Homework 1
Group 2
02/27/2019
Part 1: Overview
The purpose of this assignment is to explore, analyze and model professional baseball team performance from the years 1871 to 2006. Our objective is to build a multiple linear regression model on the provided data to predict the number of wins for the team.**
Dependencies
Replication of our work requires the following dependencies:
library(dplyr)
library(tidyr)
library(ggplot2)
library(car)
library(corrplot)
library(Hmisc)
library(psych)
library (MASS)
library(lmtest)
library(faraway)
library(knitr)
library(kableExtra)Data Preparation
We first read the training and test data from the csv files located in our repository.
train_data <- "moneyball-training-data.csv"
test_data <- "moneyball-evaluation-data.csv"
moneyball_data <- read.csv(train_data, header=TRUE, stringsAsFactors=FALSE, fileEncoding="latin1")
test_data <- read.csv(test_data, header = TRUE, stringsAsFactors = FALSE)Part 2: Data Exploration
Upon review of the dataset, we found large amounts of missing variables. We choose to replace the empty data points with the mean of that data column. The method was preferable to removing the data with omitted values, because that would have removed 90% of the provided data.
We calculated the appropriate means to compensate for the incomplete data below:
sapply(moneyball_data, function(y) sum(length(which(is.na(y)))))/nrow(moneyball_data)*100## INDEX TARGET_WINS TEAM_BATTING_H TEAM_BATTING_2B
## 0.000000 0.000000 0.000000 0.000000
## TEAM_BATTING_3B TEAM_BATTING_HR TEAM_BATTING_BB TEAM_BATTING_SO
## 0.000000 0.000000 0.000000 4.481547
## TEAM_BASERUN_SB TEAM_BASERUN_CS TEAM_BATTING_HBP TEAM_PITCHING_H
## 5.755712 33.919156 91.608084 0.000000
## TEAM_PITCHING_HR TEAM_PITCHING_BB TEAM_PITCHING_SO TEAM_FIELDING_E
## 0.000000 0.000000 4.481547 0.000000
## TEAM_FIELDING_DP
## 12.565905sapply(test_data, function(y) sum(length(which(is.na(y)))))/nrow(moneyball_data)*100## INDEX TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B
## 0.0000000 0.0000000 0.0000000 0.0000000
## TEAM_BATTING_HR TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB
## 0.0000000 0.0000000 0.7908612 0.5711775
## TEAM_BASERUN_CS TEAM_BATTING_HBP TEAM_PITCHING_H TEAM_PITCHING_HR
## 3.8224956 10.5448155 0.0000000 0.0000000
## TEAM_PITCHING_BB TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## 0.0000000 0.7908612 0.0000000 1.3620387We also choose to removed “index” and “TEAM_BATTING_HBP” columns as “TEAM_BATTING_HBP” has 92% of missing values" and “index” was just a counter.
moneyball_data<-subset(moneyball_data, select = -c(INDEX))
moneyball<-subset(moneyball_data, select = -c(TEAM_BATTING_HBP))
test_data <- subset(test_data, select = -c(INDEX))
test_data <- subset(test_data, select = -c(TEAM_BATTING_HBP))Summary Statistics
Through these steps, we replaced the missing data with the appropriate mean data.
replace_mean <- function(x){
x <- as.numeric(as.character(x))
x[is.na(x)] = mean(x, na.rm=TRUE)
return(x)
}
moneyball_filled <- apply(moneyball, 2, replace_mean)
moneyball_filled <- as.data.frame(moneyball_filled)
test_filled <- apply(test_data, 2, replace_mean)
test_filled <- as.data.frame(test_data)| TEAM_BATTING_H | 1469.3900 |
| TEAM_BATTING_2B | 241.3205 |
| TEAM_BATTING_3B | 55.9112 |
| TEAM_BATTING_HR | 95.6332 |
| TEAM_BATTING_BB | 498.9575 |
| TEAM_BATTING_SO | NA |
| TEAM_BASERUN_SB | NA |
| TEAM_BASERUN_CS | NA |
| TEAM_PITCHING_H | 1813.4633 |
| TEAM_PITCHING_HR | 102.1467 |
| TEAM_PITCHING_BB | 552.4170 |
| TEAM_PITCHING_SO | NA |
| TEAM_FIELDING_E | 249.7490 |
| TEAM_FIELDING_DP | NA |
Histogram
Now that we have a ‘good’ dataset, we can look at some histograms for each data vector. Our output suggests that most variables are fairly normally distributed and span many orders of magnitude. This tells us that our model will have some kind scaling factor between our data vectors.
