Lab 10 - Intro to Linear Regression

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

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 team’s at_bats, would you be comfortable using a linear model to predict the number of runs?

We could look at a scatterplot to compare the two variables.

plot(mlb11$at_bats, mlb11$runs)

The plot looks roughly linear with a positive correlation.

cor(mlb11$runs, mlb11$at_bats)

## [1] 0.610627

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.

The two variables have a positive correlation because as at_bats increases, so does the number of runs. The correlation is not very strong because the correlation is less than 1 and several of the points do not follow a linear pattern, there are a lot of outliers.

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 -

Using plot_ss, choose a line that does a good job of minimizing the sum of squares. Run the function several times. What waqs the smallest sum of squares that you got? How does it compare to your neighbors?

I was only able to get a sum of squares of 123721.9, it never prompted me to select points.

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 -

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?

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

y = 415.234 + 1.835(homeruns)

The slope lets us know that there is a strong correlation between the number of home runs and the team’s success.

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?

The manager would predict about 725 runs. This is an overestimate because from the data, the Phillies had 5579 at_bats but 713 runs. So the residual would be 14.

plot(m1$residuals ~ mlb11$at_bats)
abline(h = 0, lty = 3)

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?

There is no apparent pattern which confirms linearity.

hist(m1$residuals)

qqnorm(m1$residuals)
qqline(m1$residuals)

Exercise 7 -

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

Although the histogram shows some slight right skewing, the QQ plot shows the data to be approximately normal.

Exercise 8 -

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

Yes.

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 ~ hits, data = mlb11)
plot(mlb11$runs ~ mlb11$hits)
abline(m3)

I compared the number of hits to the numberof runs and there is a strong linear relationship.

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?

summary(m3)

## 
## 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

Using hits as a predictor of runs is much better than using at_bats which is evident by the higher R2 value of 0.64 compared to 0.37

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 ~ bat_avg, data = mlb11)
summary(m4)

## 
## 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$runs ~ mlb11$bat_avg)
abline(m4)

hist(m4$residuals)

plot(m4$residuals ~ mlb11$bat_avg)
abline(h = 0, lty = 3)

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

Batting average seems to predict the number of runs best with an R2 value of 0.65. We can also confirm that the linear model is reliable because it is linear which is made clear from the QQ plot, has nearly normal residuals which is confirmed through the histogram, and has constant variability.

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_obs, data = mlb11)
summary(m5)

## 
## 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

Overall these 3 new variables are much better at predicting runs. However, on base percentage plus slugging seems to be the best predictor of the number of runs with an R2 value of 0.93.

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

plot(mlb11$runs ~ mlb11$new_obs)
abline(m5)

hist(m5$residuals)

plot(m5$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)

qqnorm(m5$residuals)
qqline(m5$residuals)

Lab 10 - Intro to Linear Regression

Adeola Animasahun

11/15/2019

Exercise 1 -

We could look at a scatterplot to compare the two variables.

The plot looks roughly linear with a positive correlation.

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.

The two variables have a positive correlation because as at_bats increases, so does the number of runs. The correlation is not very strong because the correlation is less than 1 and several of the points do not follow a linear pattern, there are a lot of outliers.

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 waqs the smallest sum of squares that you got? How does it compare to your neighbors?

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?

y = 415.234 + 1.835(homeruns)

The slope lets us know that there is a strong correlation between the number of home runs and the team’s success.

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?

The manager would predict about 725 runs. This is an overestimate because from the data, the Phillies had 5579 at_bats but 713 runs. So the residual would be 14.

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?

There is no apparent pattern which confirms linearity.

Exercise 7 -

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

Although the histogram shows some slight right skewing, the QQ plot shows the data to be approximately normal.

Exercise 8 -

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

Yes.

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?

I compared the number of hits to the numberof runs and there is a strong linear relationship.

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?

Using hits as a predictor of runs is much better than using at_bats which is evident by the higher R2 value of 0.64 compared to 0.37

Batting average seems to predict the number of runs best with an R2 value of 0.65. We can also confirm that the linear model is reliable because it is linear which is made clear from the QQ plot, has nearly normal residuals which is confirmed through the histogram, and has constant variability.

Overall these 3 new variables are much better at predicting runs. However, on base percentage plus slugging seems to be the best predictor of the number of runs with an R2 value of 0.93.

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

The plot of the residuals confirms linearity and constant variability. The histogram and QQ plot show nearly normal residuals.