# Read in data
firstbase = read.csv("firstbasestats.csv")
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
## Length:23 Length:23 Length:23 Min. : 5.0
## Class :character Class :character Class :character 1st Qu.:105.5
## Mode :character Mode :character Mode :character Median :131.0
## Mean :120.2
## 3rd Qu.:152.0
## Max. :160.0
## AB H X2B HR
## Min. : 14.0 Min. : 3.0 Min. : 1.00 Min. : 0.00
## 1st Qu.:309.0 1st Qu.: 74.5 1st Qu.:13.50 1st Qu.: 8.00
## Median :465.0 Median :115.0 Median :23.00 Median :18.00
## Mean :426.9 Mean :110.0 Mean :22.39 Mean :17.09
## 3rd Qu.:558.0 3rd Qu.:146.5 3rd Qu.:28.00 3rd Qu.:24.50
## Max. :638.0 Max. :199.0 Max. :47.00 Max. :36.00
## RBI AVG OBP SLG
## Min. : 1.00 Min. :0.2020 Min. :0.2140 Min. :0.2860
## 1st Qu.: 27.00 1st Qu.:0.2180 1st Qu.:0.3030 1st Qu.:0.3505
## Median : 63.00 Median :0.2420 Median :0.3210 Median :0.4230
## Mean : 59.43 Mean :0.2499 Mean :0.3242 Mean :0.4106
## 3rd Qu.: 84.50 3rd Qu.:0.2750 3rd Qu.:0.3395 3rd Qu.:0.4690
## Max. :115.00 Max. :0.3250 Max. :0.4070 Max. :0.5780
## OPS WAR Payroll.Salary2023
## Min. :0.5000 Min. :-1.470 Min. : 720000
## 1st Qu.:0.6445 1st Qu.: 0.190 1st Qu.: 739200
## Median :0.7290 Median : 1.310 Median : 4050000
## Mean :0.7346 Mean : 1.788 Mean : 6972743
## 3rd Qu.:0.8175 3rd Qu.: 3.140 3rd Qu.: 8150000
## Max. :0.9810 Max. : 7.860 Max. :27000000Building a model to predict salaries based off of RBI as a variable, for first base players.
# 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: 0.001331Results: P-value is less than 0.05, so RBI is in fact significant. Our adjusted R-square shows 36%.
# Sum of Squared Errors
model1$residuals## 1 2 3 4 5 6
## 13654950.2 10082148.6 -5524939.3 10298631.2 1626214.0 -6731642.8
## 7 8 9 10 11 12
## -5902522.2 -10250330.7 -4711916.8 -532796.1 -6667082.5 -6696203.1
## 13 14 15 16 17 18
## 7582148.6 -4916640.9 -1898125.3 -336532.3 -995042.5 -1311618.3
## 19 20 21 22 23
## -843454.5 8050721.3 1250336.9 1847040.4 2926656.0SSE = sum(model1$residuals^2)
SSE## [1] 8.914926e+14Results: Showing residuals, expected vs. observed.
Predicting salaries based off of two variables, batting average and RBI.
# 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: 0.001636Results: Batting average was not significant, with a p-value greater than 0.05. However, the model improved slightly with a small increase in our adjusted R-squared.
# Sum of Squared Errors
SSE = sum(model2$residuals^2)
SSE## [1] 7.751841e+14Results: Errors have lowered with the inclusion of AVG, as expected with an increase in variables.
Predicting salaries using all variables.
# Linear Regression (all variables)
model3 = lm(Payroll.Salary2023 ~ HR + RBI + AVG + OBP+ OPS, data=firstbase)
summary(model3)##
## Call:
## lm(formula = Payroll.Salary2023 ~ HR + RBI + AVG + OBP + OPS,
## data = firstbase)
##
## Residuals:
## Min 1Q Median 3Q Max
## -9611440 -3338119 64016 4472451 9490309
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -31107859 11738494 -2.650 0.0168 *
## HR -341069 552069 -0.618 0.5449
## RBI 115786 113932 1.016 0.3237
## AVG -63824769 104544645 -0.611 0.5496
## OBP 27054948 131210166 0.206 0.8391
## OPS 60181012 95415131 0.631 0.5366
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 6023000 on 17 degrees of freedom
## Multiple R-squared: 0.5811, Adjusted R-squared: 0.4579
## F-statistic: 4.717 on 5 and 17 DF, p-value: 0.006951Results: Although our model may be significant, the actual predictors themselves are not significant. There is likely a high VIF because of the correlation between the variables. So, this model does not tell us the information we are looking for, which is what variables contribute to the prediction of salaries.
