title: “Introduction to linear regression”

author: “Sufian”

output:

html_document:

css: ./lab.css

highlight: pygments

theme: cerulean

pdf_document: default

Rpub links: http://rpubs.com/ssufian/548411

Batter up

The movie Moneyball focuses on the “quest for the secret of success in baseball”. It follows a low-budget team, the Oakland Athletics, who believed that underused statistics, such as a player’s ability to get on base, betterpredict the ability to score runs than typical statistics like home runs, RBIs (runs batted in), and batting average. Obtaining players who excelled in these underused statistics turned out to be much more affordable for the team.

In this lab we’ll be looking at data from all 30 Major League Baseball teams and examining the linear relationship between runs scored in a season and a number of other player statistics. Our aim will be to summarize these relationships both graphically and numerically in order to find which variable, if any, helps us best predict a team’s runs scored in a season.

The data

Let’s load up the data for the 2011 season.

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

head(mlb11,n=3)
##             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
##   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

In addition to runs scored, there are seven traditionally used variables in the data set: at-bats, hits, home runs, batting average, strikeouts, stolen bases, and wins. There are also three newer variables: on-base percentage, slugging percentage, and on-base plus slugging. For the first portion of the analysis we’ll consider the seven traditional variables. At the end of the lab, you’ll work with the newer variables on your own.

  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?
library(ggplot2)

# Same, but with different colors and add regression lines
ggplot(mlb11, aes(x=mlb11$at_bats, y=mlb11$runs)) +
    geom_point(shape=1) +
    scale_colour_hue(l=50) + # Use a slightly darker palette than normal
    geom_smooth(method=lm,   # Add linear regression lines
                se=FALSE)    # Don't add shaded confidence region

ans:

  1. I use scatter plot ‘runs’ as the response variables using the other variables as predictor variables

or (indpendent variables) for instance, the ‘at_bats’.

  1. Yes, there is a linear relationship between at_bats vs. runs

  2. This is a toughie as the saying goes “Looks can be deceiving”; looking at the dignostics:

and the residuals are all normal.

Therefore I would be OK with using it as linear model predictive tool despite some mis-givings in the

visual residual tests

mlb.lm = lm(runs ~ at_bats, data=mlb11) 
mlb.stdres = rstandard(mlb.lm)

qqnorm(mlb.stdres, 
     ylab="Standardized Residuals", 
     xlab="Normal Scores", 
     main="Q-Q Plot of residuals") 
 qqline(mlb.stdres) 

library(MASS)

fit <- lm(runs ~ at_bats, data=mlb11) 
sresid <- studres(fit) 
hist(sresid, freq=FALSE, 
   main="Distribution of Studentized Residuals")
xfit<-seq(min(sresid),max(sresid),length=40) 
yfit<-dnorm(xfit) 
lines(xfit, yfit) 

#shapiro test 1
shapiro.test(mlb11$runs) #Shapiro test for normality; making sure the parrents are really normal
## 
##  Shapiro-Wilk normality test
## 
## data:  mlb11$runs
## W = 0.94815, p-value = 0.1508
#shapiro test 2
shapiro.test(mlb11$at_bats)
## 
##  Shapiro-Wilk normality test
## 
## data:  mlb11$at_bats
## W = 0.94797, p-value = 0.1491
#Shapiro test 3 - to make sure if residuals are actually normal
sp_wlk = lm(runs ~ at_bats, data = mlb11)

shapiro.test(sp_wlk$residuals)
## 
##  Shapiro-Wilk normality test
## 
## data:  sp_wlk$residuals
## W = 0.96144, p-value = 0.337

Shapiro Wilk test to make sure Wind is really Normal

Because P-values are greater than 0.05, we can be 95% confident that run & at_bats data is normal for

sure (this is only checking parents and not the residuals)

My assumption is if the parents are normal, the residuals should behave somewhat like the parents

And the last Shapiro Wilt test, ascertained that indeed the residuals were actually normal as well

If the relationship looks linear, we can quantify the strength of the relationship with the correlation coefficient.

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

Sum of squared residuals

Think back to the way that we described the distribution of a single variable. Recall that we discussed characteristics such as center, spread, and shape. It’s also useful to be able to describe the relationship of two numerical variables, such as runs and at_bats above.

