Linear Regression Models and Descriptive Data Analysis

We will be analyzing a data frame containing 5 variables (“gameno”, “month”, “homeruns”, “playerstatus”, “player”) and 326 observations.The following below will help us understand the variables we are looking at:

gameno

an integer variable denoting the game number

month

a factor variable taking with levels “March” through “September” denoting the month of the game

homeruns

an integer vector denoting the number of homeruns hit in that game for that player

playerstatus

an integer vector equal to “0” if the player played in the game, and “1” if they did not.

player

an integer vector equal to “0” (McGwire) or “1” (Sosa)

Exploring the Data:

library(Zelig)
data("homerun")
library(survival)
str(homerun)

'data.frame':   314 obs. of  5 variables:
 $ gameno      : int  1 2 3 4 5 6 7 8 9 10 ...
 $ month       : Factor w/ 7 levels "April","August",..: 5 1 1 1 1 1 1 1 1 1 ...
 $ homeruns    : int  1 1 1 1 0 0 0 0 0 0 ...
 $ playerstatus: int  0 0 0 0 0 0 0 0 0 0 ...
 $ player      : Factor w/ 2 levels "McGwire","Sosa": 1 1 1 1 1 1 1 1 1 1 ...

Descriptive Analysis:

Bar Graphs and Histograms

library(ggplot2)
ggplot(homerun, aes(x=player)) + geom_bar(fill = "red")

The bar graph above tells us that Sammy Sosa had played in more games than Mark McGwire. There are 162 games in a season, and from the data above, Sosa played in more games.

ggplot(homerun, aes(x=player, y=homeruns)) + geom_bar(stat = "identity", fill="red")

The bar graph above shows the amount of homeruns each player had.

ggplot(homerun, aes(x=gameno, y=homeruns)) + geom_histogram(stat = "identity", fill = "blue")

Ignoring unknown parameters: binwidth, bins, pad

The histogram shows the number of home runs for both teams overall in each game (out of 162 games).

The relationship between the player and the amount of homeruns hit in that game

m1<- lm(homeruns ~ player, data=homerun)
summary(m1)

Call:
lm(formula = homeruns ~ player, data = homerun)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.4516 -0.4516 -0.4151  0.5484  2.5849 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.45161    0.05204   8.677 2.28e-16 ***
playerSosa  -0.03652    0.07314  -0.499    0.618    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.648 on 312 degrees of freedom
Multiple R-squared:  0.0007984, Adjusted R-squared:  -0.002404 
F-statistic: 0.2493 on 1 and 312 DF,  p-value: 0.6179

According to the information above it seems that Sosa has fewer homeruns than McGwire. For every 1 homerun from McGwire, Sosa has a -.03 homerun chance. There was no significant difference.

The relationship between homeruns hit and the month of each game

m2<- lm(homeruns ~ month, data=homerun)
summary(m2)

Call:
lm(formula = homeruns ~ month, data = homerun)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.5882 -0.4510 -0.3200  0.4800  2.6800 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     0.320000   0.091332   3.504 0.000527 ***
monthAugust     0.090714   0.125655   0.722 0.470888    
monthJuly      -0.005185   0.126748  -0.041 0.967395    
monthJune       0.268235   0.128528   2.087 0.037715 *  
monthMarch      0.180000   0.465703   0.387 0.699385    
monthMay        0.130980   0.128528   1.019 0.308967    
monthSeptember  0.200000   0.129163   1.548 0.122548    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6458 on 307 degrees of freedom
Multiple R-squared:  0.02329,   Adjusted R-squared:  0.004203 
F-statistic:  1.22 on 6 and 307 DF,  p-value: 0.2955

Now we are looking at the homeruns hit and each each month. We don’t really see any signifcance in here. It just shows us that each month varies. The only significance is for the month of July. I wonder if other factors like weather, place of game and time of game influence the number of homeruns possible for each month.

Multiple Regression: The Relationship between Homeruns for each player and the month of the games.

m3 <- lm(homeruns ~ player + month, data=homerun)
summary(m3)

Call:
lm(formula = homeruns ~ player + month, data = homerun)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.6083 -0.4316 -0.3338  0.4990  2.6603 

Coefficients:
                Estimate Std. Error t value Pr(>|t|)    
(Intercept)     0.339737   0.099009   3.431 0.000683 ***
playerSosa     -0.037955   0.073010  -0.520 0.603533    
monthAugust     0.089955   0.125813   0.715 0.475161    
monthJuly      -0.005944   0.126908  -0.047 0.962672    
monthJune       0.268593   0.128683   2.087 0.037693 *  
monthMarch      0.179241   0.466260   0.384 0.700932    
monthMay        0.129849   0.128699   1.009 0.313804    
monthSeptember  0.199241   0.129325   1.541 0.124442    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6466 on 306 degrees of freedom
Multiple R-squared:  0.02415,   Adjusted R-squared:  0.00183 
F-statistic: 1.082 on 7 and 306 DF,  p-value: 0.3747

On Average we can see that Sosa had fewer homeruns and that it varied in each month. The only t value that shows significant difference is again the month of July. For every one homerun McGwire had, Sosa was behind him by -.03.

More Simple Table

library(texreg)
screenreg(list(m1,m2,m3))

==================================================
                Model 1     Model 2     Model 3   
--------------------------------------------------
(Intercept)       0.45 ***    0.32 ***    0.34 ***
                 (0.05)      (0.09)      (0.10)   
playerSosa       -0.04                   -0.04    
                 (0.07)                  (0.07)   
monthAugust                   0.09        0.09    
                             (0.13)      (0.13)   
monthJuly                    -0.01       -0.01    
                             (0.13)      (0.13)   
monthJune                     0.27 *      0.27 *  
                             (0.13)      (0.13)   
monthMarch                    0.18        0.18    
                             (0.47)      (0.47)   
monthMay                      0.13        0.13    
                             (0.13)      (0.13)   
monthSeptember                0.20        0.20    
                             (0.13)      (0.13)   
--------------------------------------------------
R^2               0.00        0.02        0.02    
Adj. R^2         -0.00        0.00        0.00    
Num. obs.       314         314         314       
RMSE              0.65        0.65        0.65    
==================================================
*** p < 0.001, ** p < 0.01, * p < 0.05

Possible Interaction Effects

m4<-lm(homeruns ~ player*gameno, data = homerun)
summary(m4)

Call:
lm(formula = homeruns ~ player * gameno, data = homerun)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.5353 -0.4517 -0.3905  0.5482  2.6065 

Coefficients:
                    Estimate Std. Error t value Pr(>|t|)    
(Intercept)        4.508e-01  1.053e-01   4.282 2.47e-05 ***
playerSosa        -1.589e-01  1.473e-01  -1.079    0.282    
gameno             9.515e-06  1.102e-03   0.009    0.993    
playerSosa:gameno  1.484e-03  1.545e-03   0.960    0.338    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6481 on 310 degrees of freedom
Multiple R-squared:  0.006894,  Adjusted R-squared:  -0.002717 
F-statistic: 0.7173 on 3 and 310 DF,  p-value: 0.5423

I wanted to see if the game number had any significance on the number of homeruns each player had. The tvalue for gameno is less than .05 which does show that there is a signficance. On average Sosa had less homeruns than McGwire.

Overall, I wanted to seee what effected the players from achieving more homeruns. I would have liked to look at more data such as age, weight, years played, no. of injuries / seasons or games out. McGwire ended up beating Sosa by 4 homeruns. McGwire had 70 at the end of the season and Sosa had 66.