Target Wins Variable
To better understand the goal of our model, we examined the targeted wins variable. Below are the plots that show this variable follows a normal, unimodal distribution that is slightly skewed to the left.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.00 71.00 82.00 80.79 92.00 146.00Correlation
We then checked for correlation among our dependent variables, as all variables are numeric we will rely on correlation. Below is a correlation plot that highlights the correlation between various data vectors. Dark blue is a high, positive correlation and dark red is a large negative correlation.
Here, we notice several variables have poor correlation with the target variable (p<0.1):
- TEAM_FIELDING_E
- TEAM_BASERUN_CS
- TEAM_BATTING_SO
- TEAM_BATTING_3B
However, others have strong correlation between each others (>0.6):
- TEAM_PITCHING_HR vs TEAM_BATTING_HR (0.969)
- TEAM_BATTING_HR vs TEAM_BATTING_SO (0.693)
- TEAM_BATTING_3B vs TEAM_BATTING_SO (-0.656)
Due to co-linearity or statistical irrelevance, we can remove: TEAM_FIELDING_E, TEAM_BASERUN_CS, TEAM_BATTING_SO, TEAM_BATTING_3B, and TEAM_BATTING_HR.
Part 3: Modeling
Model 1
We started with a naive model that uses all of the data vectors. We got an adjusted \(R^2\) value of 31.4%.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR +
## TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP,
## data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -49.994 -8.576 0.136 8.345 58.628
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 2.502e+01 5.397e+00 4.636 3.75e-06 ***
## TEAM_BATTING_H 4.824e-02 3.687e-03 13.085 < 2e-16 ***
## TEAM_BATTING_2B -2.006e-02 9.152e-03 -2.192 0.028486 *
## TEAM_BATTING_3B 6.047e-02 1.676e-02 3.608 0.000315 ***
## TEAM_BATTING_HR 5.299e-02 2.743e-02 1.932 0.053488 .
## TEAM_BATTING_BB 1.042e-02 5.818e-03 1.790 0.073544 .
## TEAM_BATTING_SO -9.349e-03 2.551e-03 -3.665 0.000253 ***
## TEAM_BASERUN_SB 2.949e-02 4.462e-03 6.610 4.78e-11 ***
## TEAM_BASERUN_CS -1.188e-02 1.614e-02 -0.736 0.461905
## TEAM_PITCHING_H -7.342e-04 3.676e-04 -1.997 0.045946 *
## TEAM_PITCHING_HR 1.480e-02 2.432e-02 0.609 0.542877
## TEAM_PITCHING_BB 8.891e-05 4.145e-03 0.021 0.982891
## TEAM_PITCHING_SO 2.843e-03 9.187e-04 3.095 0.001994 **
## TEAM_FIELDING_E -2.112e-02 2.480e-03 -8.516 < 2e-16 ***
## TEAM_FIELDING_DP -1.210e-01 1.302e-02 -9.297 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.04 on 2261 degrees of freedom
## Multiple R-squared: 0.3189, Adjusted R-squared: 0.3147
## F-statistic: 75.63 on 14 and 2261 DF, p-value: < 2.2e-16Model 2
Then, we removed the least significant variable, TEAM_PITCHING_BB. This yielded a slight increase in our \(R^2\) score at 31.5%.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_BASERUN_CS + TEAM_PITCHING_H + TEAM_PITCHING_HR +
## TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP, data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -49.994 -8.576 0.136 8.345 58.626
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 25.0145796 5.3904993 4.640 3.67e-06 ***
