For this recipe, a dataset of Baseball statistics for catchers will be examined.
The question under examination is how a catacher's statistics and team association effect the number of runs a team lets up (Earned run average or ERA).
data<-read.csv("C:/baseballdata2.csv")
summary(data)
## Player League Team Runs RperG
## Min. : 1.0 AL:34 CHC : 3 Min. :583 Min. :3.60
## 1st Qu.:21.5 NL:37 CIN : 3 1st Qu.:651 1st Qu.:4.02
## Median :40.0 HOU : 3 Median :700 Median :4.32
## Mean :39.8 KC : 3 Mean :697 Mean :4.30
## 3rd Qu.:58.5 LAA : 3 3rd Qu.:734 3rd Qu.:4.53
## Max. :76.0 MIA : 3 Max. :808 Max. :4.99
## (Other):53
## ERA ARD TC PO
## Min. :3.19 Min. :-0.960 Min. : 117 Min. :110
## 1st Qu.:3.71 1st Qu.:-0.170 1st Qu.: 308 1st Qu.:285
## Median :4.01 Median : 0.500 Median : 408 Median :379
## Mean :4.00 Mean : 0.305 Mean : 471 Mean :436
## 3rd Qu.:4.30 3rd Qu.: 0.845 3rd Qu.: 574 3rd Qu.:522
## Max. :5.22 Max. : 1.180 Max. :1053 Max. :962
##
## A E DP SB
## Min. : 7.0 Min. :0.00 Min. : 0.00 Min. : 3.0
## 1st Qu.:18.0 1st Qu.:2.00 1st Qu.: 1.00 1st Qu.:22.5
## Median :25.0 Median :4.00 Median : 3.00 Median :37.0
## Mean :31.7 Mean :3.58 Mean : 3.24 Mean :38.9
## 3rd Qu.:41.0 3rd Qu.:5.00 3rd Qu.: 4.50 3rd Qu.:52.5
## Max. :88.0 Max. :8.00 Max. :12.00 Max. :87.0
##
## CS SBPct PB EPC
## Min. : 1.0 Min. :0.500 Min. : 0.00 Min. :0.970
## 1st Qu.: 7.0 1st Qu.:0.685 1st Qu.: 2.00 1st Qu.:0.990
## Median :12.0 Median :0.760 Median : 3.00 Median :0.990
## Mean :13.6 Mean :0.741 Mean : 4.18 Mean :0.992
## 3rd Qu.:18.0 3rd Qu.:0.805 3rd Qu.: 6.00 3rd Qu.:1.000
## Max. :38.0 Max. :0.870 Max. :18.00 Max. :1.000
##
## RF
## Min. :5.29
## 1st Qu.:6.79
## Median :7.38
## Mean :7.30
## 3rd Qu.:7.75
## Max. :8.90
##
The three factors being considered for this experiment are:
Errors (E)- number of errors a catcher made. Levels(4): 0-3, 3-6, 6-9, and 9 and up
Put Outs (PO)- number of opposing team players are put out by the catcher Levels(4): 0-250, 250-500, 500-750, 750 and up
Stolen Bases (SB)- Number of bases a catcher allows the opposing team to steal Levels(3): 0-30, 30-60, and 60-90
Since errors and put outs are currently continuous, we will create discrete factor levels
Creat discrete factor levels for Errors factor:
data$E[data$E<3 & data$E>=0] = "0-3"
data$E[data$E<6 & data$E>=3] = "3-6"
data$E[data$E<9 & data$E>=6] = "6-9"
data$E[data$E>=9] = "9 and up"
Creat discrete factor levels for Put Outs factor:
data$PO[data$PO<250 & data$PO>=0] = "0-250"
data$PO[data$PO<500 & data$PO>=250] = "250-500"
data$PO[data$PO<750 & data$PO>=500] = "500-750"
data$PO[data$PO>=750] = "750 and up"
Creat discrete factor levels for Stolen Bases factor:
data$SB[data$SB<30 & data$SB>=0] = "0-30"
data$SB[data$SB<60 & data$SB>=30] = "30-60"
data$SB[data$SB<90 & data$SB>=60] = "60-100"
Now that levels are set, save the 3 as factors:
Save Errors (E), Put outs (PO) and stolen basses allowed (SB) as factors:
data$E=as.factor(data$E)
data$PO=as.factor(data$PO)
data$SB=as.factor(data$SB)
The other continuous variables in the dataset are Runs, average runs per game, earned run average, average run differential, total chances, assists, double plays, caught stealing, stolen base percentage, passed balls, errors per total chances, and a range factor.
