Goal! Goal! Goal!
master yeoda
a long, long time ago in a galaxy far, far away
Directions
Regression models to predict the wages of football players.
1. Load Data
Load the data and explore them.
football <- read.csv("football_2.csv", header = FALSE)
head(football, 10)## V1 V2 V3 V4
## 1 ID Name Age Photo
## 2 207439 L. Paredes 24 https://cdn.sofifa.org/players/4/19/207439.png
## 3 156713 A. Granqvist 33 https://cdn.sofifa.org/players/4/19/156713.png
## 4 229909 A. Lunev 26 https://cdn.sofifa.org/players/4/19/229909.png
## 5 187347 I. Smolnikov 29 https://cdn.sofifa.org/players/4/19/187347.png
## 6 153260 Hilton 40 https://cdn.sofifa.org/players/4/19/153260.png
## 7 187607 A. Dzyuba 29 https://cdn.sofifa.org/players/4/19/187607.png
## 8 204341 Lu\xcc_s Neto 30 https://cdn.sofifa.org/players/4/19/204341.png
## 9 223058 D. Kuzyaev 25 https://cdn.sofifa.org/players/4/19/223058.png
## 10 183389 G. Sio 29 https://cdn.sofifa.org/players/4/19/183389.png
## V5 V6 V7 V8
## 1 Nationality Flag Overall Potential
## 2 Argentina https://cdn.sofifa.org/flags/52.png 80 85
## 3 Sweden https://cdn.sofifa.org/flags/46.png 80 80
## 4 Russia https://cdn.sofifa.org/flags/40.png 79 81
## 5 Russia https://cdn.sofifa.org/flags/40.png 79 79
## 6 Brazil https://cdn.sofifa.org/flags/54.png 78 78
## 7 Russia https://cdn.sofifa.org/flags/40.png 78 78
## 8 Portugal https://cdn.sofifa.org/flags/38.png 77 77
## 9 Russia https://cdn.sofifa.org/flags/40.png 77 80
## 10 Ivory Coast https://cdn.sofifa.org/flags/108.png 77 77
## V9 V10 V11 V12
## 1 Club Club Logo Value Wage
## 2 https://cdn.sofifa.org/flags/52.png 5684 1602
## 3 https://cdn.sofifa.org/flags/46.png 6370 3591
## 4 https://cdn.sofifa.org/flags/40.png 5675 3672
## 5 https://cdn.sofifa.org/flags/40.png 6030 1448
## 6 Montpellier HSC https://cdn.sofifa.org/teams/2/light/70.png 6405 19799
## 7 https://cdn.sofifa.org/flags/40.png 5764 1105
## 8 https://cdn.sofifa.org/flags/38.png 6075 2836
## 9 https://cdn.sofifa.org/flags/40.png 5565 2653
## 10 https://cdn.sofifa.org/flags/108.png 5275 2138
## V13 V14 V15 V16 V17
## 1 Special Preferred Foot International Reputation Weak Foot Skill Moves
## 2 2122 Right 2 4 4
## 3 1797 Right 2 4 2
## 4 1217 Right 1 3 1
## 5 2038 Right 2 3 3
## 6 1807 Right 2 3 3
## 7 1810 Right 2 3 3
## 8 1749 Right 1 3 2
## 9 2041 Right 1 3 3
## 10 1933 Left 2 3 3
## V18 V19 V20 V21 V22 V23
## 1 Work Rate Body Type Real Face Position Jersey Number Joined
## 2 Medium/ Medium Normal No CM 5
## 3 High/ Medium Normal No LCB 4
## 4 Medium/ Medium Normal No GK 12
## 5 High/ High Lean No RB 2
## 6 Medium/ Medium Normal Yes CB 4 1-Aug-11
## 7 High/ Medium Stocky No ST 22
## 8 Medium/ Medium Lean No CB 4
## 9 Medium/ High Lean No RM 7
## 10 High/ Low Normal No ST 21
## V24 V25 V26 V27 V28 V29 V30 V31 V32 V33
## 1 Loaned From Contract Valid Until Height Weight LS ST RS