DATA 621: WEEK 1 ASSIGNMENT

Introducing the dataset.

options(warn = -1)

suppressMessages(require(plotly))
library(knitr)
suppressMessages(library(RCurl))
suppressMessages(library(plyr))
suppressMessages(library(ggplot2))
suppressMessages(library(plotly))
suppressMessages(require(scatterplot3d));



training <- read.csv("https://raw.githubusercontent.com/mascotinme/MSDA-IS621/master/moneyball-training-data.csv", header = TRUE, sep = ",")

evaluation <- read.csv("https://raw.githubusercontent.com/mascotinme/MSDA-IS621/master/moneyball-training-data.csv", header = TRUE, sep = ",")

str(training)

## 'data.frame':    2276 obs. of  17 variables:
##  $ INDEX           : int  1 2 3 4 5 6 7 8 11 12 ...
##  $ TARGET_WINS     : int  39 70 86 70 82 75 80 85 86 76 ...
##  $ TEAM_BATTING_H  : int  1445 1339 1377 1387 1297 1279 1244 1273 1391 1271 ...
##  $ TEAM_BATTING_2B : int  194 219 232 209 186 200 179 171 197 213 ...
##  $ TEAM_BATTING_3B : int  39 22 35 38 27 36 54 37 40 18 ...
##  $ TEAM_BATTING_HR : int  13 190 137 96 102 92 122 115 114 96 ...
##  $ TEAM_BATTING_BB : int  143 685 602 451 472 443 525 456 447 441 ...
##  $ TEAM_BATTING_SO : int  842 1075 917 922 920 973 1062 1027 922 827 ...
##  $ TEAM_BASERUN_SB : int  NA 37 46 43 49 107 80 40 69 72 ...
##  $ TEAM_BASERUN_CS : int  NA 28 27 30 39 59 54 36 27 34 ...
##  $ TEAM_BATTING_HBP: int  NA NA NA NA NA NA NA NA NA NA ...
##  $ TEAM_PITCHING_H : int  9364 1347 1377 1396 1297 1279 1244 1281 1391 1271 ...
##  $ TEAM_PITCHING_HR: int  84 191 137 97 102 92 122 116 114 96 ...
##  $ TEAM_PITCHING_BB: int  927 689 602 454 472 443 525 459 447 441 ...
##  $ TEAM_PITCHING_SO: int  5456 1082 917 928 920 973 1062 1033 922 827 ...
##  $ TEAM_FIELDING_E : int  1011 193 175 164 138 123 136 112 127 131 ...
##  $ TEAM_FIELDING_DP: int  NA 155 153 156 168 149 186 136 169 159 ...

dim(training)

## [1] 2276   17

kable(summary(training))

INDEX	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_BATTING_HBP	TEAM_PITCHING_H	TEAM_PITCHING_HR	TEAM_PITCHING_BB	TEAM_PITCHING_SO	TEAM_FIELDING_E	TEAM_FIELDING_DP
Min. : 1.0	Min. : 0.00	Min. : 891	Min. : 69.0	Min. : 0.00	Min. : 0.00	Min. : 0.0	Min. : 0.0	Min. : 0.0	Min. : 0.0	Min. :29.00	Min. : 1137	Min. : 0.0	Min. : 0.0	Min. : 0.0	Min. : 65.0	Min. : 52.0
1st Qu.: 630.8	1st Qu.: 71.00	1st Qu.:1383	1st Qu.:208.0	1st Qu.: 34.00	1st Qu.: 42.00	1st Qu.:451.0	1st Qu.: 548.0	1st Qu.: 66.0	1st Qu.: 38.0	1st Qu.:50.50	1st Qu.: 1419	1st Qu.: 50.0	1st Qu.: 476.0	1st Qu.: 615.0	1st Qu.: 127.0	1st Qu.:131.0
Median :1270.5	Median : 82.00	Median :1454	Median :238.0	Median : 47.00	Median :102.00	Median :512.0	Median : 750.0	Median :101.0	Median : 49.0	Median :58.00	Median : 1518	Median :107.0	Median : 536.5	Median : 813.5	Median : 159.0	Median :149.0
Mean :1268.5	Mean : 80.79	Mean :1469	Mean :241.2	Mean : 55.25	Mean : 99.61	Mean :501.6	Mean : 735.6	Mean :124.8	Mean : 52.8	Mean :59.36	Mean : 1779	Mean :105.7	Mean : 553.0	Mean : 817.7	Mean : 246.5	Mean :146.4
3rd Qu.:1915.5	3rd Qu.: 92.00	3rd Qu.:1537	3rd Qu.:273.0	3rd Qu.: 72.00	3rd Qu.:147.00	3rd Qu.:580.0	3rd Qu.: 930.0	3rd Qu.:156.0	3rd Qu.: 62.0	3rd Qu.:67.00	3rd Qu.: 1682	3rd Qu.:150.0	3rd Qu.: 611.0	3rd Qu.: 968.0	3rd Qu.: 249.2	3rd Qu.:164.0
Max. :2535.0	Max. :146.00	Max. :2554	Max. :458.0	Max. :223.00	Max. :264.00	Max. :878.0	Max. :1399.0	Max. :697.0	Max. :201.0	Max. :95.00	Max. :30132	Max. :343.0	Max. :3645.0	Max. :19278.0	Max. :1898.0	Max. :228.0
NA	NA	NA	NA	NA	NA	NA	NA’s :102	NA’s :131	NA’s :772	NA’s :2085	NA	NA	NA	NA’s :102	NA	NA’s :286