We will use earned run average (ERA), which is the average number of runs a team gives up per game, as the response variable to measure the catchers performance.
The dataset is collected from all the catchers that played in the 2012 MLB season.
The data has the 15 continuous variables listed above (although in the factors and levels section above we turned errors, put outs, and stolen bases allowed into discrete levels), a player identification number, and league and team identification factors.
Structure of the baseball catchers dataset:
str(data)
## 'data.frame': 71 obs. of 18 variables:
## $ Player: int 1 2 3 5 6 7 8 9 10 11 ...
## $ League: Factor w/ 2 levels "AL","NL": 2 2 2 2 2 2 2 2 2 2 ...
## $ Team : Factor w/ 30 levels "ARI","ATL","BAL",..: 26 26 22 6 6 6 16 16 16 5 ...
## $ Runs : int 765 765 651 669 669 669 776 776 776 613 ...
## $ RperG : num 4.72 4.72 4.02 4.13 4.13 4.13 4.79 4.79 4.79 3.78 ...
## $ ERA : num 3.71 3.71 3.86 3.34 3.34 3.34 4.22 4.22 4.22 4.51 ...
## $ ARD : num 1.01 1.01 0.16 0.79 0.79 0.79 0.57 0.57 0.57 -0.73 ...
## $ TC : int 1053 280 539 393 833 141 169 755 576 388 ...
## $ PO : Factor w/ 4 levels "0-250","250-500",..: 4 2 2 2 4 1 1 3 3 2 ...
## $ A : int 88 23 41 24 48 9 7 34 47 25 ...
## $ E : Factor w/ 3 levels "0-3","3-6","6-9": 2 1 2 1 2 1 1 3 3 3 ...
## $ DP : int 12 3 1 1 6 0 0 4 6 2 ...
## $ SB : Factor w/ 3 levels "0-30","30-60",..: 2 1 3 2 2 1 1 3 2 2 ...
## $ CS : int 35 8 13 10 32 1 5 19 15 14 ...
## $ SBPct : num 0.52 0.69 0.82 0.8 0.52 0.83 0.81 0.79 0.68 0.73 ...
## $ PB : int 6 3 2 3 3 1 1 2 2 1 ...
## $ EPC : num 1 0.99 0.99 1 1 0.99 1 0.99 0.99 0.98 ...
## $ RF : num 7.72 5.91 6.62 7.4 7.54 6.67 6.26 8.5 8.26 7.33 ...
First and last six observations of the dataset:
head(data)
## Player League Team Runs RperG ERA ARD TC PO A E DP SB
## 1 1 NL STL 765 4.72 3.71 1.01 1053 750 and up 88 3-6 12 30-60
## 2 2 NL STL 765 4.72 3.71 1.01 280 250-500 23 0-3 3 0-30
## 3 3 NL PIT 651 4.02 3.86 0.16 539 250-500 41 3-6 1 60-100
## 4 5 NL CIN 669 4.13 3.34 0.79 393 250-500 24 0-3 1 30-60
## 5 6 NL CIN 669 4.13 3.34 0.79 833 750 and up 48 3-6 6 30-60
## 6 7 NL CIN 669 4.13 3.34 0.79 141 0-250 9 0-3 0 0-30
## CS SBPct PB EPC RF
## 1 35 0.52 6 1.00 7.72
## 2 8 0.69 3 0.99 5.91
## 3 13 0.82 2 0.99 6.62
## 4 10 0.80 3 1.00 7.40
## 5 32 0.52 3 1.00 7.54
## 6 1 0.83 1 0.99 6.67
tail(data)
## Player League Team Runs RperG ERA ARD TC PO A E DP SB CS
## 66 71 AL TB 697 4.30 3.19 1.11 277 250-500 12 3-6 2 0-30 5
## 67 72 AL TEX 808 4.99 3.99 1.00 552 500-750 37 3-6 1 30-60 11
## 68 73 AL TEX 808 4.99 3.99 1.00 397 250-500 18 0-3 1 30-60 9
## 69 74 AL TEX 808 4.99 3.99 1.00 371 250-500 12 0-3 1 0-30 7
## 70 75 AL TOR 716 4.42 4.64 -0.22 683 500-750 57 3-6 1 30-60 22
## 71 76 AL TOR 716 4.42 4.64 -0.22 473 250-500 38 0-3 5 0-30 20
## SBPct PB EPC RF
## 66 0.80 0 0.99 7.03
## 67 0.79 8 0.99 7.61
## 68 0.77 4 1.00 8.06
## 69 0.80 1 1.00 8.41
## 70 0.71 9 0.99 7.22
## 71 0.59 6 1.00 7.14
The experiment will be a multi-factor design with 3 factors and multiple levels to test if the variation in runs against (ERA) can be explained by the team, errors by the catcher or put outs by the catcher.
