Introduction to linear regression

The data

Let’s load up the data for the 2011 season.

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?

library(ggplot2)

load("more/mlb11.RData")

ggplot(mlb11, aes(x = runs, y = at_bats)) +
  geom_point(color='red') + 
  xlab('No of runs') + 
  ylab('No of at bats') +
  ggtitle('runs vs at_bats')

I used a scatterplot and we can say that there appears to be a linear relationship between at_bats and runs variables.

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

Sum of squared residuals

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.

Answer -There is a positive relationship between runs and at_bats - 0.610627 means a positive correlation but a weak one -There appears to be a great deal of deviation amongst the teams and the points are not centered around a specific value

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.

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?

Answer:Smallest sum of squares was 123721.9

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)

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:

\[ \hat{y} = -2789.2429 + 0.6305 * atbats \]

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 ~ at_bats, data = mlb11)

summary(m2)

## 
## 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

y^=-2789.2429+0.6305∗at_bats

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)

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?

at_bats <- 5578
Prediction<- -2789.2429 + 0.6305 * at_bats
Prediction

## [1] 727.6861

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? Answer:Based on the scatterplot, there is no determinable pattern. This does not indicate the linearity of the relationship between runs and at-bats

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? Answer:Based on the histogram and normal probability plot, the residuals appear to be normal.

Constant variability:

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

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

## 
## 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

cor(mlb11$runs, mlb11$bat_avg)

## [1] 0.8099859

ggplot(mlb11, aes(x = bat_avg, y = runs)) +
  geom_point() +
  geom_smooth(method = 'lm')

There is a positive linear relationship between bat_avg 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

m1 = lm(runs ~ at_bats,      data = mlb11)
m3 = lm(runs ~ stolen_bases, data = mlb11)

rSquaredAtBats  = summary(m1)$r.squared
rSquaredStBases = summary(m3)$r.squared

rSquaredAtBats

## [1] 0.3728654

rSquaredStBases

## [1] 0.002913993

The R2 value for variable at_bats is 0.3728654 and the R2 value for variable stolen_bases is 0.002914.R2 value of at_bats is higher than R2 value of stolen_bases, which means we can at_bats is a better predictor than stolen_bases.

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).

ggplot(mlb11, aes(x = bat_avg, y = runs)) +
  geom_point() +
  geom_smooth(method = 'lm')

m4 = lm(runs ~ bat_avg, data=mlb11)
rSquaredBatAvg  = summary(m4)$r.squared
rSquaredBatAvg

## [1] 0.6560771

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?

model1 <- lm(formula = runs ~ new_onbase, mlb11)

ggplot(mlb11, aes(x=new_onbase, y=runs)) +
  geom_point(color = "red") +
  geom_line(aes(x=new_onbase, y=predict(model1, newdata = mlb11)), color = "blue")

summary(model1)

## 
## 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

new_model1 <- lm(formula = runs ~ new_slug, mlb11)

ggplot(mlb11, aes(x=new_slug, y=runs)) +
  geom_point(color = "red") +
  geom_line(aes(x=new_slug, y=predict(new_model1, newdata = mlb11)), color = "blue")

summary(new_model1)

## 
## 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

obs_model <- lm(formula = runs ~ new_obs, mlb11)

ggplot(mlb11, aes(x=new_obs, y=runs)) +
  geom_point(color = "red") +
  geom_line(aes(x=new_obs, y=predict(obs_model, newdata = mlb11)), color = "blue")

summary(obs_model)

## 
## 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

From within these 3 new variables, the maximum R-quared and the minimum p-value are for the variable : new_obs, which are 0.9349 and 2.2e-16 respectively. Hence the best predictor of runs from all the variables is the variable : new_obs.

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

new_obs_model <- lm(runs ~ new_obs, data = mlb11)
ggplot(mlb11, aes(x = new_obs, y = new_obs_model$residuals)) +
  geom_point() +
  geom_hline(yintercept = 0, linetype = 3)

The relationship does not indicate any curvatures or abnormalities.It looks linear.