LW LF CF
## 2 5'11 165lbs 71+2 71+2 71+2 75+2 75+2 75+2
## 3 6'4 185lbs 62+2 62+2 62+2 56+2 58+2 58+2
## 4 6'2 176lbs
## 5 5'10 154lbs 70+2 70+2 70+2 73+2 72+2 72+2
## 6 2019 5'11 172lbs 58+2 58+2 58+2 58+2 59+2 59+2
## 7 6'5 201lbs 77+2 77+2 77+2 71+2 74+2 74+2
## 8 6'2 157lbs 52+2 52+2 52+2 51+2 51+2 51+2
## 9 6'0 163lbs 70+2 70+2 70+2 74+2 74+2 74+2
## 10 5'11 176lbs 75+2 75+2 75+2 75+2 75+2 75+2
## V34 V35 V36 V37 V38 V39 V40 V41 V42 V43 V44 V45 V46 V47 V48
## 1 RF RW LAM CAM RAM LM LCM CM RCM RM LWB LDM CDM RDM RWB
## 2 75+2 75+2 77+2 77+2 77+2 76+2 79+2 79+2 79+2 76+2 75+2 77+2 77+2 77+2 75+2
## 3 58+2 56+2 58+2 58+2 58+2 57+2 64+2 64+2 64+2 57+2 68+2 74+2 74+2 74+2 68+2
## 4
## 5 72+2 73+2 73+2 73+2 73+2 75+2 74+2 74+2 74+2 75+2 78+2 75+2 75+2 75+2 78+2
## 6 59+2 58+2 62+2 62+2 62+2 60+2 67+2 67+2 67+2 60+2 67+2 73+2 73+2 73+2 67+2
## 7 74+2 71+2 71+2 71+2 71+2 71+2 66+2 66+2 66+2 71+2 52+2 52+2 52+2 52+2 52+2
## 8 51+2 51+2 54+2 54+2 54+2 54+2 61+2 61+2 61+2 54+2 67+2 72+2 72+2 72+2 67+2
## 9 74+2 74+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2
## 10 75+2 75+2 74+2 74+2 74+2 74+2 67+2 67+2 67+2 74+2 53+2 52+2 52+2 52+2 53+2
## V49 V50 V51 V52 V53 V54 V55 V56 V57
## 1 LB LCB CB RCB RB Crossing Finishing HeadingAccuracy ShortPassing
## 2 74+2 72+2 72+2 72+2 74+2 76 55 60 84
## 3 70+2 79+2 79+2 79+2 70+2 49 51 81 73
## 4 16 14 17 25
## 5 78+2 73+2 73+2 73+2 78+2 73 61 69 79
## 6 68+2 76+2 76+2 76+2 68+2 60 45 79 73
## 7 48+2 48+2 48+2 48+2 48+2 61 79 86 71
## 8 69+2 75+2 75+2 75+2 69+2 42 33 80 72
## 9 74+2 70+2 70+2 70+2 74+2 67 64 51 82
## 10 50+2 46+2 46+2 46+2 50+2 68 77 71 73
## V58 V59 V60 V61 V62 V63 V64
## 1 Volleys Dribbling Curve FKAccuracy LongPassing BallControl Acceleration
## 2 73 78 79 78 82 82 75
## 3 37 49 36 40 67 63 46
## 4 13 15 18 17 32 17 58
## 5 57 72 49 46 75 72 84
## 6 51 63 42 48 72 73 33
## 7 74 71 64 60 55 77 66
## 8 40 49 52 43 77 48 57
## 9 57 78 60 61 75 79 78
## 10 73 76 73 69 67 76 78
## V65 V66 V67 V68 V69 V70 V71 V72
## 1 SprintSpeed Agility Reactions Balance ShotPower Jumping Stamina Strength
## 2 69 77 74 77 82 61 79 69
## 3 49 55 76 36 74 64 67 83
## 4 54 36 76 50 24 60 27 70
## 5 90 80 75 76 67 85 93 68
## 6 38 51 70 60 55 79 54 76
## 7 65 50 75 32 78 63 77 93
## 8 59 69 78 61 42 79 72 72
## 9 81 80 73 76 76 60 79 59
## 10 85 79 71 73 77 70 78 74
## V73 V74 V75 V76 V77 V78 V79
## 1 LongShots Aggression Interceptions Positioning Vision Penalties Composure
## 2 80 79 72 74 82 57 74
## 3 59 81 82 54 49 79 78
## 4 13 26 20 11 63 15 69
## 5 57 65 71 77 72 41 73
## 6 58 76 79 50 67 64 70
## 7 68 75 30 78 73 77 70
## 8 37 76 78 44 46 47 72
## 9 74 70 74 71 70 63 64
## 10 74 77 18 76 73 72 72
## V80 V81 V82 V83 V84 V85
## 1 Marking StandingTackle SlidingTackle GKDiving GKHandling GKKicking
## 2 73 75 72 9 14 6
## 3 82 83 79 7 9 12
## 4 18 20 12 80 73 65
## 5 76 76 80 7 12 10
## 6 83 77 76 12 7 11
## 7 21 15 19 15 12 11
## 8 80 77 78 10 15 13
## 9 71 77 76 15 16 13
## 10 40 18 12 15 9 10
## V86 V87 V88
## 1 GKPositioning GKReflexes Release Clause
## 2 9 10
## 3 10 15
## 4 77 85
## 5 8 15
## 6 12 13
## 7 11 8
## 8 15 8
## 9 7 8
## 10 15 16names(football) <- football[1,]
head(football)## ID Name Age Photo
