Lab 7

Exercises

Load data:

download.file("http://www.openintro.org/stat/data/mlb11.RData", destfile = "mlb11.RData")
load("mlb11.RData")
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
## 7       New York Yankees  867    5518 1452      222   0.263       1138
## 8      Milwaukee Brewers  721    5447 1422      185   0.261       1083
## 9       Colorado Rockies  735    5544 1429      163   0.258       1201
## 10        Houston Astros  615    5598 1442       95   0.258       1164
## 11     Baltimore Orioles  708    5585 1434      191   0.257       1120
## 12   Los Angeles Dodgers  644    5436 1395      117   0.257       1087
## 13          Chicago Cubs  654    5549 1423      148   0.256       1202
## 14       Cincinnati Reds  735    5612 1438      183   0.256       1250
## 15    Los Angeles Angels  667    5513 1394      155   0.253       1086
## 16 Philadelphia Phillies  713    5579 1409      153   0.253       1024
## 17     Chicago White Sox  654    5502 1387      154   0.252        989
## 18     Cleveland Indians  704    5509 1380      154   0.250       1269
## 19  Arizona Diamondbacks  731    5421 1357      172   0.250       1249
## 20     Toronto Blue Jays  743    5559 1384      186   0.249       1184
## 21       Minnesota Twins  619    5487 1357      103   0.247       1048
## 22       Florida Marlins  625    5508 1358      149   0.247       1244
## 23    Pittsburgh Pirates  610    5421 1325      107   0.244       1308
## 24     Oakland Athletics  645    5452 1330      114   0.244       1094
## 25        Tampa Bay Rays  707    5436 1324      172   0.244       1193
## 26        Atlanta Braves  641    5528 1345      173   0.243       1260
## 27  Washington Nationals  624    5441 1319      154   0.242       1323
## 28  San Francisco Giants  570    5486 1327      121   0.242       1122
## 29      San Diego Padres  593    5417 1284       91   0.237       1320
## 30      Seattle Mariners  556    5421 1263      109   0.233       1280
##    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
## 7           147   97      0.343    0.444   0.788
## 8            94   96      0.325    0.425   0.750
## 9           118   73      0.329    0.410   0.739
## 10          118   56      0.311    0.374   0.684
## 11           81   69      0.316    0.413   0.729
## 12          126   82      0.322    0.375   0.697
## 13           69   71      0.314    0.401   0.715
## 14           97   79      0.326    0.408   0.734
## 15          135   86      0.313    0.402   0.714
## 16           96  102      0.323    0.395   0.717
## 17           81   79      0.319    0.388   0.706
## 18           89   80      0.317    0.396   0.714
## 19          133   94      0.322    0.413   0.736
## 20          131   81      0.317    0.413   0.730
## 21           92   63      0.306    0.360   0.666
## 22           95   72      0.318    0.388   0.706
## 23          108   72      0.309    0.368   0.676
## 24          117   74      0.311    0.369   0.680
## 25          155   91      0.322    0.402   0.724
## 26           77   89      0.308    0.387   0.695
## 27          106   80      0.309    0.383   0.691
## 28           85   86      0.303    0.368   0.671
## 29          170   71      0.305    0.349   0.653
## 30          125   67      0.292    0.348   0.640

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

I would use scatterplot to display the relationship between runs and one of the other numerical variables.

plot(mlb11$runs ~ mlb11$at_bats, main = "Relationship between Runs and atBats", xlab = "At Bats", ylab = "Runs")

The relationship looks moderately linear but not strong enough to be able to comfortably use a linear model to predict the number of runs.

Since the relationship is linear we can quanitfy the strength of the relationship with the correlation coefficient

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 relationship between runs and at bats can be considered positive but moderately strong as the correlation coefficient 0.610627 turns out to be far below from +1.

we can also clearly spot several positive outliers in the plot such as a team with 5518 and 5600 at bats.

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

The smallest sum of squares that i got after running plot_ss function several times is 125153 with the coefficients x -> 0.5882 Intercept -> -2549.4628

The neigboring value deviate from the smallest value by around 4000 - 5000

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?

plot(mlb11$runs ~ mlb11$at_bats, main = "Relationship between Runs and Home runs", xlab = "Home Runs", ylab = "Runs")

Correlation Coefficient

cor(mlb11$runs, mlb11$homeruns)

## [1] 0.7915577

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 of the regression line for the relationship between Run and Home Runs

y^ = 415.2389 + 1.8345 * homeruns

By looking at the plot we can say that the relationship between runs and home runs is linear positive and relatively strong as the correlation coefficient 0.7916 is closer to +1

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?