## TEAM_BATTING_H 0.0482393 0.0036807 13.106 < 2e-16 ***
## TEAM_BATTING_2B -0.0200575 0.0091490 -2.192 0.028457 *
## TEAM_BATTING_3B 0.0604730 0.0167556 3.609 0.000314 ***
## TEAM_BATTING_HR 0.0527106 0.0240710 2.190 0.028641 *
## TEAM_BATTING_BB 0.0105175 0.0033664 3.124 0.001805 **
## TEAM_BATTING_SO -0.0093631 0.0024585 -3.809 0.000144 ***
## TEAM_BASERUN_SB 0.0295055 0.0044087 6.693 2.76e-11 ***
## TEAM_BASERUN_CS -0.0118872 0.0161276 -0.737 0.461155
## TEAM_PITCHING_H -0.0007306 0.0003283 -2.225 0.026147 *
## TEAM_PITCHING_HR 0.0150659 0.0209923 0.718 0.473025
## TEAM_PITCHING_SO 0.0028567 0.0006717 4.253 2.20e-05 ***
## TEAM_FIELDING_E -0.0211192 0.0024784 -8.521 < 2e-16 ***
## TEAM_FIELDING_DP -0.1210298 0.0130139 -9.300 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.04 on 2262 degrees of freedom
## Multiple R-squared: 0.3189, Adjusted R-squared: 0.315
## F-statistic: 81.49 on 13 and 2262 DF, p-value: < 2.2e-16Model 3
We repeated the above procedure for TEAM_BASERUN_CS, netting us a score of 31.52%. By removing two variables we were able to oh-so-slightly increase our \(R^2\) value while reducing the amount of data we have to track and the processing time for tracking it.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_SO +
## TEAM_FIELDING_E + TEAM_FIELDING_DP, data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -49.905 -8.584 0.124 8.406 58.593
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 24.2348098 5.2851330 4.585 4.78e-06 ***
## TEAM_BATTING_H 0.0482055 0.0036800 13.099 < 2e-16 ***
## TEAM_BATTING_2B -0.0203302 0.0091405 -2.224 0.026235 *
## TEAM_BATTING_3B 0.0608466 0.0167463 3.633 0.000286 ***
## TEAM_BATTING_HR 0.0543985 0.0239594 2.270 0.023274 *
## TEAM_BATTING_BB 0.0107643 0.0033494 3.214 0.001328 **
## TEAM_BATTING_SO -0.0093418 0.0024580 -3.800 0.000148 ***
## TEAM_BASERUN_SB 0.0287600 0.0042906 6.703 2.57e-11 ***
## TEAM_PITCHING_H -0.0007390 0.0003281 -2.253 0.024372 *
## TEAM_PITCHING_HR 0.0147103 0.0209846 0.701 0.483372
## TEAM_PITCHING_SO 0.0028640 0.0006716 4.265 2.08e-05 ***
## TEAM_FIELDING_E -0.0207217 0.0024188 -8.567 < 2e-16 ***
## TEAM_FIELDING_DP -0.1211603 0.0130114 -9.312 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.04 on 2263 degrees of freedom
## Multiple R-squared: 0.3188, Adjusted R-squared: 0.3152
## F-statistic: 88.25 on 12 and 2263 DF, p-value: < 2.2e-16Model 4
By removing TEAM_PITCHING_HR, we increase our \(R^2\) value one last time to 31.53%.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_SO + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -49.899 -8.568 0.091 8.397 58.651
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 23.6666983 5.2220414 4.532 6.14e-06 ***
## TEAM_BATTING_H 0.0484570 0.0036621 13.232 < 2e-16 ***
## TEAM_BATTING_2B -0.0205123 0.0091358 -2.245 0.024847 *
## TEAM_BATTING_3B 0.0624661 0.0165843 3.767 0.000170 ***
## TEAM_BATTING_HR 0.0697785 0.0096266 7.249 5.75e-13 ***
## TEAM_BATTING_BB 0.0107446 0.0033489 3.208 0.001354 **
## TEAM_BATTING_SO -0.0093019 0.0024571 -3.786 0.000157 ***
## TEAM_BASERUN_SB 0.0287708 0.0042901 6.706 2.51e-11 ***
## TEAM_PITCHING_H -0.0006920 0.0003211 -2.155 0.031253 *
## TEAM_PITCHING_SO 0.0028867 0.0006707 4.304 1.75e-05 ***
## TEAM_FIELDING_E -0.0205973 0.0024120 -8.540 < 2e-16 ***
## TEAM_FIELDING_DP -0.1210083 0.0130082 -9.302 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.03 on 2264 degrees of freedom
## Multiple R-squared: 0.3186, Adjusted R-squared: 0.3153
## F-statistic: 96.25 on 11 and 2264 DF, p-value: < 2.2e-16Evaluate Non-linearity
Below we use the crPlots() function to check for non-linearity.