## 1 ID Name Age Photo
## 2 207439 L. Paredes 24 https://cdn.sofifa.org/players/4/19/207439.png
## 3 156713 A. Granqvist 33 https://cdn.sofifa.org/players/4/19/156713.png
## 4 229909 A. Lunev 26 https://cdn.sofifa.org/players/4/19/229909.png
## 5 187347 I. Smolnikov 29 https://cdn.sofifa.org/players/4/19/187347.png
## 6 153260 Hilton 40 https://cdn.sofifa.org/players/4/19/153260.png
## Nationality Flag Overall Potential
## 1 Nationality Flag Overall Potential
## 2 Argentina https://cdn.sofifa.org/flags/52.png 80 85
## 3 Sweden https://cdn.sofifa.org/flags/46.png 80 80
## 4 Russia https://cdn.sofifa.org/flags/40.png 79 81
## 5 Russia https://cdn.sofifa.org/flags/40.png 79 79
## 6 Brazil https://cdn.sofifa.org/flags/54.png 78 78
## Club Club Logo Value Wage
## 1 Club Club Logo Value Wage
## 2 https://cdn.sofifa.org/flags/52.png 5684 1602
## 3 https://cdn.sofifa.org/flags/46.png 6370 3591
## 4 https://cdn.sofifa.org/flags/40.png 5675 3672
## 5 https://cdn.sofifa.org/flags/40.png 6030 1448
## 6 Montpellier HSC https://cdn.sofifa.org/teams/2/light/70.png 6405 19799
## Special Preferred Foot International Reputation Weak Foot Skill Moves
## 1 Special Preferred Foot International Reputation Weak Foot Skill Moves
## 2 2122 Right 2 4 4
## 3 1797 Right 2 4 2
## 4 1217 Right 1 3 1
## 5 2038 Right 2 3 3
## 6 1807 Right 2 3 3
## Work Rate Body Type Real Face Position Jersey Number Joined
## 1 Work Rate Body Type Real Face Position Jersey Number Joined
## 2 Medium/ Medium Normal No CM 5
## 3 High/ Medium Normal No LCB 4
## 4 Medium/ Medium Normal No GK 12
## 5 High/ High Lean No RB 2
## 6 Medium/ Medium Normal Yes CB 4 1-Aug-11
## Loaned From Contract Valid Until Height Weight LS ST RS LW LF CF
## 1 Loaned From Contract Valid Until Height Weight LS ST RS LW LF CF
## 2 5'11 165lbs 71+2 71+2 71+2 75+2 75+2 75+2
## 3 6'4 185lbs 62+2 62+2 62+2 56+2 58+2 58+2
## 4 6'2 176lbs
## 5 5'10 154lbs 70+2 70+2 70+2 73+2 72+2 72+2
## 6 2019 5'11 172lbs 58+2 58+2 58+2 58+2 59+2 59+2
## RF RW LAM CAM RAM LM LCM CM RCM RM LWB LDM CDM RDM RWB
## 1 RF RW LAM CAM RAM LM LCM CM RCM RM LWB LDM CDM RDM RWB
## 2 75+2 75+2 77+2 77+2 77+2 76+2 79+2 79+2 79+2 76+2 75+2 77+2 77+2 77+2 75+2
## 3 58+2 56+2 58+2 58+2 58+2 57+2 64+2 64+2 64+2 57+2 68+2 74+2 74+2 74+2 68+2
## 4
## 5 72+2 73+2 73+2 73+2 73+2 75+2 74+2 74+2 74+2 75+2 78+2 75+2 75+2 75+2 78+2
## 6 59+2 58+2 62+2 62+2 62+2 60+2 67+2 67+2 67+2 60+2 67+2 73+2 73+2 73+2 67+2
## LB LCB CB RCB RB Crossing Finishing HeadingAccuracy