Introduction to Linear Regression

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

Exercise 1

I would use a scatterplot to display the relationship between runs and another numerical variable. The function for this in R is “plot(x,y)”. The relationship looks relatively linear, and the correlation coefficient indicates the same, so I would feel comfortable using a linear model to predict runs given an adequate margin of error.

cor(mlb11$runs, mlb11$at_bats)

## [1] 0.610627

plot(mlb11$at_bats, mlb11$runs)

Exercise 2 and 3

Form: Linear
Direction: Positive
Strength: Moderate

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

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

equation: y_hat = 415.239 + 1.835*homeruns

The slope tells us that homeruns and runs are positively correlated. I don’t understand baseball so I can’t interpret what exactly this means lol. Maybe its just saying that the measures of success of the teams are related directly?

plot_ss(x = mlb11$homeruns, y = mlb11$runs, showSquares = TRUE)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##     415.239        1.835  
## 
## Sum of Squares:  73671.99

m2 <- lm(runs ~ homeruns, data = mlb11)

Exercise 5

The data shows 713 runs at 5579 at-bats, but the model could predict it to be higher by roughly 14.686 runs than it may actually be.

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

713 - (-2789.2429+0.6305*5578)

## [1] -14.6861

Exercise 6

There appears to be no pattern in the plot, and the correlation coefficient confirms this. You could use a linear model with slope zero to model the relationship but that wouldn’t mean much.

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

cor(m1$residuals, mlb11$at_bats)

## [1] 1.386089e-15

hist(m1$residuals)

shapiro.test(m1$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  m1$residuals
## W = 0.96144, p-value = 0.337

Exercise 7

Yes, it does and this is evident from the p-value of 0.337 from the Shapiro-Wilks test.

Exercise 8

The data does not appear to cluster, at least visually, so I feel comfortable saying that it is constantly variable.

On Your Own

At a glance there doesn’t appear to be some sort of linear relationship to me. The relationship is inverse compared to that of runs and at_bats (there is a negative correlation). The R^2 value is smaller, at 13.97% of the variability of runs being explained by strikeouts (as compared to 37.3%). My variable does not seem to predict runs better than at_bats because both the R^2 and correlation coefficient are lower.The variable bat_avg appears to predict runs the best with a correlation coefficient of 0.801 (out of the traditional variables). Out of the new variables, new_obs seems to predict runs the best, with a correlation coefficient of 0.967, the best of all the variables. The conditions of constant variability and normal distribution of residuals appear to be met based on the scatterplot and shapiro-wilks test of the residuals of model 4 which uses new_obs as a predictor of runs, and therefore I believe that this is the most reliable linear model for runs in the dataset.

cor(mlb11$runs, mlb11$strikeouts)

## [1] -0.4115312

plot_ss(x = mlb11$strikeouts, y = mlb11$runs, showSquares = TRUE)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##   1054.7342      -0.3141  
## 
## Sum of Squares:  163870.1

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

## 
## Call:
## lm(formula = runs ~ strikeouts, data = mlb11)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -132.27  -46.95  -11.92   55.14  169.76 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1054.7342   151.7890   6.949 1.49e-07 ***
## strikeouts    -0.3141     0.1315  -2.389   0.0239 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 76.5 on 28 degrees of freedom
## Multiple R-squared:  0.1694, Adjusted R-squared:  0.1397 
## F-statistic: 5.709 on 1 and 28 DF,  p-value: 0.02386

plot(mlb11$runs ~ mlb11$strikeouts)
abline(m3)

plot(m3$residuals ~ mlb11$strikeouts)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 0

cor(m3$residuals, mlb11$at_bats)

## [1] 0.4607368

hist(m3$residuals)

shapiro.test(m3$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  m3$residuals
## W = 0.96885, p-value = 0.5083

cor(mlb11$runs, mlb11$hits)

## [1] 0.8012108

cor(mlb11$runs, mlb11$bat_avg)

## [1] 0.8099859

cor(mlb11$runs, mlb11$stolen_bases)

## [1] 0.05398141

cor(mlb11$runs, mlb11$wins)

## [1] 0.6008088

cor(mlb11$runs, mlb11$new_onbase)

## [1] 0.9214691

cor(mlb11$runs, mlb11$new_slug)

## [1] 0.9470324

cor(mlb11$runs, mlb11$new_obs)

## [1] 0.9669163

plot_ss(x = mlb11$new_obs, y = mlb11$runs, showSquares = TRUE)

## Click two points to make a line.
                                
## Call:
## lm(formula = y ~ x, data = pts)
## 
## Coefficients:
## (Intercept)            x  
##      -686.6       1919.4  
## 
## Sum of Squares:  12837.66

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

plot(mlb11$runs ~ mlb11$new_obs)
abline(m4)

plot(m4$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 0

cor(m4$residuals, mlb11$at_bats)

## [1] 0.01368748

hist(m4$residuals)

shapiro.test(m4$residuals)

## 
##  Shapiro-Wilk normality test
## 
## data:  m4$residuals
## W = 0.98068, p-value = 0.8434