The Multiple Linear Regression Equation for the data analysis is:

\({ Y }\quad =\quad { B }_{ 0 }\quad +\quad { B }_{ 1 }{ x }_{ 1 }\quad +\quad { B }_{ 2 }{ x }_{ 2 }\quad +\quad\) ………+\(\quad { B }_{ n }{ x }_{ n }\quad\) +\(\quad { e}\\\)

Where,

\(\quad { Y }\quad\) = Reponse or Dependent Variable,

\(\quad{ x }_{ 1 }\) …..\({ x }_{ n }\quad\) = Explantory or Independent Variables

\(\quad { B }_{ 0 }\quad\) = Intercept,

\(\quad { B }_{ 1 }\quad , ...., \quad { B }_{ n }\quad\) = Slope of Independent variables or Model Parameter.

\(\quad { e}\\\) = Residual or Error term ( the difference between an actual and a predicted value of y)

Could be re-written in terms of the training dataset as:

\({ Y }\quad =\quad { B }_{ 0 }\quad +\quad { B }_{ target-wins }{ X }_{ target-wins}\quad +\quad { B }_{ team-batting-H }{ X }_{ team-batting-H }\quad +\quad\) ………+\(\quad { B }_{ team-fielding-DP }{ X }_{ team-fielding-DP }\quad\) +\(\quad { e}\\\)

A glimpse at the multiple linear regression Analysis:

fit1 <- lm(TARGET_WINS ~. -INDEX, data = training) # The Variable INDEX is intentional omitted as it has nothing to do with the analysis

summary(fit1)

## 
## Call:
## lm(formula = TARGET_WINS ~ . - INDEX, data = training)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -19.8708  -5.6564  -0.0599   5.2545  22.9274 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      60.28826   19.67842   3.064  0.00253 ** 
## TEAM_BATTING_H    1.91348    2.76139   0.693  0.48927    
## TEAM_BATTING_2B   0.02639    0.03029   0.871  0.38484    
## TEAM_BATTING_3B  -0.10118    0.07751  -1.305  0.19348    
## TEAM_BATTING_HR  -4.84371   10.50851  -0.461  0.64542    
## TEAM_BATTING_BB  -4.45969    3.63624  -1.226  0.22167    
## TEAM_BATTING_SO   0.34196    2.59876   0.132  0.89546    
## TEAM_BASERUN_SB   0.03304    0.02867   1.152  0.25071    
## TEAM_BASERUN_CS  -0.01104    0.07143  -0.155  0.87730    
## TEAM_BATTING_HBP  0.08247    0.04960   1.663  0.09815 .  
## TEAM_PITCHING_H  -1.89096    2.76095  -0.685  0.49432    
## TEAM_PITCHING_HR  4.93043   10.50664   0.469  0.63946    
## TEAM_PITCHING_BB  4.51089    3.63372   1.241  0.21612    
## TEAM_PITCHING_SO -0.37364    2.59705  -0.144  0.88577    
## TEAM_FIELDING_E  -0.17204    0.04140  -4.155 5.08e-05 ***
## TEAM_FIELDING_DP -0.10819    0.03654  -2.961  0.00349 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.467 on 175 degrees of freedom
##   (2085 observations deleted due to missingness)
## Multiple R-squared:  0.5501, Adjusted R-squared:  0.5116 
## F-statistic: 14.27 on 15 and 175 DF,  p-value: < 2.2e-16

