DATA606_Lab7 Introduction to linear regression

library("DATA606")
library("ggplot2")
download.file("http://www.openintro.org/stat/data/mlb11.RData", destfile = "mlb11.RData")
load("mlb11.RData")
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   0.725

Question 1

Choose two 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?

Answer 1

I chose runs and bat_avg to see if it is a good predictor. From the plot and summary statistics below it looks to me that the two variables fit a liner model.

y = b0 + b1X = -642.8+5242.2*bat_avg

mlb11.lm1  <- lm(runs ~ bat_avg, data = mlb11)
plot(mlb11$runs ~ mlb11$bat_avg, main = "Relationship RUNS vs BAT_AVG")
abline(mlb11.lm1 )

Question 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?

mlb11.lm2  <- lm(runs ~ at_bats, data = mlb11)
plot(mlb11$runs ~ mlb11$at_bats, main = "Relationship RUNS vs AT_BATS")
abline(mlb11.lm2 )

summary(mlb11.lm1)
## 
## 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
summary(mlb11.lm2)
## 
## 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

Answer 2

The relationship between runs and bat_avg seems to be stronger than that of runs and at_bats. The R-squared at_bats is 0.3729 while bat_avg is 0.6561. Given the higher R-squared, I could say the model using bat_avg is better predictor of runs.

Question 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).

Answer 3

The bat_avg variable seems the best predict for runs. The next best variable is hits for predictor.

mlb11.lm2=lm(runs~bat_avg,data=mlb11)
 plot(mlb11$bat_avg,mlb11$runs,xlab="Bat_Avg",ylab="Runs",main="Batting Avg Vs runs") 
 abline(mlb11.lm2)

#Hits
mlb11.lm3=lm(runs~hits,data=mlb11)
plot(mlb11$hits,mlb11$runs,xlab="Hits",ylab="Runs",main="HITS Vs RUNS") 
abline(mlb11.lm3)

summary(mlb11.lm1)
## 
## 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
summary(mlb11.lm3)
## 
## 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
 sum(mlb11.lm1$residuals^2) #Sum of squares -> bat_Avg
## [1] 67849.52
 sum(mlb11.lm2$residuals^2) #Sum of squares -> at_bats
## [1] 67849.52
 sum(mlb11.lm3$residuals^2) #Sum of squares -> Hits
## [1] 70638.75

Question 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?

Answer 4

Seems like newer variable predict runs better than old variables. The R2 value for newer variables are higher than that of old variable and the sum of square of residuals of newer variable is less than that of old variable. Given that the newer variables represent more advanced statistics of baseball it does make sense that they are better predictor of runs

n1=lm(runs~new_onbase,data=mlb11)
n2=lm(runs~new_slug,data=mlb11)
n3=lm(runs~new_obs,data=mlb11)
par(mfrow=c(1,3))
plot(mlb11$new_onbase,mlb11$runs,xlab="onbase",ylab="runs",main="onbase Vs runs") 
abline(n1)
plot(mlb11$new_slug,mlb11$runs,xlab="slug",ylab="runs",main="SLUG Vs runs") 
abline(n2)
plot(mlb11$new_obs,mlb11$runs,xlab="new_obs",ylab="runs",main="OBS Vs runs") 
abline(n3)

summary(n1)
## 
## 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
summary(n2)
## 
## 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
summary(n3)
## 
## 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
sum(n1$residuals^2) 
## [1] 29768.7
sum(n2$residuals^2) 
## [1] 20345.54
sum(n3$residuals^2) 
## [1] 12837.65

Question 5

Check the model diagnostics for the regression model with the variable you decided was the best predictor for runs. ####Answer 5 new_obs is the best predictor for runs. The model built using new_obs has R-squared value of 0.93 which is higher than the models built using other variable. The residual sum of errors is 20345.54 which is lowest compared to models built using other variables

summary(n3)
## 
## 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
sum(n3$residuals^2)
## [1] 12837.65