An ANOVA model will be created and analyzed to test the null hypothesis that the variation in runs against (ERA) cannot be explained by anything other than variation (there are no differences in means among the samples).
Analyze diff of means of groups see variation among and between groups
This dataset does not have a randomization scheme as it is a set of all statistics for each catcher in the 2012 season.
Some of the observations are averages, such as stolen base percentage and ERA, so the replicates can be considered the 162 games the data was averaged over.
Only the 3 catchers per team that played the most games were considered in the dataset to block the catchers who only played a game or two.
Summary Statistics for entire baseball catchers dataset:
summary(data)
## Player League Team Runs RperG
## Min. : 1.0 AL:34 CHC : 3 Min. :583 Min. :3.60
## 1st Qu.:21.5 NL:37 CIN : 3 1st Qu.:651 1st Qu.:4.02
## Median :40.0 HOU : 3 Median :700 Median :4.32
## Mean :39.8 KC : 3 Mean :697 Mean :4.30
## 3rd Qu.:58.5 LAA : 3 3rd Qu.:734 3rd Qu.:4.53
## Max. :76.0 MIA : 3 Max. :808 Max. :4.99
## (Other):53
## ERA ARD TC PO
## Min. :3.19 Min. :-0.960 Min. : 117 0-250 :13
## 1st Qu.:3.71 1st Qu.:-0.170 1st Qu.: 308 250-500 :37
## Median :4.01 Median : 0.500 Median : 408 500-750 :13
## Mean :4.00 Mean : 0.305 Mean : 471 750 and up: 8
## 3rd Qu.:4.30 3rd Qu.: 0.845 3rd Qu.: 574
## Max. :5.22 Max. : 1.180 Max. :1053
##
## A E DP SB CS
## Min. : 7.0 0-3:26 Min. : 0.00 0-30 :26 Min. : 1.0
## 1st Qu.:18.0 3-6:29 1st Qu.: 1.00 30-60 :33 1st Qu.: 7.0
## Median :25.0 6-9:16 Median : 3.00 60-100:12 Median :12.0
## Mean :31.7 Mean : 3.24 Mean :13.6
## 3rd Qu.:41.0 3rd Qu.: 4.50 3rd Qu.:18.0
## Max. :88.0 Max. :12.00 Max. :38.0
##
## SBPct PB EPC RF
## Min. :0.500 Min. : 0.00 Min. :0.970 Min. :5.29
## 1st Qu.:0.685 1st Qu.: 2.00 1st Qu.:0.990 1st Qu.:6.79
## Median :0.760 Median : 3.00 Median :0.990 Median :7.38
## Mean :0.741 Mean : 4.18 Mean :0.992 Mean :7.30
## 3rd Qu.:0.805 3rd Qu.: 6.00 3rd Qu.:1.000 3rd Qu.:7.75
## Max. :0.870 Max. :18.00 Max. :1.000 Max. :8.90
##
Mean ERA:
mean(data$ERA)
## [1] 3.997
Boxplots by errors and put outs:
boxplot(data$ERA~data$E, xlab="Number of Errors", ylab="ERA")
boxplot(data$ERA~data$PO, xlab="Number of Put Outs", ylab="ERA")
boxplot(data$ERA~data$SB, xlab="Number of Stolen Bases Allowed", ylab="ERA")
Examining the errors plot, it is hard to say if the number of errors is the cause of the variation in ERA, although it does appear that the larger level of errors may lead to the largest ERA.
As for the put outs box plot, it does not appear that the first 3 levels would explain the variation in ERA, but again the last level of 750+ put outs by the catcher appears to result in a lower ERA, which makes sense in the game of baseball.
From the stolen bases boxplot, it seems unlikely that the number of stolen bases allowed will exaplin the variation in ERA.