par(mfrow=c(2,2))
plot(fit1)

A scatter Plot and Correlation co-efficient between TARGET_WINS and TEAM_BATTING_2B

plot_ly(data = training, x = TEAM_BATTING_2B , y = TARGET_WINS, mode = "markers",
        color = "blue", line = list(shape = "linear"))

plot(TARGET_WINS~TEAM_BATTING_2B, training)

fitline <- lm(training$TARGET_WINS~training$TEAM_BATTING_2B)
abline(fitline)

cor(training$TARGET_WINS, training$TEAM_BATTING_2B)

## [1] 0.2891036

A 3D Scatterplot display for TARGET_WINS, TEAM_BATTING_2B and TEAM_BATTING_BB

attach(training);

#Run the this query to display it in 3D

scatterplot3d(TARGET_WINS, TEAM_BATTING_2B, TEAM_BATTING_BB ,pch = 20, highlight.3d = TRUE, type = "h", main = "3D ScatterPlots");

A check for Normality.

hist(training$TARGET_WINS, col="green")

hist(training$TEAM_FIELDING_DP, col="blue")

We can deduce from the above model that some of the variables are not comtributing meaningfully to the analysis, we therefore proceeded by using a statistical tool for selecting the best model for the analysis.
We shall select the best model by using both forward and backward selection process

stepwise <- step(fit1, direction = "both") # Model Selection using both FORWARD AND BACKWARD selection.