ShortPassing
## 1 LB LCB CB RCB RB Crossing Finishing HeadingAccuracy ShortPassing
## 2 74+2 72+2 72+2 72+2 74+2 76 55 60 84
## 3 70+2 79+2 79+2 79+2 70+2 49 51 81 73
## 4 16 14 17 25
## 5 78+2 73+2 73+2 73+2 78+2 73 61 69 79
## 6 68+2 76+2 76+2 76+2 68+2 60 45 79 73
## Volleys Dribbling Curve FKAccuracy LongPassing BallControl Acceleration
## 1 Volleys Dribbling Curve FKAccuracy LongPassing BallControl Acceleration
## 2 73 78 79 78 82 82 75
## 3 37 49 36 40 67 63 46
## 4 13 15 18 17 32 17 58
## 5 57 72 49 46 75 72 84
## 6 51 63 42 48 72 73 33
## SprintSpeed Agility Reactions Balance ShotPower Jumping Stamina Strength
## 1 SprintSpeed Agility Reactions Balance ShotPower Jumping Stamina Strength
## 2 69 77 74 77 82 61 79 69
## 3 49 55 76 36 74 64 67 83
## 4 54 36 76 50 24 60 27 70
## 5 90 80 75 76 67 85 93 68
## 6 38 51 70 60 55 79 54 76
## LongShots Aggression Interceptions Positioning Vision Penalties Composure
## 1 LongShots Aggression Interceptions Positioning Vision Penalties Composure
## 2 80 79 72 74 82 57 74
## 3 59 81 82 54 49 79 78
## 4 13 26 20 11 63 15 69
## 5 57 65 71 77 72 41 73
## 6 58 76 79 50 67 64 70
## Marking StandingTackle SlidingTackle GKDiving GKHandling GKKicking
## 1 Marking StandingTackle SlidingTackle GKDiving GKHandling GKKicking
## 2 73 75 72 9 14 6
## 3 82 83 79 7 9 12
## 4 18 20 12 80 73 65
## 5 76 76 80 7 12 10
## 6 83 77 76 12 7 11
## GKPositioning GKReflexes Release Clause
## 1 GKPositioning GKReflexes Release Clause
## 2 9 10
## 3 10 15
## 4 77 85
## 5 8 15
## 6 12 13football <- football[-c(1),]
head(football)## ID Name Age Photo
## 2 207439 L. Paredes 24 https://cdn.sofifa.org/players/4/19/207439.png
## 3 156713 A. Granqvist 33 https://cdn.sofifa.org/players/4/19/156713.png
## 4 229909 A. Lunev 26 https://cdn.sofifa.org/players/4/19/229909.png
## 5 187347 I. Smolnikov 29 https://cdn.sofifa.org/players/4/19/187347.png
## 6 153260 Hilton 40 https://cdn.sofifa.org/players/4/19/153260.png
## 7 187607 A. Dzyuba 29 https://cdn.sofifa.org/players/4/19/187607.png
## Nationality Flag Overall Potential
## 2 Argentina https://cdn.sofifa.org/flags/52.png 80 85
## 3 Sweden https://cdn.sofifa.org/flags/46.png 80 80
## 4 Russia https://cdn.sofifa.org/flags/40.png 79 81
## 5 Russia https://cdn.sofifa.org/flags/40.png 79 79
## 6 Brazil https://cdn.sofifa.org/flags/54.png 78 78
## 7 Russia https://cdn.sofifa.org/flags/40.png 78 78
## Club Club Logo Value Wage
## 2 https://cdn.sofifa.org/flags/52.png 5684 1602
## 3 https://cdn.sofifa.org/flags/46.png 6370 3591
## 4 https://cdn.sofifa.org/flags/40.png 5675 3672
## 5 https://cdn.sofifa.org/flags/40.png 6030 1448
## 6 Montpellier HSC https://cdn.sofifa.org/teams/2/light/70.png 6405 19799
## 7 https://cdn.sofifa.org/flags/40.png 5764 1105
## Special Preferred Foot International Reputation Weak Foot Skill Moves
## 2 2122 Right 2 4 4
## 3 