics: Introduction to Linear Regression
Brian Liles
February 5, 2019
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 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.
Libraries
library(tidyverse)## -- Attaching packages ---------------------------------- tidyverse 1.2.1 --## v ggplot2 3.1.0 v purrr 0.2.5
## v tibble 1.4.2 v dplyr 0.7.7
## v tidyr 0.8.2 v stringr 1.3.1
## v readr 1.1.1 v forcats 0.3.0## -- Conflicts ------------------------------------- tidyverse_conflicts() --
## x dplyr::filter() masks stats::filter()
## x dplyr::lag() masks stats::lag()The Data
Let’s load up the data for the 2011 season.
download.file("http://www.openintro.org/stat/data/mlb11.RData", destfile = "mlb11.RData")
load("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.
glimpse(mlb11)## Observations: 30
## Variables: 12
## $ team <fct> Texas Rangers, Boston Red Sox, Detroit Tigers, Ka...
## $ runs <int> 855, 875, 787, 730, 762, 718, 867, 721, 735, 615,...
## $ at_bats <int> 5659, 5710, 5563, 5672, 5532, 5600, 5518, 5447, 5...
## $ hits <int> 1599, 1600, 1540, 1560, 1513, 1477, 1452, 1422, 1...
## $ homeruns <int> 210, 203, 169, 129, 162, 108, 222, 185, 163, 95, ...
## $ bat_avg <dbl> 0.283, 0.280, 0.277, 0.275, 0.273, 0.264, 0.263, ...
## $ strikeouts <int> 930, 1108, 1143, 1006, 978, 1085, 1138, 1083, 120...
## $ stolen_bases <int> 143, 102, 49, 153, 57, 130, 147, 94, 118, 118, 81...
## $ wins <int> 96, 90, 95, 71, 90, 77, 97, 96, 73, 56, 69, 82, 7...
## $ new_onbase <dbl> 0.340, 0.349, 0.340, 0.329, 0.341, 0.335, 0.343, ...
## $ new_slug <dbl> 0.460, 0.461, 0.434, 0.415, 0.425, 0.391, 0.444, ...
## $ new_obs <dbl> 0.800, 0.810, 0.773, 0.744, 0.766, 0.725, 0.788, ...summary(mlb11)## team runs at_bats hits
## Arizona Diamondbacks: 1 Min. :556.0 Min. :5417 Min. :1263
## Atlanta Braves : 1 1st Qu.:629.0 1st Qu.:5448 1st Qu.:1348
## Baltimore Orioles : 1 Median :705.5 Median :5516 Median :1394
## Boston Red Sox : 1 Mean :693.6 Mean :5524 Mean :1409
## Chicago Cubs : 1 3rd Qu.:734.0 3rd Qu.:5575 3rd Qu.:1441
## Chicago White Sox : 1 Max. :875.0 Max. :5710 Max. :1600
## (Other) :24
## homeruns bat_avg strikeouts stolen_bases
## Min. : 91.0 Min. :0.2330 Min. : 930 Min. : 49.00
## 1st Qu.:118.0 1st Qu.:0.2447 1st Qu.:1085 1st Qu.: 89.75
## Median :154.0 Median :0.2530 Median :1140 Median :107.00
## Mean :151.7 Mean :0.2549 Mean :1150 Mean :109.30
## 3rd Qu.:172.8 3rd Qu.:0.2602 3rd Qu.:1248 3rd Qu.:130.75
## Max. :222.0 Max. :0.2830 Max. :1323 Max. :170.00
##
## wins new_onbase new_slug new_obs
## Min. : 56.00 Min. :0.2920 Min. :0.3480 Min. :0.6400
## 1st Qu.: 72.00 1st Qu.:0.3110 1st Qu.:0.3770 1st Qu.:0.6920
## Median : 80.00 Median :0.3185 Median :0.3985 Median :0.7160
## Mean : 80.97 Mean :0.3205 Mean :0.3988 Mean :0.7191
## 3rd Qu.: 90.00 3rd Qu.:0.3282 3rd Qu.:0.4130 3rd Qu.:0.7382
## Max. :102.00 Max. :0.3490 Max. :0.4610 Max. :0.8100
## head(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
## 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 0.725Exercise 1
What type of plot would you use to display the relationship between runs and one of the other numerical variables? A scatter plot is useful for observing the relationship between two numeric variables.
