# Read in data
firstbase = read.csv("C:/Users/auris/Downloads/firstbasestats.csv",header = TRUE, sep = ",")
str(firstbase)'data.frame': 23 obs. of 15 variables:
$ Player : chr "Freddie Freeman" "Jose Abreu" "Nate Lowe" "Paul Goldschmidt" ...
$ Pos : chr "1B" "1B" "1B" "1B" ...
$ Team : chr "LAD" "CHW" "TEX" "STL" ...
$ GP : int 159 157 157 151 160 140 160 145 146 143 ...
$ AB : int 612 601 593 561 638 551 583 555 545 519 ...
$ H : int 199 183 179 178 175 152 141 139 132 124 ...
$ X2B : int 47 40 26 41 35 27 25 28 40 23 ...
$ HR : int 21 15 27 35 32 20 36 22 8 18 ...
$ RBI : int 100 75 76 115 97 84 94 85 53 63 ...
$ AVG : num 0.325 0.305 0.302 0.317 0.274 0.276 0.242 0.251 0.242 0.239 ...
$ OBP : num 0.407 0.379 0.358 0.404 0.339 0.34 0.327 0.305 0.288 0.319 ...
$ SLG : num 0.511 0.446 0.492 0.578 0.48 0.437 0.477 0.423 0.36 0.391 ...
$ OPS : num 0.918 0.824 0.851 0.981 0.818 0.777 0.804 0.729 0.647 0.71 ...
$ WAR : num 5.77 4.19 3.21 7.86 3.85 3.07 5.05 1.32 -0.33 1.87 ...
$ Payroll.Salary2023: num 27000000 19500000 4050000 26000000 14500000 ...summary(firstbase) Player Pos Team GP AB
Length:23 Length:23 Length:23 Min. : 5.0 Min. : 14.0
Class :character Class :character Class :character 1st Qu.:105.5 1st Qu.:309.0
Mode :character Mode :character Mode :character Median :131.0 Median :465.0
Mean :120.2 Mean :426.9
3rd Qu.:152.0 3rd Qu.:558.0
Max. :160.0 Max. :638.0
H X2B HR RBI AVG OBP
Min. : 3.0 Min. : 1.00 Min. : 0.00 Min. : 1.00 Min. :0.2020 Min. :0.2140
1st Qu.: 74.5 1st Qu.:13.50 1st Qu.: 8.00 1st Qu.: 27.00 1st Qu.:0.2180 1st Qu.:0.3030
Median :115.0 Median :23.00 Median :18.00 Median : 63.00 Median :0.2420 Median :0.3210
Mean :110.0 Mean :22.39 Mean :17.09 Mean : 59.43 Mean :0.2499 Mean :0.3242
3rd Qu.:146.5 3rd Qu.:28.00 3rd Qu.:24.50 3rd Qu.: 84.50 3rd Qu.:0.2750 3rd Qu.:0.3395
Max. :199.0 Max. :47.00 Max. :36.00 Max. :115.00 Max. :0.3250 Max. :0.4070
SLG OPS WAR Payroll.Salary2023
Min. :0.2860 Min. :0.5000 Min. :-1.470 Min. : 720000
1st Qu.:0.3505 1st Qu.:0.6445 1st Qu.: 0.190 1st Qu.: 739200
Median :0.4230 Median :0.7290 Median : 1.310 Median : 4050000
Mean :0.4106 Mean :0.7346 Mean : 1.788 Mean : 6972743
3rd Qu.:0.4690 3rd Qu.:0.8175 3rd Qu.: 3.140 3rd Qu.: 8150000
Max. :0.5780 Max. :0.9810 Max. : 7.860 Max. :27000000 # Linear Regression (one variable)
model1 = lm(Payroll.Salary2023 ~ RBI, data=firstbase)
summary(model1)
Call:
lm(formula = Payroll.Salary2023 ~ RBI, data = firstbase)
Residuals:
Min 1Q Median 3Q Max
-10250331 -5220790 -843455 2386848 13654950
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -2363744 2866320 -0.825 0.41883
RBI 157088 42465 3.699 0.00133 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6516000 on 21 degrees of freedom
Multiple R-squared: 0.3945, Adjusted R-squared: 0.3657
F-statistic: 13.68 on 1 and 21 DF, p-value: 0.001331In the model 1, the linear model with one variable (RBI) can be expressed as:
Payroll.Salary2023 = −2,363,744 + 157,088 × RBI
This model fits a linear relationship where Payroll.Salary2023 (the salary of players in 2023) is the dependent variable and RBI (Runs Batted In) is the independent variable.