1797 Right 2 4 2
## 4 1217 Right 1 3 1
## 5 2038 Right 2 3 3
## 6 1807 Right 2 3 3
## 7 1810 Right 2 3 3
## Work Rate Body Type Real Face Position Jersey Number Joined
## 2 Medium/ Medium Normal No CM 5
## 3 High/ Medium Normal No LCB 4
## 4 Medium/ Medium Normal No GK 12
## 5 High/ High Lean No RB 2
## 6 Medium/ Medium Normal Yes CB 4 1-Aug-11
## 7 High/ Medium Stocky No ST 22
## Loaned From Contract Valid Until Height Weight LS ST RS LW LF CF
## 2 5'11 165lbs 71+2 71+2 71+2 75+2 75+2 75+2
## 3 6'4 185lbs 62+2 62+2 62+2 56+2 58+2 58+2
## 4 6'2 176lbs
## 5 5'10 154lbs 70+2 70+2 70+2 73+2 72+2 72+2
## 6 2019 5'11 172lbs 58+2 58+2 58+2 58+2 59+2 59+2
## 7 6'5 201lbs 77+2 77+2 77+2 71+2 74+2 74+2
## RF RW LAM CAM RAM LM LCM CM RCM RM LWB LDM CDM RDM RWB
## 2 75+2 75+2 77+2 77+2 77+2 76+2 79+2 79+2 79+2 76+2 75+2 77+2 77+2 77+2 75+2
## 3 58+2 56+2 58+2 58+2 58+2 57+2 64+2 64+2 64+2 57+2 68+2 74+2 74+2 74+2 68+2
## 4
## 5 72+2 73+2 73+2 73+2 73+2 75+2 74+2 74+2 74+2 75+2 78+2 75+2 75+2 75+2 78+2
## 6 59+2 58+2 62+2 62+2 62+2 60+2 67+2 67+2 67+2 60+2 67+2 73+2 73+2 73+2 67+2
## 7 74+2 71+2 71+2 71+2 71+2 71+2 66+2 66+2 66+2 71+2 52+2 52+2 52+2 52+2 52+2
## LB LCB CB RCB RB Crossing Finishing HeadingAccuracy ShortPassing
## 2 74+2 72+2 72+2 72+2 74+2 76 55 60 84
## 3 70+2 79+2 79+2 79+2 70+2 49 51 81 73
## 4 16 14 17 25
## 5 78+2 73+2 73+2 73+2 78+2 73 61 69 79
## 6 68+2 76+2 76+2 76+2 68+2 60 45 79 73
## 7 48+2 48+2 48+2 48+2 48+2 61 79 86 71
## Volleys Dribbling Curve FKAccuracy LongPassing BallControl Acceleration
## 2 73 78 79 78 82 82 75
## 3 37 49 36 40 67 63 46
## 4 13 15 18 17 32 17 58
## 5 57 72 49 46 75 72 84
## 6 51 63 42 48 72 73 33
## 7 74 71 64 60 55 77 66
## SprintSpeed Agility Reactions Balance ShotPower Jumping Stamina Strength
## 2 69 77 74 77 82 61 79 69
## 3 49 55 76 36 74 64 67 83
## 4 54 36 76 50 24 60 27 70
## 5 90 80 75 76 67 85 93 68
## 6 38 51 70 60 55 79 54 76
## 7 65 50 75 32 78 63 77 93
## LongShots Aggression Interceptions Positioning Vision Penalties Composure
## 2 80 79 72 74 82 57 74
## 3 59 81 82 54 49 79 78
## 4 13 26 20 11 63 15 69
## 5 57 65 71 77 72 41 73
## 6 58 76 79 50 67 64 70
## 7 68 75 30 78 73 77 70
## Marking StandingTackle SlidingTackle GKDiving GKHandling GKKicking
## 2 73 75 72 9 14 6
## 3 82 83 79 7 9 12
## 4 18 20 12 80 73 65
## 5 76 76 80 7 12 10
## 6 83 77 76 12 7 11
## 7 21 15 19 15 12 11
## GKPositioning GKReflexes Release Clause
## 2 9 10
## 3 10 15
## 4 77 85
## 5 8 15
## 6 12 13
## 7 11 8nrow(football)## [1] 18207table(football$Position)##
## CAM CB CDM CF CM GK LAM LB LCB LCM LDM LF LM LS LW
## 60 958 1778 948 74 1394 2025 21 1322 648 395 243 15 1095 207 381
## LWB RAM RB RCB RCM RDM RF RM RS RW RWB ST
## 78 21 1291 662 391 248 16 1124 203 370 87 21522. Scatter Plot
2.1 Filter for strikers
Strikers are defined in the dataset as Position = “ST”.
