load("more/mlb11.RData")
  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?

I would use a scatterplot to display the relationship between two numeric variables.

plot(mlb11$at_bats,mlb11$runs)
abline(lm(runs~at_bats,data=mlb11),col = "red")

The relationship looks weakly linear. A linear model would be reasonable to use at bats to predict the number of runs.

  1. 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.

There is a weak positive linear relationship between at bats and runs. There is one outlier who has over 850 runs while having less than 5550 at bats.

  1. 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_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

The smallest sum of squares is 123721.9.

  1. 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)

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

\[ \hat{y} = 415.2389 + 1.8345 * homeruns \]

According to the slope of the model, as homeruns increase by 1, runs increase by 1.8345.

  1. 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?
unname(m1$coefficients[1] + m1$coefficients[2] * 5578)
## [1] 727.965
mlb11[mlb11$at_bats == 5579,]
##                     team runs at_bats hits homeruns bat_avg strikeouts
## 16 Philadelphia Phillies  713    5579 1409      153   0.253       1024
##    stolen_bases wins new_onbase new_slug new_obs
## 16           96  102      0.323    0.395   0.717

The manager would predict 727.965 runs from a team with 5,578 at-bats. The team with the closest number of at-bats in this dataset is the Philidelphia Phillies with 5,579 at-bats. The Phillies had 713 runs, so this prediction would be an overestimate. The residual is 14.965.

  1. 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

There does not appear to be any particular pattern in the residual plot. This indicates that the linearity assumption is satisfied between runs and at-bats.

  1. 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

The histogram looks closely normal. The points do not deviate too much from the QQ line, which is a good indication of normality of the residuals.

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

Since the variance remains constant as the number of at-bats changes, the constant variability condition appears to be met.

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)

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
plot(mlb11$bat_avg,mlb11$runs)
abline(m3, col = "red")

There seems to be a linear relationship between batting average and runs.

2). 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?

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
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

There appears to be a stronger relationship between batting average and runs comapred to at-bats and runs. The R-squared value of model 3 is 0.6561 while the value for model 1 is 0.3729. Since the R-squared value is higher for batting average than for at-bats, it is a better predictor of runs.

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).

library(corrplot)
## corrplot 0.84 loaded
corrplot.mixed(cor(mlb11[,2:12]))

hist(m3$residuals)

plot(m3$residuals)
abline(h = 0)

qqnorm(m3$residuals)
qqline(m3$residuals)

Out of all the traditional variables, batting average best predicts runs. The correlation value for batting average is highest compared to all other traditional variables. The residuals for this model are normal, random, and have constant variance.

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?

corrplot.mixed(cor(mlb11[,2:12]))

The three new variables do a much better job at predicting runs compared to the traditional statistics. According to the correlation plot, the correlation for these new variables are all over 0.9, which indicates that they are good predictors. New_obs appears to be the best predictor of runs due to its high correlation of 0.97, which is the highest out of all the variables. This result makes sense.

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

m4 = lm(runs~new_obs, data = mlb11)

summary(m4)
## 
## 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

This model is a good model.