TEAM_PITCHING_H, TEAM_PITCHING_SO did not pass the check for non-linearity. So, we will transform them and refit the model. We are using a log10 transform because these numbers span many orders of magnitude.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_SO + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.500 -8.353 0.050 8.276 63.152
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -18.385299 13.648651 -1.347 0.178102
## TEAM_BATTING_H 0.041874 0.003784 11.065 < 2e-16 ***
## TEAM_BATTING_2B -0.020476 0.009106 -2.249 0.024630 *
## TEAM_BATTING_3B 0.087638 0.016862 5.197 2.20e-07 ***
## TEAM_BATTING_HR 0.058540 0.009697 6.037 1.83e-09 ***
## TEAM_BATTING_BB 0.012944 0.003388 3.821 0.000137 ***
## TEAM_BATTING_SO -0.001186 0.002534 -0.468 0.639742
## TEAM_BASERUN_SB 0.031437 0.004300 7.311 3.65e-13 ***
## TEAM_PITCHING_H 17.140905 4.594173 3.731 0.000195 ***
## TEAM_PITCHING_SO -2.656620 0.914734 -2.904 0.003717 **
## TEAM_FIELDING_E -0.030455 0.002954 -10.309 < 2e-16 ***
## TEAM_FIELDING_DP -0.120505 0.013004 -9.267 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.03 on 2264 degrees of freedom
## Multiple R-squared: 0.319, Adjusted R-squared: 0.3157
## F-statistic: 96.42 on 11 and 2264 DF, p-value: < 2.2e-16
Now we remove TEAM_BATTING_SO because it has a p-value > 0.05.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BASERUN_SB +
## TEAM_PITCHING_H + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP,
## data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -53.382 -8.328 0.025 8.211 62.933
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -20.516367 12.864856 -1.595 0.110905
## TEAM_BATTING_H 0.042628 0.003424 12.448 < 2e-16 ***
## TEAM_BATTING_2B -0.021583 0.008792 -2.455 0.014167 *
## TEAM_BATTING_3B 0.089227 0.016513 5.403 7.23e-08 ***
## TEAM_BATTING_HR 0.055774 0.007688 7.255 5.50e-13 ***
## TEAM_BATTING_BB 0.013293 0.003304 4.023 5.93e-05 ***
## TEAM_BASERUN_SB 0.030879 0.004130 7.476 1.09e-13 ***
## TEAM_PITCHING_H 17.440250 4.548668 3.834 0.000129 ***
## TEAM_PITCHING_SO -2.847123 0.819083 -3.476 0.000519 ***
## TEAM_FIELDING_E -0.030494 0.002953 -10.328 < 2e-16 ***
## TEAM_FIELDING_DP -0.119878 0.012932 -9.270 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 13.03 on 2265 degrees of freedom
## Multiple R-squared: 0.319, Adjusted R-squared: 0.316
## F-statistic: 106.1 on 10 and 2265 DF, p-value: < 2.2e-16Eliminating Outliers
Then, we used Cook’s distance to identify extreme values, removing them as necessary. 
Then, we re-fit the model to the new data, yielding our highest \(R^2\) value of 31.57%.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_SO + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = moneyball_filled)
##
## Residuals:
## Min 1Q Median 3Q Max
## -47.117 -8.396 0.026 8.238 64.496
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -18.936246 14.300248 -1.324 0.185574
## TEAM_BATTING_H 0.039743 0.003901 10.187 < 2e-16 ***
## TEAM_BATTING_2B -0.021808 0.009043 -2.412 0.015957 *
## TEAM_BATTING_3B 0.101540 0.016844 6.028 1.93e-09 ***
## TEAM_BATTING_HR 0.065471 0.009640 6.791 1.41e-11 ***
## TEAM_BATTING_BB 0.012322 0.003354 3.674 0.000245 ***
## TEAM_BATTING_SO -0.001465 0.002507 -0.584 0.559106
## TEAM_BASERUN_SB 0.032302 0.004275 7.556 5.99e-14 ***
## TEAM_PITCHING_H 18.926032 4.961698 3.814 0.000140 ***
## TEAM_PITCHING_SO -3.560746 0.932952 -3.817 0.000139 ***
## TEAM_FIELDING_E -0.031367 0.003048 -10.291 < 2e-16 ***
## TEAM_FIELDING_DP -0.120207 0.012872 -9.339 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 12.89 on 2261 degrees of freedom
## Multiple R-squared: 0.3245, Adjusted R-squared: 0.3212
## F-statistic: 98.72 on 11 and 2261 DF, p-value: < 2.2e-16Checking for Colinearity
## TEAM_BATTING_H TEAM_BATTING_2B TEAM_BATTING_3B TEAM_BATTING_HR
## 4.322088 2.438948 3.007291 4.650721
## TEAM_BATTING_BB TEAM_BATTING_SO TEAM_BASERUN_SB TEAM_PITCHING_H
## 2.294133 5.045176 1.812464 5.617980
## TEAM_PITCHING_SO TEAM_FIELDING_E TEAM_FIELDING_DP
## 1.722976 6.490171 1.364366
Model 5
TEAM_FIELDING_E is withing the range 5-10 (suggesting co linearity with other variables), but eliminating TEAM_FIELDING_E does not improve the model. This yields our highest \(R^2\) value with 40% of the variance explained by our model.