#Model 1: ANOVA Model for number of Errors effect on ERA First we will test to see the effect the Number of errors a catcher has can have on the ERA
model1=aov(data$ERA~data$E)
anova(model1)
## Analysis of Variance Table
##
## Response: data$ERA
## Df Sum Sq Mean Sq F value Pr(>F)
## data$E 2 0.78 0.391 1.77 0.18
## Residuals 68 15.07 0.222
The probability that the variation in ERA among teams is caused by randomization is 0.1788. It is likely that randomization is causing the variation in ERA. We fail to reject the null hypothesis. Number of errors by a catcher alone is not enough to predict the ERA of a team.
#Model 2:ANOVA Model for number of put outs effect on ERA
model2=aov(data$ERA~data$PO)
anova(model2)
## Analysis of Variance Table
##
## Response: data$ERA
## Df Sum Sq Mean Sq F value Pr(>F)
## data$PO 3 1.02 0.339 1.53 0.21
## Residuals 67 14.84 0.221
Again, we fail to reject the null hypothesis. Put outs by a catcher alone does not explain the variation among ERA.
#Model 3:ANOVA Model for number of passed balls effect on ERA
model3=aov(data$ERA~data$SB)
anova(model3)
## Analysis of Variance Table
##
## Response: data$ERA
## Df Sum Sq Mean Sq F value Pr(>F)
## data$SB 2 0.37 0.185 0.81 0.45
## Residuals 68 15.48 0.228
It is even more unlikely that stolen bases can explain the varaition in ERA. We fail to reject the null hypothesis that the variation is due to randomization.
#Model 4: ANOVA model to test if interation of all three factors (Errors, put outs, and stolen bases) effect on ERA
Although each factor individually could not explain the variation among ERA, this model will test to see if it is likely that the factors together can explain the variation in ERA:
model4=aov(data$ERA~data$E*data$PO*data$SB)
anova(model4)
## Analysis of Variance Table
##
## Response: data$ERA
## Df Sum Sq Mean Sq F value Pr(>F)
## data$E 2 0.78 0.391 1.76 0.182
## data$PO 3 1.53 0.510 2.30 0.088 .
## data$SB 2 0.89 0.443 1.99 0.146
## data$E:data$PO 5 0.27 0.054 0.24 0.941
## data$E:data$SB 3 0.11 0.036 0.16 0.920
## data$PO:data$SB 2 0.51 0.254 1.14 0.327
## Residuals 53 11.77 0.222
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
This analysis gives statistics for each factor individual, for the combinations of each pair, and finally for all three together. It is unlikely than any of the combinations can explain the variation in ERA.
#Check Normality We must check that our data follows the normallity assumption of the ANOVA model:
qqnorm(residuals(model4))
qqline(residuals(model4))
Based on this plot, the dataset looks somewhat normal, although there is a chance it may not be. This means the results of the above analysis may not accurate.
plot(fitted(model4),residuals(model4))
The residuals are somewhat clustered in the middle but they still seem to be equally distributed along the horizontal zero, indicating a decent fit of the model.
#Tukey Test
The Tukey test takes the anova model created and compares the means of each level between all the different factors. The null hypothesis of the test is that there is no difference between the means of the data pair.
