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, better predict the ability to score runs than typical statistics like home runs, RBIs (runs battled in), and batting averages. 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 relationshipt between runs scored in a season and a number of other player statistics. Our aim will be to summarize these relationships both graphically and numericallyin 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.

download.file("http://www.openintro.org/stat/data/mlb11.RData", destfile = "mlb11.RData")
load("mlb11.RData")

In addition to runs score, there are seven traditionally used variable 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.

Exercise 1

What type of plot would you use to display the relationship between runs and one of the other numberical 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 scatterplot to display the relationship between the two variables.

plot(mlb11$runs ~ mlb11$at_bats, main = "Runs and at_bats relationship")

The relationship looks somewhat linear with a positive correlation but not enough to where I would feel comfortable using a linear model to predict the number of runs.

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

cor(mlb11$runs, mlb11$at_bats)
## [1] 0.610627

Sum of square 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 numberical variables such as runs and at_bats above.

Exercise 2

Look at your plot from the previous exercise, describe the relationship between these teo variables. Make sure to discuss the form, direction, and strength of the relationship as well as any unusual observations.

Using the plot above, it looks to have a positive correlation, meaning that as the number of at bats increases, the number of runs will increase. There seems to be a moderately strong correlation, which is confirmed by the 0.61 correlation coefficient. There are a few outliers that can be seen in the plot which could skew the results. It appears that most teams are below 5600.

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 beset 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: ei=yi−ŷ

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.

Excersice 3

Using the 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’m not sure if I did the above incorrectly, but I was never prompted to cick two points for a line, a line always appeared automatically. I always got the same SS when I ran the function more than once.

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 taht 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 informationwe 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 for the linear model:

ŷ =−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.

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?

plot_ss(x = mlb11$runs, y = mlb11$homeruns, showSquares = TRUE)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##    -85.1566       0.3415  
## 
## Sum of Squares:  13715.52
cor(mlb11$runs, mlb11$homeruns)
## [1] 0.7915577
m2 <- lm(runs ~ homeruns, data = mlb11)
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

The equation for the line of regression for the relationship between runs and home runs is: y^ = 415.2389 + 1.8345 * homeruns. It looks like there is a positive and relatively strong relationship between runs and home runs. This is supported by the correlation coefficient which is relatively close to 1 at 0.7916.

Predictions 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 providng 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 reange 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?

Equation for runs and at_bats is: ŷ =−2789.2429+0.6305∗atbats

replace atbats with 5,578

y_hat <- -2789.2429 + 0.6305 * 5578
y_hat
## [1] 727.6861

If the team manager used the regression line, they would estimate 728 runs based on 5578 at-bats. Looking at the data, we see that the Phillies have 5579 at-bats with 713 runs. This means that we can draw a conclusion that the looking at the regression line would give us an overestimate. The residual for this prediction would be 713 - 728, or -15.

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

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 does not appear to be any apparent pattern in the residuals plot. This indicates that the relationship between runs and at-bats is linear.

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 ot 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?

The qq plot looks to be a little bit step wise and the histogram looks a bit skewed to the right, but overall I do think that the nearly normal residuals condition has been met.

Constant variability:

Exercise 8

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

Yes, I do think that based on the plot the variability condition has been met.