Introduction to Linear Regression

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

Exercise 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?

runs <- mlb11$runs
at_bats <- mlb11$at_bats
plot(runs ~ at_bats, ylab = "Runs", xlab = "At-Bats")

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

Answer: I would use a scatterplot to display the relationship between runs and one of the other numerical variables. Visually, the relationship between runs and at-bats does not look linear and therefore I would not be comfortable using a linear model to predict the number of runs using a team’s at-bats. However, the correlation coefficient is 0.61 which indicates that there is a somewhat positive linear relationship between runs and at-bats. Therefore, you could use a linear model to predict runs using a teams at-bats but not with perfect accuracy.

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.

Answer: There is a positive and moderately strong correlation between the variables as the coefficient is positive and is closer to 1 than it is to zero (0.61). This means that you can expect that in general, the greater the value for an at-bat, the larger the value of a run will be. It is difficult to describe the form of the scatterplot as even though there is a somewhat linear relationship, the points are broadly scattered.

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?

plot(runs ~ at_bats, ylab = "Runs", xlab = "At-Bats")

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

Answer: The sum of squares I got was 123721.9.

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?

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
homeruns <- mlb11$homeruns
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

Answer: The equation of the regression line is y^ = 415.2389 + 1.8345(homeruns), where 62.6% of the variability in runs is explained by homeruns. The positive slope tells us that there is a positive linear association between the success of a team and its home runs.

plot(mlb11$runs ~ mlb11$at_bats)
abline(m1)

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?

Answer: The team manager would predict that a team with 5,578 at-bats would have about 720 runs. The residual for this prediction is 66.47, and as it is positive then the regression line is an overestimate (as residual = observed - predicted).

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?

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

Answer: There appears to be a random pattern in the residual plot which supports a linear model between runs and at-bats.

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

hist(m1$residuals)

qqnorm(m1$residuals)
qqline(m1$residuals)  # adds diagonal line to the normal prob plot

Answer: In observing the plots, it does appear that the nearly normal residuals condition is met.

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

Answer: Yes, as the residuals appear to be scattered randomly around zero.

On Your Own 1: 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)
plot(mlb11$runs ~ mlb11$bat_avg)
abline(m3)

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

Answer: There does appear to be a linear relationship between runs and batting average.

On Your Own 2: How 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?

Answer: The R2 value of the summary between runs and batting average is 0.656 while the R2 value of the summary between runs and at-bats is 0.373. It appears that batting average predicts runs better than at-bats as batting average is responsible for the larger value (65.6%) of the variability in runs.

On Your Own 3: 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).

m4 <- lm(runs ~ hits, data = mlb11)
plot(mlb11$runs ~ mlb11$hits)
abline(m4)

summary(m4)
## 
## 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-07

Answer: Hits best predicts runs as it has a visiually positive linear relationship with runs. It also has the largest R2 out of the other variables meaning that it accounts for the most variability in runs compared to the others.

On Your Own 4: 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?

m5 <- lm(runs ~ new_onbase, data = mlb11)
plot(mlb11$runs ~ mlb11$new_onbase)
abline(m5)

summary(m5)
## 
## 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
m6 <- lm(runs ~ new_slug, data = mlb11)
plot(mlb11$runs ~ mlb11$new_slug)
abline(m6)

summary(m6)
## 
## 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
m7 <- lm(runs ~ new_obs, data = mlb11)
plot(mlb11$runs ~ mlb11$new_obs)
abline(m7)

summary(m7)
## 
## 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

Answer: It does appear that the new variables are more effective at predicting runs than the old variables as they create a more linear distribution and higher R2 values. Using these same observations of all ten variables, new_obs is the best predictor of runs. I am unsure of what new_obs refers to in baseball.

On Your Own 5: Check the model diagnostics for the regression model with the variable you decided was the best predictor for runs.

plot(m7$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)

Answer: It does appear that there is a random pattern (no pattern) to the residuals plot for the new_obs variable plot. This suggests that there really is a linear relationship between runs and new_obs.