Plot this relationship using the variable at_bats as the predictor. Does the relationship look linear?
#base R scatter plot
plot(mlb11$at_bats,mlb11$runs,xlab="Number of At Bats",
ylab="Number of Runs",main="Number of Bats vs. Number of Runs MLB 2011 Season")
ggplot(data=mlb11, aes(x=at_bats, y=runs)) +
geom_point() +
xlab("Number of At Bats") +
ylab("Number of Runs") +
ggtitle("Number of Bats vs. Number of Runs MLB 2011 Season")
Based off the plots, the relationship appears to have a positive linear trend.
If you knew a team’s at_bats, would you be comfortable using a linear model to predict the number of runs? Yes, because it allows one to make a better prediction.
If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient.
#cor computes the correlation of x and y if these are vectors. If x and y are matrices then the correlations between the columns of x and the columns of y are computed.
cor(mlb11$runs,mlb11$at_bats)## [1] 0.610627In statistics, the correlation coefficient r measures the strength and direction of a linear relationship between two variables on a scatterplot. The value of r is always between +1 and -1. To interpret its value, see which of the following values your correlation r is closest to:
Exactly -1: a perfect downhill (negative) linear relationship
-70: a strong downhill (negative) linear relationship
-50: a moderate downhill (negative) linear relationship
-30: a weak downhill (negative) linear relationship
0: no linear relationship
+30: a weak uphill (positive) linear relationship
+50: a moderate uphill (positive) linear relationship
+70: a strongh uphill (positive) linear relationship
+1: a perfect uphill (positive) linear relationship
If the scatterplot doesn’t indicate there’s at least somewhat of a linear relationship, the correlation doesn’t mean much. Why measure the amount of linear relationship if there isn’t enough of one to speak of? However, you can take the idea of no linear relationship two ways: 1) If no relationship at all exists, calculating the correlation doesn’t make sense because correlation only applies to linear relationships; and 2) If a strong relationship exists but it’s not linear, the correlation may be misleading, because in some cases a strong curved relationship exists. That’s why it’s critical to examine the scatterplot first.
Many folks make the mistake of thinking that a correlation of -1 is a bad thing, indicating no relationship. Just the opposite is true! A correlation of -1 means the data are lined up in a perfect straight line, the strongest negative linear relationship you can get. The “-” (minus) sign just happens to indicate a negative relationship, a downhill line.
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.
Exercise 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 two variables apper to be positively correlated (r=0.61) with a slope of approximately 1. The variation appears constant. There are a few outliers, though they appear to follow the general linear trend.
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: 123721.9After 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: 123721.9Note 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.
Exercise 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?
The smallest sum of squares: 123721.9
The Linear Model
It is rather cumbersome (burdensome; troublesome) 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).
#lm is used to fit linear models. It can be used to carry out regression, single stratum analysis of variance and analysis of covariance
m1 <- lm(runs ~ at_bats, data = mlb11) #make sure to place y before xThe 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: 0.0003388Let’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, R2. The R2 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.
Exercise 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?
#marked m1_n for new version
m1_n <- lm(runs ~ homeruns, data = mlb11)
summary(m1_n)##
## 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\[\hat{y} = 415.2389 + 1.835 * homeruns\]
For this model, 62.7% of the variability in runs is explained by home runs.
Prediction and Prediction Errors
Let’s create a scatterplot with the least squares line laid on top.
plot(mlb11$runs ~ mlb11$at_bats, xlab="Number of At Bats",ylab="Number of Runs",
main="Number of At Bats vs Number of Runs")
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.
Exercise 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?
Based on the graph, the line predicts that a team with 5,578 at bats would make 725 runs.
#residuals is a generic function which extracts model residuals from objects returned by modeling functions.
resid(m1)## 1 2 3 4 5 6
## 75.960476 63.802426 68.493275 -57.236674 63.040325 -23.837074
## 7 8 9 10 11 12
## 176.868025 75.637074 28.473725 -125.575974 -24.378825 5.573124
## 13 14 15 16 17 18
## -55.679025 -14.403674 -19.979225 -15.595525 -26.043175 19.542975
## 19 20 21 22 23 24
## 102.031374 27.015475 -51.584925 -58.826475 -18.968626 -3.515676
## 25 26 27 28 29 30
## 68.573124 -55.437475 -17.579626 -99.954375 -33.446426 -72.968626Model 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, xlab = "Number of At Bats",
ylab = "Residuals",main = "Residuals vs. Number of At Bats")
abline(h = 0, lty = 3) #adds a horizontal dashed line @ y = 0
Exercise 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?