  1. 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.

ans:

The relationship between the variables at_bats and runs appears to be positive and linear. The strength

of the relationship is weak to moderate. There are few outliers with at_bats above 5500 and runs above

  1. Other than that there are no unusual observations.

Just as we used the mean and standard deviation to summarize a single variable, we can summarize the relationship between these two variables by finding the line that best follows their association. Use the following interactive function to select the line that you think does the best job of going through the cloud of points.

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

After running this command, you’ll be prompted to click two points on the plot to define a line. Once you’ve done that, the line you specified will be shown in black and the residuals in blue. Note that there are 30 residuals, one for each of the 30 observations. Recall that the residuals are the difference between the observed values and the values predicted by the line:

\[ e_i = y_i - \hat{y}_i \]

The most common way to do linear regression is to select the line that minimizes the sum of squared residuals. To visualize the squared residuals, you can rerun the plot command and add the argument showSquares = TRUE.

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

Note that the output from the plot_ss function provides you with the slope and intercept of your line as well as the sum of squares.

  1. 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 linear model

It is rather cumbersome to try to get the correct least squares line, i.e. the line that minimizes the sum of squared residuals, through trial and error. Instead we can use the lm function in R to fit the linear model (a.k.a. regression line).

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

The first argument in the function lm is a formula that takes the form y ~ x. Here it can be read that we want to make a linear model of runs as a function of at_bats. The second argument specifies that R should look in the mlb11 data frame to find the runs and at_bats variables.

The output of lm is an object that contains all of the information we need about the linear model that was just fit. We can access this information using the summary function.

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

Let’s consider this output piece by piece. First, the formula used to describe the model is shown at the top. After the formula you find the five-number summary of the residuals. The “Coefficients” table shown next is key; its first column displays the linear model’s y-intercept and the coefficient of at_bats. With this table, we can write down the least squares regression line for the linear model:

\[ \hat{y} = -2789.2429 + 0.6305 * atbats \]

One last piece of information we will discuss from the summary output is the Multiple R-squared, or more simply, \(R^2\). The \(R^2\) value represents the proportion of variability in the response variable that is explained by the explanatory variable. For this model, 37.3% of the variability in runs is explained by at-bats.

  1. 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

Linear equation: runs=415.2388849+1.8345416∗homeruns

Prediction and prediction errors

Let’s create a scatterplot with the least squares line laid on top.

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

The function abline plots a line based on its slope and intercept. Here, we used a shortcut by providing the model m1, which contains both parameter estimates. This line can be used to predict \(y\) at any value of \(x\). When predictions are made for values of \(x\) that are beyond the range of the observed data, it is referred to as extrapolation and is not usually recommended. However, predictions made within the range of the data are more reliable. They’re also used to compute the residuals.

  1. 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?
atbats <- 5578
yPredicted <- -2789.2429 + 0.6305 * atbats
yPredicted
## [1] 727.6861
library(dplyr)
## 
## Attaching package: 'dplyr'
## The following object is masked from 'package:MASS':
## 
##     select
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
#Find the actuals

actual_df <- mlb11 %>% 
  filter(at_bats == 5579)
yactual <- actual_df$runs

yactual
## [1] 713
residual <- yactual - yPredicted

residual
## [1] -14.6861

ans:

The model predicted higher than the actual; therefore, it was an over-estimation by 14.68

Model diagnostics

To assess whether the linear model is reliable, we need to check for (1) linearity, (2) nearly normal residuals, and (3) constant variability.

Linearity: You already checked if the relationship between runs and at-bats is linear using a scatterplot. We should also verify this condition with a plot of the residuals vs. at-bats. Recall that any code following a # is intended to be a comment that helps understand the code but is ignored by R.

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

  1. Is there any apparent pattern in the residuals plot? What does this indicate about the linearity of the relationship between runs and at-bats?

ans:

Like I said, the the previous question, “Looks can be deceiving”. There appears to be more points to

the left but then the number of points above and below zero seems random, which indicate a nearly normal

residual set but not a “slam dunk” normal behavior

Nearly normal residuals: To check this condition, we can look at a histogram

hist(m1$residuals)

or a normal probability plot of the residuals.