## Start:  AIC=831.31
## TARGET_WINS ~ (INDEX + 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_BATTING_HBP + TEAM_PITCHING_H + TEAM_PITCHING_HR + 
##     TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP) - 
##     INDEX
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BATTING_SO   1      1.24 12547 829.33
## - TEAM_PITCHING_SO  1      1.48 12547 829.33
## - TEAM_BASERUN_CS   1      1.71 12548 829.34
## - TEAM_BATTING_HR   1     15.23 12561 829.54
## - TEAM_PITCHING_HR  1     15.79 12562 829.55
## - TEAM_PITCHING_H   1     33.63 12580 829.82
## - TEAM_BATTING_H    1     34.42 12580 829.83
## - TEAM_BATTING_2B   1     54.41 12600 830.14
## - TEAM_BASERUN_SB   1     95.22 12641 830.76
## - TEAM_BATTING_BB   1    107.84 12654 830.95
## - TEAM_PITCHING_BB  1    110.48 12656 830.99
## - TEAM_BATTING_3B   1    122.16 12668 831.16
## <none>                          12546 831.31
## - TEAM_BATTING_HBP  1    198.21 12744 832.31
## - TEAM_FIELDING_DP  1    628.49 13174 838.65
## - TEAM_FIELDING_E   1   1237.79 13784 847.28
## 
## Step:  AIC=829.33
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B + 
##     TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_BASERUN_CS + 
##     TEAM_BATTING_HBP + TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB + 
##     TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BASERUN_CS   1      1.59 12549 827.35
## - TEAM_BATTING_HR   1     15.82 12563 827.57
## - TEAM_PITCHING_HR  1     16.39 12564 827.58
## - TEAM_BATTING_2B   1     53.47 12601 828.14
## - TEAM_PITCHING_H   1     88.45 12636 828.67
## - TEAM_BATTING_H    1     90.30 12637 828.70
## - TEAM_BASERUN_SB   1     94.19 12641 828.76
## - TEAM_BATTING_BB   1    107.95 12655 828.97
## - TEAM_PITCHING_BB  1    110.60 12658 829.01
## - TEAM_BATTING_3B   1    122.20 12669 829.18
## <none>                          12547 829.33
## - TEAM_BATTING_HBP  1    197.11 12744 830.31
## + TEAM_BATTING_SO   1      1.24 12546 831.31
## - TEAM_FIELDING_DP  1    630.68 13178 836.70
## - TEAM_FIELDING_E   1   1240.80 13788 845.34
## - TEAM_PITCHING_SO  1   1312.89 13860 846.34
## 
## Step:  AIC=827.35
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B + 
##     TEAM_BATTING_HR + TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_BATTING_HBP + 
##     TEAM_PITCHING_H + TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BATTING_HR   1     16.06 12565 825.60
## - TEAM_PITCHING_HR  1     16.64 12565 825.61
## - TEAM_BATTING_2B   1     53.05 12602 826.16
## - TEAM_PITCHING_H   1     90.24 12639 826.72
## - TEAM_BATTING_H    1     92.13 12641 826.75
## - TEAM_BATTING_BB   1    110.31 12659 827.03
## - TEAM_PITCHING_BB  1    113.00 12662 827.07
## - TEAM_BASERUN_SB   1    123.42 12672 827.22
## - TEAM_BATTING_3B   1    129.33 12678 827.31
## <none>                          12549 827.35
## - TEAM_BATTING_HBP  1    197.23 12746 828.33
## + TEAM_BASERUN_CS   1      1.59 12547 829.33
## + TEAM_BATTING_SO   1      1.12 12548 829.34
## - TEAM_FIELDING_DP  1    635.62 13184 834.79
## - TEAM_PITCHING_SO  1   1311.88 13861 844.35
## - TEAM_FIELDING_E   1   1322.05 13871 844.49
## 
## Step:  AIC=825.6
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_2B + TEAM_BATTING_3B + 
##     TEAM_BATTING_BB + TEAM_BASERUN_SB + TEAM_BATTING_HBP + TEAM_PITCHING_H + 
##     TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BATTING_2B   1     55.48 12620 824.44
## - TEAM_PITCHING_H   1     89.26 12654 824.95
## - TEAM_BATTING_H    1     91.97 12657 824.99
## - TEAM_BATTING_BB   1    104.58 12669 825.18
## - TEAM_PITCHING_BB  1    107.19 12672 825.22
## <none>                          12565 825.60
## - TEAM_BATTING_3B   1    137.48 12702 825.68
## - TEAM_BASERUN_SB   1    146.90 12712 825.82
## - TEAM_BATTING_HBP  1    200.36 12765 826.62
## + TEAM_BATTING_HR   1     16.06 12549 827.35
## + TEAM_BASERUN_CS   1      1.83 12563 827.57
## + TEAM_BATTING_SO   1      1.67 12563 827.57
## - TEAM_FIELDING_DP  1    628.95 13194 832.93
## - TEAM_PITCHING_HR  1    853.54 13418 836.15
## - TEAM_PITCHING_SO  1   1316.68 13882 842.63
## - TEAM_FIELDING_E   1   1333.15 13898 842.86
## 
## Step:  AIC=824.44
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B + TEAM_BATTING_BB + 