There seems to be no pattern in the residuals plot.
Nearly normal residuals: To check this condition, we can look at a histogram
hist(m1$residuals, xlab="Residuals",main = "Histogram of Residuals", col = "beige")
or a normal probability plot of the residuals.
#qqnorm(y, ylim, main = "Normal Q-Q Plot",xlab = "Theoretical Quantiles", ylab = "Sample Quantiles",plot.it = TRUE, datax = FALSE, ...)
#qqline(y, datax = FALSE, distribution = qnorm,probs = c(0.25, 0.75), qtype = 7, ...)
qqnorm(m1$residuals)
qqline(m1$residuals) #adds diagonal line to the normal prob plot
Exercise 7
Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?
Yes
Exercise 8
Based on the plot in (1), does the constant variability condition appear to be met? Based on the plot, there appears to be constant variability.
On Your Own
Choose another traditional variable from mlb11 that you think might be a good predictor of runs. Produce a scatterplot of the two variables and fit a linear model. At a glance, does there seem to be a linear relationship?
plot(runs ~ hits, data = mlb11, xlab = "Number of Hits",
ylab = "Number of Runs", main = "Number of Hits vs. Number of Runs MLB 2011")
m2 <- lm(runs ~ hits, data = mlb11)
abline(m2)
There is linearity between the two variables. Next, we will check the correlation using the cor function.
cor(mlb11$hits,mlb11$runs)## [1] 0.8012108How does this relationship compare to the relationship between runs and at_bats? Use the R2 values from the two model summaries to compare. Does your variable seem to predict runs better than at_bats? How can you tell?
summary(m2)##
## Call:
## lm(formula = runs ~ hits, data = mlb11)
##
## Residuals:
## Min 1Q Median 3Q Max
## -103.718 -27.179 -5.233 19.322 140.693
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -375.5600 151.1806 -2.484 0.0192 *
## hits 0.7589 0.1071 7.085 1.04e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 50.23 on 28 degrees of freedom
## Multiple R-squared: 0.6419, Adjusted R-squared: 0.6292
## F-statistic: 50.2 on 1 and 28 DF, p-value: 1.043e-07Since at_bats R-squared/R(cor) (0.3729/0.61) vs hits R-squared/R(cor) (0.6419/.80), hits is a better predictor for runs than at_bats
Now that you can summarize the linear relationship between two variables, investigate the relationships between runs and each of the other five traditional variables. Which variable best predicts runs? Support your conclusion using the graphical and numerical methods we’ve discussed (for the sake of conciseness, only include output for the best variable, not all five).
preds = c("at_bats","hits","homeruns","bat_avg","strikeouts","stolen_bases","wins")
for (p in preds) {
R = cor(mlb11[, p], mlb11$runs)
print(paste(p, ': ', R ^ 2, sep=''))
}## [1] "at_bats: 0.372865390186806"
## [1] "hits: 0.641938767239419"
## [1] "homeruns: 0.626563569566283"
## [1] "bat_avg: 0.656077134646863"
## [1] "strikeouts: 0.169357932236312"
## [1] "stolen_bases: 0.00291399266657398"
## [1] "wins: 0.360971179446681"plot(runs ~ bat_avg, data = mlb11,xlab="Batting Average",ylab = "Number of Runs",
main="Batting Average vs. Number of Runs")
m3 <- lm(runs ~ bat_avg, data = mlb11)
abline(m3)
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?
Check the model diagnostics for the regression model with the variable you decided was the best predictor for runs.
preds = c("new_onbase","new_slug","new_obs")
for (p in preds) {
R = cor(mlb11[, p], mlb11$runs)
print(paste(p, ': ', R ^ 2, sep=''))
}## [1] "new_onbase: 0.849105251446139"
## [1] "new_slug: 0.896870368409638"
## [1] "new_obs: 0.934927126351814"plot(runs ~ new_obs, data = mlb11,xlab="On Base Plus Slugging",ylab = "Number of Runs",
main="On Base Plus Slugging vs. Number of Runs")
m4 <- lm(runs ~ new_obs, data = mlb11)
abline(m4)
hist(m4$residuals, xlab="Residuals",main = "Histogram of Residuals", col = "beige")
qqnorm(m4$residuals)
qqline(m4$residuals)