TukeyHSD(model4)
## Tukey multiple comparisons of means
## 95% family-wise confidence level
##
## Fit: aov(formula = data$ERA ~ data$E * data$PO * data$SB)
##
## $`data$E`
## diff lwr upr p adj
## 3-6-0-3 -0.1298 -0.4367 0.1771 0.5677
## 6-9-0-3 0.1425 -0.2185 0.5036 0.6101
## 6-9-3-6 0.2723 -0.0815 0.6262 0.1617
##
## $`data$PO`
## diff lwr upr p adj
## 250-500-0-250 0.09166 -0.3113 0.49465 0.9306
## 500-750-0-250 0.14492 -0.3453 0.63517 0.8614
## 750 and up-0-250 -0.33997 -0.9016 0.22168 0.3843
## 500-750-250-500 0.05325 -0.3497 0.45624 0.9851
## 750 and up-250-500 -0.43164 -0.9190 0.05571 0.0999
## 750 and up-500-750 -0.48489 -1.0465 0.07676 0.1133
##
## $`data$SB`
## diff lwr upr p adj
## 30-60-0-30 0.0857 -0.2123 0.3837 0.7683
## 60-100-0-30 0.2308 -0.1658 0.6273 0.3466
## 60-100-30-60 0.1451 -0.2380 0.5281 0.6344
##
## $`data$E:data$PO`
## diff lwr upr p adj
## 3-6:0-250-0-3:0-250 -0.18087 -1.4284 1.0667 1.0000
## 6-9:0-250-0-3:0-250 0.22740 -1.4618 1.9166 1.0000
## 0-3:250-500-0-3:0-250 0.10747 -0.5500 0.7650 1.0000
## 3-6:250-500-0-3:0-250 -0.06502 -0.6942 0.5642 1.0000
## 6-9:250-500-0-3:0-250 0.30907 -0.7511 1.3693 0.9972
## 0-3:500-750-0-3:0-250 NA NA NA NA
## 3-6:500-750-0-3:0-250 -0.08976 -0.9215 0.7419 1.0000
## 6-9:500-750-0-3:0-250 0.37489 -0.4188 1.1686 0.8960
## 0-3:750 and up-0-3:0-250 -0.71075 -2.3999 0.9784 0.9501
## 3-6:750 and up-0-3:0-250 -0.31157 -1.5591 0.9360 0.9993
## 6-9:750 and up-0-3:0-250 -0.18868 -1.0708 0.6935 0.9998
## 6-9:0-250-3-6:0-250 0.40828 -1.5643 2.3808 0.9999
## 0-3:250-500-3-6:0-250 0.28834 -0.9241 1.5007 0.9996
## 3-6:250-500-3-6:0-250 0.11585 -1.0814 1.3131 1.0000
## 6-9:250-500-3-6:0-250 0.48994 -0.9803 1.9602 0.9912
## 0-3:500-750-3-6:0-250 NA NA NA NA
## 3-6:500-750-3-6:0-250 0.09111 -1.2239 1.4061 1.0000
## 6-9:500-750-3-6:0-250 0.55576 -0.7356 1.8471 0.9420
## 0-3:750 and up-3-6:0-250 -0.52988 -2.5024 1.4427 0.9986
## 3-6:750 and up-3-6:0-250 -0.13069 -1.7413 1.4799 1.0000
## 6-9:750 and up-3-6:0-250 -0.00781 -1.3553 1.3397 1.0000
## 0-3:250-500-6-9:0-250 -0.11993 -1.7833 1.5435 1.0000
## 3-6:250-500-6-9:0-250 -0.29243 -1.9448 1.3600 1.0000
## 6-9:250-500-6-9:0-250 0.08166 -1.7781 1.9414 1.0000
## 0-3:500-750-6-9:0-250 NA NA NA NA
## 3-6:500-750-6-9:0-250 -0.31717 -2.0568 1.4225 1.0000
## 6-9:500-750-6-9:0-250 0.14748 -1.5743 1.8693 1.0000
## 0-3:750 and up-6-9:0-250 -0.93816 -3.2158 1.3395 0.9568
## 3-6:750 and up-6-9:0-250 -0.53897 -2.5115 1.4336 0.9984
## 6-9:750 and up-6-9:0-250 -0.41608 -2.1804 1.3482 0.9996
## 3-6:250-500-0-3:250-500 -0.17250 -0.7288 0.3838 0.9952
## 6-9:250-500-0-3:250-500 0.20160 -0.8170 1.2202 0.9999
## 0-3:500-750-0-3:250-500 NA NA NA NA
## 3-6:500-750-0-3:250-500 -0.19723 -0.9752 0.5807 0.9992
## 6-9:500-750-0-3:250-500 0.26742 -0.4698 1.0046 0.9829
## 0-3:750 and up-0-3:250-500 -0.81822 -2.4816 0.8452 0.8686
## 3-6:750 and up-0-3:250-500 -0.41904 -1.6314 0.7934 0.9883
## 6-9:750 and up-0-3:250-500 -0.29615 -1.1278 0.5355 0.9852
## 6-9:250-500-3-6:250-500 0.37409 -0.6265 1.3747 0.9785
## 0-3:500-750-3-6:250-500 NA NA NA NA
## 3-6:500-750-3-6:250-500 -0.02474 -0.7790 0.7295 1.0000
## 6-9:500-750-3-6:250-500 0.43991 -0.2722 1.1520 0.6172
## 0-3:750 and up-3-6:250-500 -0.64573 -2.2981 1.0067 0.9702
## 3-6:750 and up-3-6:250-500 -0.24654 -1.4438 0.9507 0.9999
## 6-9:750 and up-3-6:250-500 -0.12366 -0.9332 0.6859 1.0000
## 0-3:500-750-6-9:250-500 NA NA NA NA
## 3-6:500-750-6-9:250-500 -0.39883 -1.5377 0.7400 0.9870
## 6-9:500-750-6-9:250-500 0.06582 -1.0456 1.1772 1.0000
## 0-3:750 and up-6-9:250-500 -1.01982 -2.8795 0.8399 0.7691
## 3-6:750 and up-6-9:250-500 -0.62063 -2.0909 0.8496 0.9490
## 6-9:750 and up-6-9:250-500 -0.49775 -1.6739 0.6784 0.9481
## 3-6:500-750-0-3:500-750 NA NA NA NA
## 6-9:500-750-0-3:500-750 NA NA NA NA
## 0-3:750 and up-0-3:500-750 NA NA NA NA
## 3-6:750 and up-0-3:500-750 NA NA NA NA
## 6-9:750 and up-0-3:500-750 NA NA NA NA
## 6-9:500-750-3-6:500-750 0.46465 -0.4314 1.3607 0.8251
## 0-3:750 and up-3-6:500-750 -0.62099 -2.3606 1.1186 0.9849
## 3-6:750 and up-3-6:500-750 -0.22180 -1.5368 1.0932 1.0000
## 6-9:750 and up-3-6:500-750 -0.09892 -1.0742 0.8763 1.0000
## 0-3:750 and up-6-9:500-750 -1.08564 -2.8074 0.6361 0.5875
## 3-6:750 and up-6-9:500-750 -0.68645 -1.9778 0.6049 0.8015
## 6-9:750 and up-6-9:500-750 -0.56357 -1.5066 0.3795 0.6634
## 3-6:750 and up-0-3:750 and up 0.39919 -1.5734 2.3717 0.9999
## 6-9:750 and up-0-3:750 and up 0.52207 -1.2422 2.2864 0.9968
## 6-9:750 and up-3-6:750 and up 0.12288 -1.2246 1.4704 1.0000
##
## $`data$E:data$SB`
## diff lwr upr p adj
## 3-6:0-30-0-3:0-30 -0.16356 -0.8168 0.4897 0.9961
## 6-9:0-30-0-3:0-30 0.08809 -1.4797 1.6558 1.0000
## 0-3:30-60-0-3:0-30 0.11882 -0.5654 0.8030 0.9997
## 3-6:30-60-0-3:0-30 -0.04421 -0.5529 0.4644 1.0000
## 6-9:30-60-0-3:0-30 0.13659 -0.5476 0.8208 0.9992
## 0-3:60-100-0-3:0-30 0.44672 -0.6922 1.5857 0.9361
## 3-6:60-100-0-3:0-30 0.15422 -0.9847 1.2932 1.0000
## 6-9:60-100-0-3:0-30 0.28728 -0.3659 0.9405 0.8843
## 6-9:0-30-3-6:0-30 0.25165 -1.3644 1.8676 0.9999
## 0-3:30-60-3-6:0-30 0.28238 -0.5061 1.0709 0.9617
## 3-6:30-60-3-6:0-30 0.11935 -0.5228 0.7615 0.9995
## 6-9:30-60-3-6:0-30 0.30014 -0.4884 1.0887 0.9458
## 0-3:60-100-3-6:0-30 0.61028 -0.5942 1.8148 0.7796
## 3-6:60-100-3-6:0-30 0.31778 -0.8867 1.5223 0.9944
## 6-9:60-100-3-6:0-30 0.45084 -0.3110 1.2126 0.6078
## 0-3:30-60-6-9:0-30 0.03074 -1.5980 1.6595 1.0000
## 3-6:30-60-6-9:0-30 -0.13229 -1.6955 1.4309 1.0000
## 6-9:30-60-6-9:0-30 0.04850 -1.5803 1.6773 1.0000
## 0-3:60-100-6-9:0-30 0.35863 -1.5074 2.2246 0.9994
## 3-6:60-100-6-9:0-30 0.06614 -1.7999 1.9321 1.0000
## 6-9:60-100-6-9:0-30 0.19919 -1.4168 1.8152 1.0000
## 3-6:30-60-0-3:30-60 -0.16303 -0.8367 0.5106 0.9969
## 6-9:30-60-0-3:30-60 0.01776 -0.7966 0.8321 1.0000
## 0-3:60-100-0-3:30-60 0.32790 -0.8937 1.5495 0.9938
## 3-6:60-100-0-3:30-60 0.03540 -1.1862 1.2570 1.0000
## 6-9:60-100-0-3:30-60 0.16846 -0.6201 0.9570 0.9987
## 6-9:30-60-3-6:30-60 0.18079 -0.4928 0.8544 0.9938
## 0-3:60-100-3-6:30-60 0.49093 -0.6417 1.6235 0.8923
## 3-6:60-100-3-6:30-60 