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionfootball_st <- football %>% filter(Position == "ST")
head(football_st)## ID Name Age Photo
## 1 187607 A. Dzyuba 29 https://cdn.sofifa.org/players/4/19/187607.png
## 2 183389 G. Sio 29 https://cdn.sofifa.org/players/4/19/183389.png
## 3 245683 K. Fofana 26 https://cdn.sofifa.org/players/4/19/245683.png
## 4 190461 B. Sigur̡arson 27 https://cdn.sofifa.org/players/4/19/190461.png
## 5 225900 J. Sambenito 26 https://cdn.sofifa.org/players/4/19/225900.png
## 6 246405 B. Angulo 22 https://cdn.sofifa.org/players/4/19/246405.png
## Nationality Flag Overall Potential Club
## 1 Russia https://cdn.sofifa.org/flags/40.png 78 78
## 2 Ivory Coast https://cdn.sofifa.org/flags/108.png 77 77
## 3 Ivory Coast https://cdn.sofifa.org/flags/108.png 75 75
## 4 Iceland https://cdn.sofifa.org/flags/24.png 73 74
## 5 Paraguay https://cdn.sofifa.org/flags/58.png 71 74
## 6 Ecuador https://cdn.sofifa.org/flags/57.png 71 77
## Club Logo Value Wage Special Preferred Foot
## 1 https://cdn.sofifa.org/flags/40.png 5764 1105 1810 Right
## 2 https://cdn.sofifa.org/flags/108.png 5275 2138 1933 Left
## 3 https://cdn.sofifa.org/flags/108.png 5589 3875 1877 Right
## 4 https://cdn.sofifa.org/flags/24.png 5629 3661 1893 Right
## 5 https://cdn.sofifa.org/flags/58.png 6113 2445 1651 Right
## 6 https://cdn.sofifa.org/flags/57.png 5057 2216 1628 Right
## International Reputation Weak Foot Skill Moves Work Rate Body Type
## 1 2 3 3 High/ Medium Stocky
## 2 2 3 3 High/ Low Normal
## 3 1 3 3 Medium/ Medium Normal
## 4 1 4 3 High/ High Normal
## 5 1 3 2 High/ Medium Lean
## 6 1 4 3 High/ Low Normal
## Real Face Position Jersey Number Joined Loaned From Contract Valid Until
## 1 No ST 22
## 2 No ST 21
## 3 No ST 22
## 4 No ST 9
## 5 No ST 9
## 6 No ST 19
## Height Weight LS ST RS LW LF CF RF RW LAM CAM RAM LM
## 1 6'5 201lbs 77+2 77+2 77+2 71+2 74+2 74+2 74+2 71+2 71+2 71+2 71+2 71+2
## 2 5'11 176lbs 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 74+2 74+2 74+2 74+2
## 3 6'2 179lbs 73+2 73+2 73+2 71+2 72+2 72+2 72+2 71+2 71+2 71+2 71+2 71+2
## 4 6'1 190lbs 72+2 72+2 72+2 71+2 71+2 71+2 71+2 71+2 70+2 70+2 70+2 71+2
## 5 6'0 190lbs 70+2 70+2 70+2 64+2 67+2 67+2 67+2 64+2 63+2 63+2 63+2 62+2
## 6 6'0 154lbs 70+2 70+2 70+2 67+2 68+2 68+2 68+2 67+2 63+2 63+2 63+2 65+2
## LCM CM RCM RM LWB LDM CDM RDM RWB LB LCB CB RCB RB
## 1 66+2 66+2 66+2 71+2 52+2 52+2 52+2 52+2 52+2 48+2 48+2 48+2 48+2 48+2
## 2 67+2 67+2 67+2 74+2 53+2 52+2 52+2 52+2 53+2 50+2 46+2 46+2 46+2 50+2
## 3 67+2 67+2 67+2 71+2 59+2 57+2 57+2 57+2 59+2 57+2 52+2 52+2 52+2 57+2
## 4 64+2 64+2 64+2 71+2 59+2 55+2 55+2 55+2 59+2 56+2 53+2 53+2 53+2 56+2
## 5 55+2 55+2 55+2 62+2 43+2 41+2 41+2 41+2 43+2 41+2 38+2 38+2 38+2 41+2
## 6 54+2 54+2 54+2 65+2 47+2 39+2 39+2 39+2 47+2 44+2 36+2 36+2 36+2 44+2
## Crossing Finishing HeadingAccuracy ShortPassing Volleys Dribbling Curve
## 1 61 79 86 71 74 71 64
## 2 68 77 71 73 73 76 73
## 3 66 75 72 74 74 72 63
## 4 66 71 68 68 65 73 63
## 5 40 74 72 57 72 60 64
## 6 50 78 69 56 46 76 58
## FKAccuracy LongPassing BallControl Acceleration SprintSpeed Agility Reactions
## 1 60 55 77 66 65 50 75
## 2 69 67 76 78 85 79 71
## 3 59 58 75 59 77 63 72
## 4 48 44 73 78 79 83 74
## 5 42 42 63 79 72 61 69
## 6 58 33 71 82 79 78 73
## Balance ShotPower Jumping Stamina Strength LongShots Aggression Interceptions
## 1 32 78 63 77 93 68 75 30
## 2 73 77 70 78 74 74 77 18
## 3 60 78 69 83 77 73 67 40
## 4 76 68 78 90 85 66 73 42
## 5 64 73 69 67 72 67 49 14
## 6 64 72 69 77 69 54 28 16
## Positioning Vision Penalties Composure Marking StandingTackle SlidingTackle
## 1 78 73 77 70 21 15 19
## 2 76 73 72 72 40 18 12
## 3 72 69 74 83 23 37 46
## 4 73 64 69 76 31 39 24
## 5 75 60 67 74 15 16 16
## 6 62 45 82 51 11 18 12
## GKDiving GKHandling GKKicking GKPositioning GKReflexes Release Clause
## 1 15 12 11 11 8
## 2 15 9 10 15 16
## 3 7 11 7 11 14
## 4 9 12 10 15 16
## 5 15 16 15 7 7
## 6 11 8 10 7 6nrow(football_st)## [1] 21522.2 Scatter Plot
convert to numeric.