H0: there is no relationship between RBI and Payroll.Salary2023. H1: there is a relationship between RBI and Payroll.Salary2023
The p-value associated with RBI is 0.001331. Since the p-value is less than 0.05, we reject the null hypothesis. This provides sufficient evidence to conclude that there is a statistically significant positive relationship between RBI and Payroll.Salary2023 in the dataset represented by firstbase. An increase in RBI corresponds to a proportional increase in salary.
# Sum of Squared Errors
model1$residuals 1 2 3 4 5 6 7 8
13654950.2 10082148.6 -5524939.3 10298631.2 1626214.0 -6731642.8 -5902522.2 -10250330.7
9 10 11 12 13 14 15 16
-4711916.8 -532796.1 -6667082.5 -6696203.1 7582148.6 -4916640.9 -1898125.3 -336532.3
17 18 19 20 21 22 23
-995042.5 -1311618.3 -843454.5 8050721.3 1250336.9 1847040.4 2926656.0 SSE = sum(model1$residuals^2)
SSE[1] 8.914926e+14The large discrepancy (8.914926e+14 )between the observed data and the values predicted by the model indicates that the model’s predictions deviate significantly from the observed values. This suggests a potential need for model improvement or further investigation into the data.
# Linear Regression (two variables)
model2 = lm(Payroll.Salary2023 ~ AVG + RBI, data=firstbase)
summary(model2)
Call:
lm(formula = Payroll.Salary2023 ~ AVG + RBI, data = firstbase)
Residuals:
Min 1Q Median 3Q Max
-9097952 -4621582 -33233 3016541 10260245
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -18083756 9479037 -1.908 0.0709 .
AVG 74374031 42934155 1.732 0.0986 .
RBI 108850 49212 2.212 0.0388 *
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6226000 on 20 degrees of freedom
Multiple R-squared: 0.4735, Adjusted R-squared: 0.4209
F-statistic: 8.994 on 2 and 20 DF, p-value: 0.001636In the model2, the linear model with two variables RBI and AVG can be expressed as:
Payroll.Salary2023 = -18,083,756 + 74,374,031 x AVG + 108,850 × RBI
This model fits a linear relationship where Payroll.Salary2023 (the salary of players in 2023) is the dependent variable, and RBI (Runs Batted In) and AVG (batting average) are the independent variables. It aims to explain the variation in player salaries based on their batting averages and RBIs.
H0: There is no relationship between the independent variables AVG and RBI and the dependent variable Payroll.Salary2023. H1: There is a relationship between the independent variables AVG and RBI and the dependent variable Payroll.Salary2023.
Since the p-value for AVG is 0.096 is greater than 0.05, we fail to reject the null hypothesis for AVG. This suggests that there is not enough evidence to conclude that AVG has a statistically significant relationship with Payroll.Salary2023 at the 5% significance level.
and Since the p-value for RBI is0.0388 is less than 0.05, we reject the null hypothesis for RBI. This suggests that there is enough evidence to conclude that RBI has a statistically significant relationship with Payroll.Salary2023 at the 5% significance level.
The p-value for AVG (0.0986) suggests that there is not enough evidence to conclude a significant relationship between AVG and player salaries, whereas the p-value for RBI (0.0388) indicates a significant relationship between RBI and player salaries.
# Sum of Squared Errors
SSE = sum(model2$residuals^2)
SSE[1] 7.751841e+14The model 1 has large discrepancy of 8.914926e+14 between the observed data and the values predicted by the model. Similarly, Model 2 also has a large discrepancy of 7.751841e+14. However, the discrepancy is smaller in Model 2, indicating a better fit compared to Model 1.