0.19843 -0.9342 1.3310 0.9997
## 6-9:60-100-3-6:30-60 0.33149 -0.3106 0.9736 0.7619
## 0-3:60-100-6-9:30-60 0.31014 -0.9114 1.5317 0.9957
## 3-6:60-100-6-9:30-60 0.01764 -1.2039 1.2392 1.0000
## 6-9:60-100-6-9:30-60 0.15070 -0.6378 0.9392 0.9994
## 3-6:60-100-0-3:60-100 -0.29250 -1.8161 1.2311 0.9994
## 6-9:60-100-0-3:60-100 -0.15944 -1.3639 1.0451 1.0000
## 6-9:60-100-3-6:60-100 0.13306 -1.0714 1.3376 1.0000
##
## $`data$PO:data$SB`
## diff lwr upr p adj
## 250-500:0-30-0-250:0-30 -0.09430 -0.7390 0.5504 1.0000
## 500-750:0-30-0-250:0-30 0.47864 -1.1977 2.1550 0.9976
## 750 and up:0-30-0-250:0-30 NA NA NA NA
## 0-250:30-60-0-250:0-30 0.05047 -1.6259 1.7268 1.0000
## 250-500:30-60-0-250:0-30 0.16769 -0.4151 0.7505 0.9975
## 500-750:30-60-0-250:0-30 0.03364 -0.6766 0.7438 1.0000
## 750 and up:30-60-0-250:0-30 -0.55843 -1.7885 0.6717 0.9183
## 0-250:60-100-0-250:0-30 NA NA NA NA
## 250-500:60-100-0-250:0-30 0.41325 -0.6264 1.4529 0.9664
## 500-750:60-100-0-250:0-30 0.38432 -0.6553 1.4239 0.9803
## 750 and up:60-100-0-250:0-30 -0.26198 -1.0673 0.5433 0.9928
## 500-750:0-30-250-500:0-30 0.57295 -1.0984 2.2443 0.9890
## 750 and up:0-30-250-500:0-30 NA NA NA NA
## 0-250:30-60-250-500:0-30 0.14478 -1.5266 1.8161 1.0000
## 250-500:30-60-250-500:0-30 0.26199 -0.3064 0.8304 0.9102
## 500-750:30-60-250-500:0-30 0.12795 -0.5704 0.8263 1.0000
## 750 and up:30-60-250-500:0-30 -0.46413 -1.6874 0.7592 0.9760
## 0-250:60-100-250-500:0-30 NA NA NA NA
## 250-500:60-100-250-500:0-30 0.50755 -0.5240 1.5391 0.8684
## 500-750:60-100-250-500:0-30 0.47862 -0.5530 1.5102 0.9066
## 750 and up:60-100-250-500:0-30 -0.16767 -0.9626 0.6272 0.9999
## 750 and up:0-30-500-750:0-30 NA NA NA NA
## 0-250:30-60-500-750:0-30 -0.42817 -2.7059 1.8495 1.0000
## 250-500:30-60-500-750:0-30 -0.31096 -1.9594 1.3375 1.0000
## 500-750:30-60-500-750:0-30 -0.44500 -2.1427 1.2527 0.9989
## 750 and up:30-60-500-750:0-30 -1.03707 -3.0096 0.9355 0.8123
## 0-250:60-100-500-750:0-30 NA NA NA NA
## 250-500:60-100-500-750:0-30 -0.06539 -1.9251 1.7943 1.0000
## 500-750:60-100-500-750:0-30 -0.09433 -1.9541 1.7654 1.0000
## 750 and up:60-100-500-750:0-30 -0.74062 -2.4802 0.9990 0.9460
## 0-250:30-60-750 and up:0-30 NA NA NA NA
## 250-500:30-60-750 and up:0-30 NA NA NA NA
## 500-750:30-60-750 and up:0-30 NA NA NA NA
## 750 and up:30-60-750 and up:0-30 NA NA NA NA
## 0-250:60-100-750 and up:0-30 NA NA NA NA
## 250-500:60-100-750 and up:0-30 NA NA NA NA
## 500-750:60-100-750 and up:0-30 NA NA NA NA
## 750 and up:60-100-750 and up:0-30 NA NA NA NA
## 250-500:30-60-0-250:30-60 0.11721 -1.5313 1.7657 1.0000
## 500-750:30-60-0-250:30-60 -0.01683 -1.7145 1.6809 1.0000
## 750 and up:30-60-0-250:30-60 -0.60890 -2.5814 1.3636 0.9954
## 0-250:60-100-0-250:30-60 NA NA NA NA
## 250-500:60-100-0-250:30-60 0.36278 -1.4970 2.2225 0.9999
## 500-750:60-100-0-250:30-60 0.33385 -1.5259 2.1936 1.0000
## 750 and up:60-100-0-250:30-60 -0.31245 -2.0521 1.4272 1.0000
## 500-750:30-60-250-500:30-60 -0.13404 -0.7757 0.5076 0.9999
## 750 and up:30-60-250-500:30-60 -0.72612 -1.9180 0.4657 