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

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

ans:

deviations in the line in the middle as mention before

Constant variability:

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

ans:

Condition for constant variability says that the variability of points around the least squares line

remains roughly constant. It appears that the constant variability condition is met here.

On Your Own

# Same, but with different colors and add regression lines
ggplot(mlb11, aes(x=mlb11$bat_avg, y=mlb11$runs)) +
    geom_point(shape=1) +
    scale_colour_hue(l=50) + # Use a slightly darker palette than normal
    geom_smooth(method=lm,   # Add linear regression lines
                se=FALSE)    # Don't add shaded confidence region

ans:

Using Batting avg. as predictor variables

And Yes, at first glance, it appears to be very linear and have a positive correlational relationship

# runs vs. at_bats
at_bats_rho <- cor(mlb11$runs, mlb11$at_bats)

# runs vs. batting avg
batting_avg_rho <- cor(mlb11$runs, mlb11$bat_avg)

R_square_at_batts <- at_bats_rho*at_bats_rho 
R_square_batting_avg <- batting_avg_rho*batting_avg_rho
R_square_at_batts
## [1] 0.3728654
R_square_batting_avg
## [1] 0.6560771

ans:

Using R square, batting avg is a better fit as it explains more of the deviations of the runs

as compared to the at_bats variables

m1 = lm(runs ~ at_bats,      data = mlb11)
m3 = lm(runs ~ stolen_bases, data = mlb11)

rSquaredAtBats  = summary(m1)$r.squared
rSquaredStBases = summary(m3)$r.squared

rSquaredAtBats
## [1] 0.3728654
rSquaredStBases
## [1] 0.002913993

ans:

All answers are for the R squares:

-value for variable at_bats is 0.3728654

-value for variable stolen_bases is 0.002914

value of stolen_bases, we can say that at_bats is a better predictor than stolen_bases.

The best predictor is batting avg.

The highest R square is 0.65, see R code above

m1  = lm(runs ~ at_bats,        data = mlb11)
m2  = lm(runs ~ hits,           data = mlb11)
m3  = lm(runs ~ homeruns,       data = mlb11)
m4  = lm(runs ~ bat_avg,        data = mlb11)
m5  = lm(runs ~ strikeouts,     data = mlb11)
m6  = lm(runs ~ stolen_bases,   data = mlb11)
m7  = lm(runs ~ wins,           data = mlb11)
m8  = lm(runs ~ new_onbase,     data = mlb11)
m9  = lm(runs ~ new_slug,       data = mlb11)
m10 = lm(runs ~ new_obs,        data = mlb11) # this one has the highest R square score

rSquared1  = summary(m1)$r.squared
rSquared2  = summary(m2)$r.squared
rSquared3  = summary(m3)$r.squared
rSquared4  = summary(m4)$r.squared
rSquared5  = summary(m5)$r.squared
rSquared6  = summary(m6)$r.squared
rSquared7  = summary(m7)$r.squared
rSquared8  = summary(m8)$r.squared
rSquared9  = summary(m9)$r.squared
rSquared10 = summary(m10)$r.squared

xR = c(rSquared1, rSquared2, rSquared3, rSquared4, rSquared5, rSquared6, rSquared7, rSquared8, rSquared9, rSquared10)
xR
##  [1] 0.372865390 0.641938767 0.626563570 0.656077135 0.169357932
##  [6] 0.002913993 0.360971179 0.849105251 0.896870368 0.934927126
max(xR)
## [1] 0.9349271

ans:

Of all the ten variables, the three new variables have highest R values to predict a team’s success.

And within these 3 new variables, the obs variable has the highest R square score

m10 = lm(runs ~ new_obs,        data = mlb11)
plot(m10$residuals ~ mlb11$new_obs)
abline(h = 0, lty = 3)  # adds a horizontal dashed line at y = 0

hist(m10$residuals)

qqnorm(m10$residuals)
qqline(m10$residuals)  # adds diagonal line to the normal prob plot

ans:

The diagnostics checks out; The residuals seems scatter, histogram appears quite normal and the Q-Q

plots looks good with no kinks are extreme deviations