# Linear Regression (all variables)
model3 = lm(Payroll.Salary2023 ~ HR + RBI + AVG + OBP+ OPS, data=firstbase)
summary(model3)
Call:
lm(formula = Payroll.Salary2023 ~ HR + RBI + AVG + OBP + OPS,
data = firstbase)
Residuals:
Min 1Q Median 3Q Max
-9611440 -3338119 64016 4472451 9490309
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -31107859 11738494 -2.650 0.0168 *
HR -341069 552069 -0.618 0.5449
RBI 115786 113932 1.016 0.3237
AVG -63824769 104544645 -0.611 0.5496
OBP 27054948 131210166 0.206 0.8391
OPS 60181012 95415131 0.631 0.5366
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 6023000 on 17 degrees of freedom
Multiple R-squared: 0.5811, Adjusted R-squared: 0.4579
F-statistic: 4.717 on 5 and 17 DF, p-value: 0.006951In the model3, the linear model with all variables can be expressed as:
Payroll.Salary2023 = -31,107,859 -341,069 x HR + 115,786 x RBI - 63,824,769 x AVG + 27,054,948 x OBP + 60,181,012 OPS
This model fits a linear relationship where Payroll.Salary2023 (the salary of players in 2023) is the dependent variable, and HR (home runs), RBI (runs batted in), AVG (batting average), OBP (on-base percentage), and OPS (on-base plus slugging percentage) are the independent variables. It aims to explain the variation in player salaries based on these independent variables.
However, the p-values for HR (0.5449), RBI (0.3237), AVG (0.5496), OBP (0.8391), and OPS (0.5366) are all greater than 0.05. Therefore, we fail to reject the null hypothesis for each variable, indicating that there is insufficient evidence to conclude a statistically significant relationship between these variables and player salaries.
# Sum of Squared Errors
SSE = sum(model3$residuals^2)
SSE[1] 6.167793e+14The model 1 shows a discrepancy of 8.914926e+14 between the observed data and the values predicted by the model. Similarly, Model 2 has a discrepancy of 7.751841e+14, and model 3, which includes all variables, has a discrepancy of 6.167793e+14. While Model 3 has the smallest discrepancy among the three, indicating a better fit compared to Models 1 and 2, it is also more complex due to the inclusion of additional variables.
# Remove HR
model4 = lm(Payroll.Salary2023 ~ RBI + AVG + OBP+OPS, data=firstbase)
summary(model4)
Call:
lm(formula = Payroll.Salary2023 ~ RBI + AVG + OBP + OPS, data = firstbase)
Residuals:
Min 1Q Median 3Q Max
-9399551 -3573842 98921 3979339 9263512
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -29466887 11235931 -2.623 0.0173 *
RBI 71495 87015 0.822 0.4220
AVG -11035457 59192453 -0.186 0.8542
OBP 86360720 87899074 0.982 0.3389
OPS 9464546 47788458 0.198 0.8452
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 5919000 on 18 degrees of freedom
Multiple R-squared: 0.5717, Adjusted R-squared: 0.4765
F-statistic: 6.007 on 4 and 18 DF, p-value: 0.00298In the model4, the linear model without HR can be expressed as:
Payroll.Salary2023 = -29,466,887 + 71,495 x RBI - 11,035,457 x AVG + 86,360,720 x OBP + 9,464,546 OPS
This model fits a linear relationship where Payroll.Salary2023 (the salary of players in 2023) is the dependent variable, and RBI (runs batted in), AVG (batting average), OBP (on-base percentage), and OPS (on-base plus slugging percentage) are the independent variables. It aims to explain the variation in player salaries based on these independent variables.
However, the p-values for RBI (0.4220), AVG (0.8542), OBP (0.3389), and OPS (0.8452) are all greater than 0.05. Therefore, we fail to reject the null hypothesis for each variable, indicating that there is insufficient evidence to conclude a statistically significant relationship between these variables and player salaries.