##     TEAM_BASERUN_SB + TEAM_BATTING_HBP + TEAM_PITCHING_H + TEAM_PITCHING_HR + 
##     TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_PITCHING_H   1     84.47 12705 823.71
## - TEAM_BATTING_H    1     87.79 12708 823.76
## - TEAM_BATTING_BB   1     98.92 12719 823.93
## - TEAM_PITCHING_BB  1    101.48 12722 823.97
## - TEAM_BASERUN_SB   1    109.27 12730 824.09
## <none>                          12620 824.44
## - TEAM_BATTING_3B   1    147.01 12767 824.65
## - TEAM_BATTING_HBP  1    204.39 12825 825.51
## + TEAM_BATTING_2B   1     55.48 12565 825.60
## + TEAM_BATTING_HR   1     18.48 12602 826.16
## + TEAM_BASERUN_CS   1      1.38 12619 826.42
## + TEAM_BATTING_SO   1      0.55 12620 826.43
## - TEAM_FIELDING_DP  1    649.12 13269 832.02
## - TEAM_PITCHING_HR  1    812.92 13433 834.36
## - TEAM_PITCHING_SO  1   1262.90 13883 840.66
## - TEAM_FIELDING_E   1   1379.34 14000 842.25
## 
## Step:  AIC=823.71
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B + TEAM_BATTING_BB + 
##     TEAM_BASERUN_SB + TEAM_BATTING_HBP + TEAM_PITCHING_HR + TEAM_PITCHING_BB + 
##     TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BATTING_BB   1     32.85 12738 822.21
## - TEAM_PITCHING_BB  1     43.42 12748 822.37
## - TEAM_BASERUN_SB   1    105.16 12810 823.29
## <none>                          12705 823.71
## - TEAM_BATTING_3B   1    153.13 12858 824.00
## + TEAM_PITCHING_H   1     84.47 12620 824.44
## - TEAM_BATTING_HBP  1    183.82 12888 824.46
## + TEAM_BATTING_SO   1     62.04 12643 824.78
## + TEAM_BATTING_2B   1     50.69 12654 824.95
## + TEAM_BATTING_HR   1     12.25 12692 825.53
## + TEAM_BASERUN_CS   1      3.11 12702 825.67
## - TEAM_BATTING_H    1    504.11 13209 829.15
## - TEAM_FIELDING_DP  1    602.80 13308 830.57
## - TEAM_PITCHING_HR  1    850.25 13555 834.09
## - TEAM_PITCHING_SO  1   1259.72 13964 839.77
## - TEAM_FIELDING_E   1   1419.39 14124 841.94
## 
## Step:  AIC=822.21
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B + TEAM_BASERUN_SB + 
##     TEAM_BATTING_HBP + TEAM_PITCHING_HR + TEAM_PITCHING_BB + 
##     TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BASERUN_SB   1    109.99 12848 821.85
## <none>                          12738 822.21
## - TEAM_BATTING_3B   1    156.45 12894 822.54
## - TEAM_BATTING_HBP  1    186.58 12924 822.98
## + TEAM_BATTING_2B   1     48.63 12689 823.48
## + TEAM_BATTING_BB   1     32.85 12705 823.71
## + TEAM_BATTING_HR   1     22.99 12715 823.86
## + TEAM_PITCHING_H   1     18.40 12719 823.93
## + TEAM_BATTING_SO   1     17.51 12720 823.94
## + TEAM_BASERUN_CS   1      3.86 12734 824.15
## - TEAM_BATTING_H    1    485.67 13223 827.35
## - TEAM_FIELDING_DP  1    623.19 13361 829.33
## - TEAM_PITCHING_HR  1    843.83 13581 832.46
## - TEAM_PITCHING_SO  1   1267.25 14005 838.32
## - TEAM_FIELDING_E   1   1395.02 14133 840.06
## - TEAM_PITCHING_BB  1   2364.81 15102 852.73
## 
## Step:  AIC=821.85
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_3B + TEAM_BATTING_HBP + 
##     TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## - TEAM_BATTING_3B   1    133.47 12981 821.82
## <none>                          12848 821.85
## + TEAM_BASERUN_SB   1    109.99 12738 822.21
## - TEAM_BATTING_HBP  1    177.11 13025 822.46
## + TEAM_BATTING_BB   1     37.69 12810 823.29
## + TEAM_BATTING_HR   1     30.72 12817 823.39
## + TEAM_BASERUN_CS   1     23.16 12824 823.51
## + TEAM_PITCHING_H   1     22.34 12825 823.52
## + TEAM_BATTING_SO   1     21.53 12826 823.53
## + TEAM_BATTING_2B   1     14.11 12834 823.64
## - TEAM_BATTING_H    1    566.11 13414 828.09
## - TEAM_FIELDING_DP  1    737.46 13585 830.51
## - TEAM_PITCHING_HR  1    756.49 13604 830.78
## - TEAM_PITCHING_SO  1   1257.91 14106 837.69
## - TEAM_FIELDING_E   1   1330.40 14178 838.67
## - TEAM_PITCHING_BB  1   2371.12 15219 852.20
## 
## Step:  AIC=821.82
## TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HBP + TEAM_PITCHING_HR + 
##     TEAM_PITCHING_BB + TEAM_PITCHING_SO + TEAM_FIELDING_E + TEAM_FIELDING_DP
## 
##                    Df Sum of Sq   RSS    AIC
## <none>                          12981 821.82
## + TEAM_BATTING_3B   1    133.47 12848 821.85
## + TEAM_BASERUN_SB   1     87.02 12894 822.54
## - TEAM_BATTING_HBP  1    228.70 13210 823.16
## + TEAM_BATTING_BB   1     40.42 12941 823.23
## + TEAM_BATTING_HR   1     33.83 12947 823.33
## + TEAM_PITCHING_H   1     23.95 12957 823.47
## + TEAM_BATTING_SO   1     23.13 12958 823.48
## + TEAM_BATTING_2B   1     21.28 12960 823.51
## + TEAM_BASERUN_CS   1      7.07 12974 823.72
## - TEAM_BATTING_H    1    449.87 13431 826.33
## - TEAM_FIELDING_DP  1    813.17 13794 831.43
## - TEAM_PITCHING_HR  1    990.20 13971 833.86
## - TEAM_PITCHING_SO  1   1316.56 14298 838.27
## - TEAM_FIELDING_E   1   1334.60 14316 838.52
## - TEAM_PITCHING_BB  1   2583.00 15564 854.49