0.6369
## 0-250:60-100-250-500:30-60 NA NA NA NA
## 250-500:60-100-250-500:30-60 0.24556 -0.7485 1.2396 0.9994
## 500-750:60-100-250-500:30-60 0.21663 -0.7774 1.2107 0.9998
## 750 and up:60-100-250-500:30-60 -0.42967 -1.1752 0.3159 0.7107
## 750 and up:30-60-500-750:30-60 -0.59207 -1.8511 0.6670 0.8987
## 0-250:60-100-500-750:30-60 NA NA NA NA
## 250-500:60-100-500-750:30-60 0.37961 -0.6941 1.4533 0.9860
## 500-750:60-100-500-750:30-60 0.35067 -0.7230 1.4244 0.9926
## 750 and up:60-100-500-750:30-60 -0.29562 -1.1445 0.5532 0.9875
## 0-250:60-100-750 and up:30-60 NA NA NA NA
## 250-500:60-100-750 and up:30-60 0.97168 -0.4986 2.4419 0.5170
## 500-750:60-100-750 and up:30-60 0.94275 -0.5275 2.4130 0.5626
## 750 and up:60-100-750 and up:30-60 0.29645 -1.0186 1.6115 0.9997
## 250-500:60-100-0-250:60-100 NA NA NA NA
## 500-750:60-100-0-250:60-100 NA NA NA NA
## 750 and up:60-100-0-250:60-100 NA NA NA NA
## 500-750:60-100-250-500:60-100 -0.02893 -1.3440 1.2861 1.0000
## 750 and up:60-100-250-500:60-100 -0.67523 -1.8141 0.4636 0.6740
## 750 and up:60-100-500-750:60-100 -0.64630 -1.7851 0.4925 0.7296
A low p-value means a low probability that the means are not likely to have a significant difference in means.
tukey<-TukeyHSD(model4)
plot(tukey)
The first 3 plots suggest there is no difference in ERA means between any of the group combinations within each factor (demonstrated by the zero line intersecting with each plot) . The other plots comparing all the combinations between factors become difficult to read as there are so many combinations.
#Interaction Plot
We create an interaction plot to view the interactions between the factors.
To run the plot, we must re-save the factors as numeric:
data$E=as.numeric(data$E)
data$PO=as.numeric(data$PO)
data$SB=as.numeric(data$SB)
interaction.plot(data$E, data$PO, data$SB)
If there were no interaction between factors, the lines would be parallel and they would not intersect. The intersection and non-parallel lines shows there is interaction between the 3 factors.
None used.
Since it was unclear if the data was normally distributed, we can use the Kruskal-Wallis non-parametric analysis of variance to test the difference between groups. The null hypothesis is that the variation between groups cannot be explained by anything other than variation.
kruskal.test(data$ERA, data$E)
##
## Kruskal-Wallis rank sum test
##
## data: data$ERA and data$E
## Kruskal-Wallis chi-squared = 3.675, df = 2, p-value = 0.1593
kruskal.test(data$ERA, data$PO)
##
## Kruskal-Wallis rank sum test
##
## data: data$ERA and data$PO
## Kruskal-Wallis chi-squared = 5.902, df = 3, p-value = 0.1165
kruskal.test(data$ERA, data$SB)
##
## Kruskal-Wallis rank sum test
##
## data: data$ERA and data$SB
## Kruskal-Wallis chi-squared = 1.184, df = 2, p-value = 0.5533
Since we have large p-values for each result, we fail to reject the null. Like the other tests suggested, there is not evidence that the errors, put outs, or passed balls of a catcher can explain the variation among the ERA's.
All included above.