##
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B +
## TEAM_BATTING_3B + TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BATTING_SO +
## TEAM_BASERUN_SB + TEAM_PITCHING_H + TEAM_PITCHING_SO + TEAM_FIELDING_E +
## TEAM_FIELDING_DP, data = moneyball_data)
##
## Residuals:
## Min 1Q Median 3Q Max
## -32.317 -7.199 0.121 7.045 29.766
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 58.312951 6.019406 9.687 < 2e-16 ***
## TEAM_BATTING_H -0.010007 0.010615 -0.943 0.34594
## TEAM_BATTING_2B -0.049989 0.008875 -5.633 2.05e-08 ***
## TEAM_BATTING_3B 0.181788 0.018982 9.577 < 2e-16 ***
## TEAM_BATTING_HR 0.100845 0.009158 11.012 < 2e-16 ***
## TEAM_BATTING_BB 0.034055 0.003133 10.870 < 2e-16 ***
## TEAM_BATTING_SO 0.045928 0.016420 2.797 0.00521 **
## TEAM_BASERUN_SB 0.069889 0.005535 12.626 < 2e-16 ***
## TEAM_PITCHING_H 0.037438 0.009239 4.052 5.29e-05 ***
## TEAM_PITCHING_SO -0.065427 0.015514 -4.217 2.59e-05 ***
## TEAM_FIELDING_E -0.116444 0.007029 -16.566 < 2e-16 ***
## TEAM_FIELDING_DP -0.112850 0.012279 -9.190 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 10.19 on 1823 degrees of freedom
## (441 observations deleted due to missingness)
## Multiple R-squared: 0.4045, Adjusted R-squared: 0.4009
## F-statistic: 112.6 on 11 and 1823 DF, p-value: < 2.2e-16Part 4: Model Evaluation
Using our finished model above, we can predict the number of wins for each team. We rounded to a whole number so that the finished values have some real world analogue. The F-statistics has a p value of basically 0, so we can determine that our model is statistically significant.
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
## 65 66 74 86 NA NA NA 76 72 73 69 82 82 81 84 76 75 79
## 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
## NA 90 82 83 82 71 81 86 NA 74 84 74 90 85 82 84 81 86
## 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
## 76 91 86 92 81 90 NA NA NA NA NA 78 70 80 76 83 79 74
## 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72
## 75 79 94 76 NA NA 88 75 88 85 83 NA 77 82 NA 91 88 69
## 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
## 77 89 81 86 83 83 NA NA 83 89 97 74 86 78 82 83 87 90
## 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108
## NA NA 75 NA NA NA 90 104 88 88 80 73 83 84 80 NA NA 78
## 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126
## 87 NA 84 84 93 91 81 78 86 81 75 NA NA NA NA NA 71 88
## 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144
## 92 78 93 92 86 78 79 87 88 NA 75 77 86 80 67 NA 91 74
## 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162
## 72 72 78 78 78 82 82 81 NA 70 77 71 90 NA 96 NA 106 107
## 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180
## 95 104 98 90 82 80 73 79 NA 90 81 93 83 74 77 71 74 79
## 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198
## 86 89 85 85 NA NA NA NA NA NA NA NA 76 76 79 67 78 85
## 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
## 80 86 78 81 76 90 80 84 79 78 NA NA NA NA 85 65 69 84
## 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234
## 80 93 77 80 79 75 81 74 NA 76 81 81 82 NA NA 93 79 88
## 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252
## 80 76 83 78 NA 74 90 86 82 81 61 85 80 85 72 84 82 NA
## 253 254 255 256 257 258 259
## NA NA 69 76 83 82 78Additionally, we can use a residual plot to verify our model. We can see that our model’s residuals are fairly normal and randomly distributed. They also are centered and zero.
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -47.11665 -8.39565 0.02552 0.00000 8.23798 64.49592