The table below depicts the summary of the selected model:

summary(stepwise)

## 
## Call:
## lm(formula = TARGET_WINS ~ TEAM_BATTING_H + TEAM_BATTING_HBP + 
##     TEAM_PITCHING_HR + TEAM_PITCHING_BB + TEAM_PITCHING_SO + 
##     TEAM_FIELDING_E + TEAM_FIELDING_DP, data = training)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.2248  -5.6294  -0.0212   5.0439  21.3065 
## 
## Coefficients:
##                  Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      60.95454   19.10292   3.191 0.001670 ** 
## TEAM_BATTING_H    0.02541    0.01009   2.518 0.012648 *  
## TEAM_BATTING_HBP  0.08712    0.04852   1.796 0.074211 .  
## TEAM_PITCHING_HR  0.08945    0.02394   3.736 0.000249 ***
## TEAM_PITCHING_BB  0.05672    0.00940   6.034 8.66e-09 ***
## TEAM_PITCHING_SO -0.03136    0.00728  -4.308 2.68e-05 ***
## TEAM_FIELDING_E  -0.17218    0.03970  -4.338 2.38e-05 ***
## TEAM_FIELDING_DP -0.11904    0.03516  -3.386 0.000869 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.422 on 183 degrees of freedom
##   (2085 observations deleted due to missingness)
## Multiple R-squared:  0.5345, Adjusted R-squared:  0.5167 
## F-statistic: 30.02 on 7 and 183 DF,  p-value: < 2.2e-16

The above selection process depicts the best model for the analysis.

A look at the plots.

fit2 <- training[, c("TEAM_BATTING_H", "TEAM_PITCHING_HR" , "TEAM_PITCHING_BB", "TEAM_PITCHING_SO", "TEAM_FIELDING_E", "TEAM_FIELDING_DP", "TARGET_WINS", "TEAM_BATTING_HBP")]

par(mfrow=c(2,2))
plot(fit2)

NOTE: One of the variable (TEAM_BATTING_HBP) is not contributing significantly to the analysis, we therefore remove the variable and see the effect on the other variables.

fit3 <- lm(TARGET_WINS ~. -TEAM_BATTING_HBP, data = fit2)


summary(fit3)

