Lab 10 - Introduction to linear regression

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

Exercise 1

A scatterplot could be used since they are both numeric. I do not think the relationship looks linear and there for I would not be comfortableusing a linear model to predict the number of runs.

plot(mlb11$runs ~ mlb11$at_bats)

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

Exercise 2

If there were less outliers I could maybe see a trend?

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
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

Exercise 3

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

The smallest sum of squares I got when I ran the code a few times was 123721.9

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

Exercise 4

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

Equation y=415.2389+1.8345*homeruns For every homerun, they make about 1.8 runs, slope is positive.

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

Exercise 5

Using the equation we would end up with the line predicting about 728 runs for 5575 atbats.The phillies had 713 so it would be an overestimate of 15. -15 residual

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

Exercise 6

There doesn’t seem to be a pattern, Could be alinear relationship

hist(m1$residuals)

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

Exercise 7

Seems to be fairly normal

Exercise 8

the plot appears to show constant variability

On your own

Number 1

I will use bat_avg and runs

plot(mlb11$runs ~ mlb11$bat_avg)
m3 <- lm(runs ~ bat_avg, data = mlb11)
abline(m3)

It seems to be fairly linear.

Number 2

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

The relationship between runs and at bats which produces R2 of 37.29% The relationship between runs and bat avg which produces R2 of 65.61%.Therefore, bat_avg predicts runs more accurately than atbats.

Number 3

cor(mlb11$stolen_bases,mlb11$runs)
## [1] 0.05398141
cor(mlb11$wins,mlb11$runs)
## [1] 0.6008088
cor(mlb11$bat_avg,mlb11$runs)
## [1] 0.8099859
cor(mlb11$hits,mlb11$runs)
## [1] 0.8012108
cor(mlb11$strikeouts,mlb11$runs)
## [1] -0.4115312

Batting average best predicts runs.

Number 4

cor(mlb11$runs, mlb11$new_slug)
## [1] 0.9470324
cor(mlb11$runs, mlb11$new_obs)
## [1] 0.9669163
cor(mlb11$runs, mlb11$new_onbase)
## [1] 0.9214691
summary(lm(runs ~ new_slug, data = mlb11))
## 
## 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(lm(runs ~ new_obs, data = mlb11))
## 
## 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
summary(lm(runs ~ new_onbase, data = mlb11))
## 
## 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

The highest rsquared would be with obs, therefore it would be the best predictor of runs.

Number 5

m4 <- lm(runs ~ new_obs, data = mlb11)
plot(m4$residuals ~ mlb11$bat_avg)
abline(h = 0, lty = 3)

hist(m4$residuals)

qqnorm(m4$residuals)
qqline(m4$residuals)