#Remove columns indexed by (1:3) from the firstbase dataset, retaining all columns except the first three originally present in firstbase dataset.
firstbase<-firstbase[,-(1:3)]
# Correlations
cor(firstbase$RBI, firstbase$Payroll.Salary2023)[1] 0.6281239This computes the correlation coefficient between Runs Batted In (RBI) and Salary in 2023. The correlation coefficient, which measures the strength and direction of their linear relationship, is 0.63, indicating a strong positive correlation between these variables.
cor(firstbase$AVG, firstbase$OBP)[1] 0.8028894This computes the correlation coefficient between batting average (AVG) and Salary in 2023, which is 0.8. The higher correlation coefficient of AVG compared to RBI and Salary 2023 (0.63).
#it computes the pairwise correlation coefficients between all pairs of numeric columns in the firstbase data frame. Each element in the resulting correlation matrix represents the correlation coefficient between two variables.
cor(firstbase) GP AB H X2B HR RBI AVG OBP
GP 1.0000000 0.9779421 0.9056508 0.8446267 0.7432552 0.8813917 0.4430808 0.4841583
AB 0.9779421 1.0000000 0.9516701 0.8924632 0.7721339 0.9125839 0.5126292 0.5026125
H 0.9056508 0.9516701 1.0000000 0.9308318 0.7155225 0.9068893 0.7393167 0.6560021
X2B 0.8446267 0.8924632 0.9308318 1.0000000 0.5889699 0.8485911 0.6613085 0.5466537
HR 0.7432552 0.7721339 0.7155225 0.5889699 1.0000000 0.8929048 0.3444242 0.4603408
RBI 0.8813917 0.9125839 0.9068893 0.8485911 0.8929048 1.0000000 0.5658479 0.5704463
AVG 0.4430808 0.5126292 0.7393167 0.6613085 0.3444242 0.5658479 1.0000000 0.8028894
OBP 0.4841583 0.5026125 0.6560021 0.5466537 0.4603408 0.5704463 0.8028894 1.0000000
SLG 0.6875270 0.7471949 0.8211406 0.7211259 0.8681501 0.8824090 0.7254274 0.7617499
OPS 0.6504483 0.6980141 0.8069779 0.6966830 0.7638721 0.8156612 0.7989005 0.8987390
WAR 0.5645243 0.6211558 0.7688712 0.6757470 0.6897677 0.7885666 0.7855945 0.7766375
Payroll.Salary2023 0.4614889 0.5018820 0.6249911 0.6450730 0.5317619 0.6281239 0.5871543 0.7025979
SLG OPS WAR Payroll.Salary2023
GP 0.6875270 0.6504483 0.5645243 0.4614889
AB 0.7471949 0.6980141 0.6211558 0.5018820
H 0.8211406 0.8069779 0.7688712 0.6249911
X2B 0.7211259 0.6966830 0.6757470 0.6450730
HR 0.8681501 0.7638721 0.6897677 0.5317619
RBI 0.8824090 0.8156612 0.7885666 0.6281239
AVG 0.7254274 0.7989005 0.7855945 0.5871543
OBP 0.7617499 0.8987390 0.7766375 0.7025979
SLG 1.0000000 0.9686752 0.8611140 0.6974086
OPS 0.9686752 1.0000000 0.8799893 0.7394981
WAR 0.8611140 0.8799893 1.0000000 0.8086359
Payroll.Salary2023 0.6974086 0.7394981 0.8086359 1.0000000We can observe that the strongest relationship among the independent variables is between AB (At Bats) and GP (Games Played). Additionally, the strongest correlation with salary is seen with WAR (Wins Above Replacement), while AVG (Batting Average) shows the weakest correlation with salary.
#Removing AVG
model5 = lm(Payroll.Salary2023 ~ RBI + OBP+OPS, data=firstbase)
summary(model5)
Call:
lm(formula = Payroll.Salary2023 ~ RBI + OBP + OPS, data = firstbase)
Residuals:
Min 1Q Median 3Q Max
-9465449 -3411234 259746 4102864 8876798
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -29737007 10855411 -2.739 0.013 *
RBI 72393 84646 0.855 0.403
OBP 82751360 83534224 0.991 0.334
OPS 7598051 45525575 0.167 0.869
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 5767000 on 19 degrees of freedom
Multiple R-squared: 0.5709, Adjusted R-squared: 0.5031
F-statistic: 8.426 on 3 and 19 DF, p-value: 0.000913In the model 5, the linear model without HR & AVG can be expressed as:
Payroll.Salary2023 = -29,737,007 + 72,393 x RBI + 82,751,360 x OBP + 7,598,051 OPS
This model fits a linear relationship where Payroll.Salary2023 (the salary of players in 2023) is the dependent variable, and RBI (runs batted in), OBP (on-base percentage), and OPS (on-base plus slugging percentage) are the independent variables. It aims to explain the variation in player salaries based on these independent variables.