# Sum of Squared Errors
SSE = sum(model3$residuals^2)
SSE## [1] 6.167793e+14Removing HR because of its high correlation to other 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: 0.00298Results: There was almost no change by removing HR from the model, all other variables remain insignificant.
Removing the first three columns from data, as they are non-numerical, to start finding correlations.
firstbase<-firstbase[,-(1:3)]
firstbaseGathering correlations among the data.
# Correlations
cor(firstbase$RBI, firstbase$Payroll.Salary2023)## [1] 0.6281239cor(firstbase$AVG, firstbase$OBP)## [1] 0.8028894cor(firstbase)## GP AB H X2B HR RBI
## GP 1.0000000 0.9779421 0.9056508 0.8446267 0.7432552 0.8813917
## AB 0.9779421 1.0000000 0.9516701 0.8924632 0.7721339 0.9125839
## H 0.9056508 0.9516701 1.0000000 0.9308318 0.7155225 0.9068893
## X2B 0.8446267 0.8924632 0.9308318 1.0000000 0.5889699 0.8485911
## HR 0.7432552 0.7721339 0.7155225 0.5889699 1.0000000 0.8929048
## RBI 0.8813917 0.9125839 0.9068893 0.8485911 0.8929048 1.0000000
## AVG 0.4430808 0.5126292 0.7393167 0.6613085 0.3444242 0.5658479
## OBP 0.4841583 0.5026125 0.6560021 0.5466537 0.4603408 0.5704463
## SLG 0.6875270 0.7471949 0.8211406 0.7211259 0.8681501 0.8824090
## OPS 0.6504483 0.6980141 0.8069779 0.6966830 0.7638721 0.8156612
## WAR 0.5645243 0.6211558 0.7688712 0.6757470 0.6897677 0.7885666
## Payroll.Salary2023 0.4614889 0.5018820 0.6249911 0.6450730 0.5317619 0.6281239
## AVG OBP SLG OPS WAR
## GP 0.4430808 0.4841583 0.6875270 0.6504483 0.5645243
## AB 0.5126292 0.5026125 0.7471949 0.6980141 0.6211558
## H 0.7393167 0.6560021 0.8211406 0.8069779 0.7688712
## X2B 0.6613085 0.5466537 0.7211259 0.6966830 0.6757470
## HR 0.3444242 0.4603408 0.8681501 0.7638721 0.6897677
## RBI 0.5658479 0.5704463 0.8824090 0.8156612 0.7885666
## AVG 1.0000000 0.8028894 0.7254274 0.7989005 0.7855945
## OBP 0.8028894 1.0000000 0.7617499 0.8987390 0.7766375
## SLG 0.7254274 0.7617499 1.0000000 0.9686752 0.8611140
## OPS 0.7989005 0.8987390 0.9686752 1.0000000 0.8799893
## WAR 0.7855945 0.7766375 0.8611140 0.8799893 1.0000000
## Payroll.Salary2023 0.5871543 0.7025979 0.6974086 0.7394981 0.8086359
## Payroll.Salary2023
## GP 0.4614889
## AB 0.5018820
## H 0.6249911
## X2B 0.6450730
## HR 0.5317619
## RBI 0.6281239
## AVG 0.5871543
## OBP 0.7025979
## SLG 0.6974086
## OPS 0.7394981
## WAR 0.8086359
## Payroll.Salary2023 1.0000000Results: Average has an 80% correlation to OBP.
Removing average because of its high correlation to other variables.
#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: 0.000913Results: Removing AVG did not make the other variables significant.
Removing OPS from the model.
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: 0.0002149Results: This has been the best model so far, with a very low p-value, and highest adjusted R-square that we have achieved through experimenting.
Testing with unseen data.
# Read in test set
firstbaseTest = read.csv("firstbasestats_test.csv")
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# Compute R-squared
SSE = sum((firstbaseTest$Payroll.Salary2023 - predictTest)^2)
SST = sum((firstbaseTest$Payroll.Salary2023 - mean(firstbase$Payroll.Salary2023))^2)
1 - SSE/SST## [1] 0.5477734Results: Our model was not good enough to make salary predictions. However, unknown factors such as overpay may be at play.