str(football_st$Wage)## chr [1:2152] "1105" "2138" "3875" "3661" "2445" "2216" "4457" "3370" ...str(football_st$Value)## chr [1:2152] "5764" "5275" "5589" "5629" "6113" "5057" "6561" "6146" ...football_st$Wage <- as.numeric(football_st$Wage)
football_st$Value <- as.numeric(football_st$Value)library(ggplot2)
library(ggpubr)
ggplot(football_st) + aes(x = Wage, y = Value) +
geom_point(shape = 2, colour = "black") +
xlab("Wage") + ylab("Value") +
ggtitle("Wage and Value") +
geom_smooth(method = lm) +
stat_regline_equation(label.x = 150000, label.y = 1700) +
stat_cor(method = "pearson", label.x = 300000, label.y = 1600)## `geom_smooth()` using formula = 'y ~ x'
3. Simple Linear Regression
value_simple <- lm(football_st$Value ~ football_st$Wage)
summary(value_simple)##
## Call:
## lm(formula = football_st$Value ~ football_st$Wage)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17073527 -633009 -209153 198333 38355242
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4.175e+05 7.060e+04 -5.913 3.91e-09 ***
## football_st$Wage 2.179e+02 2.721e+00 80.068 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2838000 on 2150 degrees of freedom
## Multiple R-squared: 0.7489, Adjusted R-squared: 0.7487
## F-statistic: 6411 on 1 and 2150 DF, p-value: < 2.2e-16confint(value_simple, level = 0.95)## 2.5 % 97.5 %
## (Intercept) -555911.3195 -278995.9221
## football_st$Wage 212.5681 223.24224. Residuals
value_simple_stdresiduals <- rstandard(value_simple)
head(value_simple_stdresiduals)## 1 2 3 4 5 6
## 0.06430004 -0.01520939 -0.14850129 -0.13205208 -0.03849210 -0.02127676Standard residuals.
football_st_comb <- cbind(football_st, value_simple_stdresiduals)
head(football_st_comb)## ID Name Age Photo
## 1 187607 A. Dzyuba 29 https://cdn.sofifa.org/players/4/19/187607.png
## 2 183389 G. Sio 29 https://cdn.sofifa.org/players/4/19/183389.png
## 3 245683 K. Fofana 26 https://cdn.sofifa.org/players/4/19/245683.png
## 4 190461 B. Sigur̡arson 27 https://cdn.sofifa.org/players/4/19/190461.png
## 5 225900 J. Sambenito 26 https://cdn.sofifa.org/players/4/19/225900.png
## 6 246405 B. Angulo 22 https://cdn.sofifa.org/players/4/19/246405.png
## Nationality Flag Overall Potential Club
## 1 Russia https://cdn.sofifa.org/flags/40.png 78 78
## 2 Ivory Coast https://cdn.sofifa.org/flags/108.png 77 77
## 3 Ivory Coast https://cdn.sofifa.org/flags/108.png 75 75
## 4 Iceland https://cdn.sofifa.org/flags/24.png 73 74
## 5 Paraguay https://cdn.sofifa.org/flags/58.png 71 74
## 6 Ecuador https://cdn.sofifa.org/flags/57.png 71 77
## Club Logo Value Wage Special Preferred Foot
## 1 https://cdn.sofifa.org/flags/40.png 5764 1105 1810 Right
## 2 https://cdn.sofifa.org/flags/108.png 5275 2138 1933 Left
## 3 https://cdn.sofifa.org/flags/108.png 5589 3875 1877 Right
## 4 https://cdn.sofifa.org/flags/24.png 5629 3661 1893 Right
## 5 https://cdn.sofifa.org/flags/58.png 6113 2445 1651 Right
## 6 https://cdn.sofifa.org/flags/57.png 5057 2216 1628 Right
## International Reputation Weak Foot Skill Moves Work Rate Body Type
## 1 2 3 3 High/ Medium Stocky
## 2 2 3 3 High/ Low Normal
## 3 1 3 3 Medium/ Medium Normal
## 4 1 4 3 High/ High Normal
## 5 1 3 2 High/ Medium Lean
## 6 1 4 3 High/ Low Normal