Least Square Regression line for runs vs at_bats

y^ = -2789.2429 + 0.6305 * atbats

If atbats is 5,578

Predicted Runs is y^ = -2789.2429 + 0.6305 * 5578 y^ = 727.6861

The estimated number of runs for 5578 at bats based on the linear regression formula above is 728. A team with 5578 at bats cannot be found in the data but we can see the team Philadelphia Phillies has 5579 at bats with 713 runs. Therefore we can conclude that the model may have overestimated the runs for a team with 5578 at bats by 728 - 713 = 15 runs.

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?

m1 <- lm(runs ~ at_bats, data = mlb11)
plot(m1$residuals ~ mlb11$at_bats)
abline(h = 0, lty = 3)

Based on the plot we can clearly say that there is no apparent pattern in the distribution as the numbers appear to be scattered unevenly around the dashed line and appear to be skewed. But it can be considered as a linear relationship.

Exercise 7 Based on the histogram and the normal probability plot, does the nearly normal residuals condition appear to be met?

m1 <- lm(runs ~ at_bats, data = mlb11)
hist(m1$residuals)

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

Looking at the histogram and the plot I would say that the nearly normal residuals condition has been met.

Exercise 8 Based on the plot in (1), does the constant variability condition appear to be met?

The variation of points around the least squares line appear to be reasonably constant thus an inference can be made that the constant variability condition has been met.

Exercise 9 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?

Lets us take bat_avg as the predictor variable as I think it might also be a good predictor of runs.

plot(mlb11$runs ~ mlb11$bat_avg, main = "Relationship between Runs and Batting Avg", xlab = "Batting Avg", ylab = "Runs")
m3 <- lm(runs ~ bat_avg, data = mlb11)
abline(m3)

Correlation Coefficient:

cor(mlb11$runs, mlb11$bat_avg)

## [1] 0.8099859

Linear Regression Line Formula: y^ = -642 + 5242.2 * bat_avg

Based on the plot, linear model statistics and correlation coefficient for the relationship between runs and batting average it is evident that the relationship is positive, linear and relatively strong.

Exercise 10 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?

R2 is the percentage of the variance in the dependent variable that can be explained by a linear model. R2 is always in the range between 0% - 100% and the higher the value the better the linear model explains the dependant variable and lower the value weaker the predictability of the dependant variable.

Let m1 be the model for the relationship between runs and at bats which produces R2 of 37.29% Let m2 be the model for the relationship between runs and bat avg which produces R2 of 65.61%

Looking at the R2s of both models we can clearly see that the the R2 value of the model m2 is far greater than that of the model m2 so it is clear that the variable bat_avg predicts runs better than at bats.

Exercise 11 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).

After running the summary statistics for all the variables, the variable which best predicts the runs based on R2 happened to be bat_avg

plot(mlb11$runs ~ mlb11$bat_avg, main = "Relationship between Runs and Batting Avg", xlab = "Batting Avg", ylab = "Runs")
m4 <- lm(runs ~ bat_avg, data = mlb11)
abline(m4)

hist(m4$residuals)

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

Correlation Coefficient:

cor(mlb11$runs, mlb11$bat_avg)

## [1] 0.8099859

Summary statistics:

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

Exercise 12 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?

Examining the three newer variables: new_onbase, new_slug and new_obs

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

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

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

plot(mlb11$runs ~ mlb11$bat_avg, main = "Relationship between Runs and Batting Avg", xlab = "Batting Avg", ylab = "Runs")
m4 <- lm(runs ~ bat_avg, data = mlb11)
abline(m4)

After examining the summary statistics and correlation coefficients of all three new predictors new_onbase, new_slug and new_obs, the relationship between runs and new_obs variable has the highest R2 and coefficient correlation values and appears to be the best and most effective predictor of the runs.

Exercise 13 Check the model diagnostics for the regression model with the variable you decided was the best predictor for runs.

Model diagnostics for the regression model with the best predictor bat_avg for runs

m5 <- lm(runs ~ new_obs, data = mlb11)

(1) Linearity:

The relationship looks linear based on a residual plot as the variability of residuals is approximately constant across the distribution but does not indicate any curvatures or any indication of non-normality.

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

(2) Nearly normal residuals:

If the residuals are approximately normaly distributed then the normal quantile-quantile plot of the residuals will result in an approximately straight line.

As you can clearly see the normal quantile-quantile plot of the residuals indicates a pretty straight line so we can safely say that the residuals are approximately normaly distributed and the model meets the nearly normal residuals condition.

hist(m5$residuals)

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

(3) Constant variability:

Based on the plot the variability of points around the least squares line remains roughly constant so the condition constant variability has been met.