## 
## Call:
## lm(formula = TARGET_WINS ~ . - TEAM_BATTING_HBP, data = fit2)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.8415  -6.0133  -0.0886   5.3245  22.1650 
## 
## Coefficients:
##                   Estimate Std. Error t value Pr(>|t|)    
## (Intercept)      63.466852  19.166412   3.311 0.001117 ** 
## TEAM_BATTING_H    0.025806   0.010150   2.543 0.011829 *  
## TEAM_PITCHING_HR  0.091740   0.024051   3.814 0.000186 ***
## TEAM_PITCHING_BB  0.056080   0.009450   5.935 1.44e-08 ***
## TEAM_PITCHING_SO -0.028885   0.007191  -4.017 8.59e-05 ***
## TEAM_FIELDING_E  -0.173892   0.039923  -4.356 2.20e-05 ***
## TEAM_FIELDING_DP -0.121696   0.035340  -3.444 0.000711 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 8.473 on 184 degrees of freedom
##   (2085 observations deleted due to missingness)
## Multiple R-squared:  0.5263, Adjusted R-squared:  0.5109 
## F-statistic: 34.07 on 6 and 184 DF,  p-value: < 2.2e-16

par(mfrow=c(2,2))
plot(fit3)

\(\hat { \quad y } =\quad \hat { { \beta }_{ 0 } } \quad +\quad \hat { { \beta }_{ 1 }{ x }_{ 1 } } \quad +\quad \hat { { \beta }_{ 2 }{ x }_{ 2 } } +....+\quad \hat { { \beta }_{ n }{ x }_{ n } } + \quad \hat {\quad e}\)

where \(\hat { \quad y }\) is the predicted value of y, and \({ \beta }_{ 0 },\quad { \beta }_{ 1 },\quad { \beta }_{ 2 }\)

are the estimated co-effients.

INTERPRETATIONS:

The R Squared:

The Initial Adjusted Rsquare before model selection was 0.5126, while the Adjusted Rsquared after the was .5167. A variable called TEAM_BATTING_HBP was not contributing significantly and was removed, the final Adjusted Rsquare is 0.5109 which shows the model is significance and that the removal of TEAM_BATTING_BP doesnt have any meaniful effect on other variables.

The P-Value

P-Value is sigficance at 2.2e^6, which is very less than the table value (0.05).

The least square prediction is:

\(\hat { \quad y } =\quad 63.4669\quad +\quad 0.0258TEAM_{ B }ATTING_{ H }\quad +\quad 0.0917TEAM_{ P }ITCHING_{ H }R\quad +\quad 0.0561TEAM-{ P }ITCHING-{ B }B\quad -\quad 0.0289TEAM-{ P }ITCHING-{ S }O\quad -\quad 0.1739TEAM-{ F }IELDING-{ E }\quad -\quad 0.1217TEAM-FIELDING-DP\)

The Co-efficient interpretations: First-Order Quantative Variables

If we increase the TEAM_BATTING_H by one unit, keeping the other variables constant, the mean value of Y increases by 0.0258. Same is applicable for other variables.

Analysis of Variance (ANOVA) is adopted here to show the effect and interaction between the variables.

We shall also obtain there respective confident intervals.

anova(fit3, test= "F")

## Analysis of Variance Table
## 
## Response: TARGET_WINS
##                   Df  Sum Sq Mean Sq F value    Pr(>F)    
## TEAM_BATTING_H     1  6158.8  6158.8  85.787 < 2.2e-16 ***
## TEAM_PITCHING_HR   1  1853.7  1853.7  25.820 9.161e-07 ***
## TEAM_PITCHING_BB   1  2573.7  2573.7  35.849 1.095e-08 ***
## TEAM_PITCHING_SO   1  1698.6  1698.6  23.660 2.459e-06 ***
## TEAM_FIELDING_E    1  1541.1  1541.1  21.466 6.800e-06 ***
## TEAM_FIELDING_DP   1   851.3   851.3  11.858 0.0007108 ***
## Residuals        184 13209.8    71.8                      
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The 95% and 5% confident interval of the variables to check if any of the variable is equal to zero.

confint(fit3)

##                         2.5 %       97.5 %
## (Intercept)      25.652660685 101.28104402
## TEAM_BATTING_H    0.005781302   0.04583131
## TEAM_PITCHING_HR  0.044287901   0.13919156
## TEAM_PITCHING_BB  0.037436027   0.07472318
## TEAM_PITCHING_SO -0.043071941  -0.01469730
## TEAM_FIELDING_E  -0.252657614  -0.09512704
## TEAM_FIELDING_DP -0.191420320  -0.05197243

None of them is equal to zero.

DATA 621: WEEK 1 ASSIGNMENT

MUSA T. GANIYU

June 8, 2016