However, the p-values for RBI (0.403), OBP (0.334), and OPS (0.869) are all greater than 0.05. Therefore, we fail to reject the null hypothesis for each variable, indicating that there is insufficient evidence to conclude a statistically significant relationship between these variables and player salaries.
model6 = lm(Payroll.Salary2023 ~ RBI + OBP, data=firstbase)
summary(model6)
Call:
lm(formula = Payroll.Salary2023 ~ RBI + OBP, data = firstbase)
Residuals:
Min 1Q Median 3Q Max
-9045497 -3487008 139497 4084739 9190185
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -28984802 9632560 -3.009 0.00693 **
RBI 84278 44634 1.888 0.07360 .
OBP 95468873 33385182 2.860 0.00969 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 5625000 on 20 degrees of freedom
Multiple R-squared: 0.5703, Adjusted R-squared: 0.5273
F-statistic: 13.27 on 2 and 20 DF, p-value: 0.0002149In the model 6 , the linear model without HR,AVG & OPS can be expressed as:
Payroll.Salary2023 = -28,984,802 + 84,278 x RBI + 95,468,873 x OBP
This model fits a linear relationship where Payroll.Salary2023 (the salary of players in 2023) is the dependent variable, and RBI (runs batted in) and OBP (on-base percentage), are the independent variables. It aims to explain the variation in player salaries based on these independent variables.
However, The p-value for RBI (0.07360) suggests that there is not enough evidence to conclude a significant relationship between RBI and player salaries, whereas the p-value for OBP (0.00969) indicates a significant relationship between OBP and player salaries.
# Read in test set
firstbaseTest = read.csv("C:/Users/auris/Downloads/firstbasestats_test.csv",header = TRUE, sep = ",")
str(firstbaseTest)'data.frame': 2 obs. of 15 variables:
$ Player : chr "Matt Olson" "Josh Bell"
$ Pos : chr "1B" "1B"
$ Team : chr "ATL" "SD"
$ GP : int 162 156
$ AB : int 616 552
$ H : int 148 147
$ X2B : int 44 29
$ HR : int 34 17
$ RBI : int 103 71
$ AVG : num 0.24 0.266
$ OBP : num 0.325 0.362
$ SLG : num 0.477 0.422
$ OPS : num 0.802 0.784
$ WAR : num 3.29 3.5
$ Payroll.Salary2023: num 21000000 16500000# Make test set predictions
predictTest = predict(model6, newdata=firstbaseTest)
predictTest 1 2
10723186 11558647 The predicted salaries for the predictTest dataset are $10,723,186 for Matt Olson and $11,558,647 for Josh Bell.
# Compute R-squared
#R-squared is calculated using the formula: (1 − (SSE/SST))
#SSE is the sum of the squared differences between the actual values and the predicted values
SSE = sum((firstbaseTest$Payroll.Salary2023 - predictTest)^2)
SSE[1] 1.300299e+14#SST measures the total variance of the dependent variable (Payroll.Salary2023) around its mean
SST = sum((firstbaseTest$Payroll.Salary2023 - mean(firstbase$Payroll.Salary2023))^2)
SST[1] 2.875325e+141 - SSE/SST[1] 0.5477734We can observe that Model 6 has an R-squared value of 0.55, indicating that 55% of the variance in the dependent variable (Payroll.Salary2023) is explained by the independent variables (RBI and OBP). This level of R-squared is generally considered moderate to good, suggesting that the model effectively captures a significant portion of the variability in salary based on Runs Batted In (RBI) and On-Base Percentage (OBP).