## Real Face Position Jersey Number Joined Loaned From Contract Valid Until
## 1 No ST 22
## 2 No ST 21
## 3 No ST 22
## 4 No ST 9
## 5 No ST 9
## 6 No ST 19
## Height Weight LS ST RS LW LF CF RF RW LAM CAM RAM LM
## 1 6'5 201lbs 77+2 77+2 77+2 71+2 74+2 74+2 74+2 71+2 71+2 71+2 71+2 71+2
## 2 5'11 176lbs 75+2 75+2 75+2 75+2 75+2 75+2 75+2 75+2 74+2 74+2 74+2 74+2
## 3 6'2 179lbs 73+2 73+2 73+2 71+2 72+2 72+2 72+2 71+2 71+2 71+2 71+2 71+2
## 4 6'1 190lbs 72+2 72+2 72+2 71+2 71+2 71+2 71+2 71+2 70+2 70+2 70+2 71+2
## 5 6'0 190lbs 70+2 70+2 70+2 64+2 67+2 67+2 67+2 64+2 63+2 63+2 63+2 62+2
## 6 6'0 154lbs 70+2 70+2 70+2 67+2 68+2 68+2 68+2 67+2 63+2 63+2 63+2 65+2
## LCM CM RCM RM LWB LDM CDM RDM RWB LB LCB CB RCB RB
## 1 66+2 66+2 66+2 71+2 52+2 52+2 52+2 52+2 52+2 48+2 48+2 48+2 48+2 48+2
## 2 67+2 67+2 67+2 74+2 53+2 52+2 52+2 52+2 53+2 50+2 46+2 46+2 46+2 50+2
## 3 67+2 67+2 67+2 71+2 59+2 57+2 57+2 57+2 59+2 57+2 52+2 52+2 52+2 57+2
## 4 64+2 64+2 64+2 71+2 59+2 55+2 55+2 55+2 59+2 56+2 53+2 53+2 53+2 56+2
## 5 55+2 55+2 55+2 62+2 43+2 41+2 41+2 41+2 43+2 41+2 38+2 38+2 38+2 41+2
## 6 54+2 54+2 54+2 65+2 47+2 39+2 39+2 39+2 47+2 44+2 36+2 36+2 36+2 44+2
## Crossing Finishing HeadingAccuracy ShortPassing Volleys Dribbling Curve
## 1 61 79 86 71 74 71 64
## 2 68 77 71 73 73 76 73
## 3 66 75 72 74 74 72 63
## 4 66 71 68 68 65 73 63
## 5 40 74 72 57 72 60 64
## 6 50 78 69 56 46 76 58
## FKAccuracy LongPassing BallControl Acceleration SprintSpeed Agility Reactions
## 1 60 55 77 66 65 50 75
## 2 69 67 76 78 85 79 71
## 3 59 58 75 59 77 63 72
## 4 48 44 73 78 79 83 74
## 5 42 42 63 79 72 61 69
## 6 58 33 71 82 79 78 73
## Balance ShotPower Jumping Stamina Strength LongShots Aggression Interceptions
## 1 32 78 63 77 93 68 75 30
## 2 73 77 70 78 74 74 77 18
## 3 60 78 69 83 77 73 67 40
## 4 76 68 78 90 85 66 73 42
## 5 64 73 69 67 72 67 49 14
## 6 64 72 69 77 69 54 28 16
## Positioning Vision Penalties Composure Marking StandingTackle SlidingTackle
## 1 78 73 77 70 21 15 19
## 2 76 73 72 72 40 18 12
## 3 72 69 74 83 23 37 46
## 4 73 64 69 76 31 39 24
## 5 75 60 67 74 15 16 16
## 6 62 45 82 51 11 18 12
## GKDiving GKHandling GKKicking GKPositioning GKReflexes Release Clause
## 1 15 12 11 11 8
## 2 15 9 10 15 16
## 3 7 11 7 11 14
## 4 9 12 10 15 16
## 5 15 16 15 7 7
## 6 11 8 10 7 6
## value_simple_stdresiduals
## 1 0.06430004
## 2 -0.01520939
## 3 -0.14850129
## 4 -0.13205208
## 5 -0.03849210
## 6 -0.02127676Plot residuals.
ggplot(football_st_comb) + aes(x = football_st_comb$Value, y = football_st_comb$value_simple_stdresiduals) +
geom_point() +
xlab("Value") + ylab("Standard Residuals") +
ggtitle("Wage and Value Prediction, Residuals")## Warning: Use of `football_st_comb$Value` is discouraged.
## ℹ Use `Value` instead.## Warning: Use of `football_st_comb$value_simple_stdresiduals` is discouraged.
## ℹ Use `value_simple_stdresiduals` instead.
4.1 Normality
ggplot(football_st) + aes(x = Value) +
geom_histogram() +
ylab("Count") +
ggtitle("Distribution of Value")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Using the Shapiro-Wilks test.
H-0: normal distribution.
H-1: distribution is different from a normal distribution.
shapiro.test(football_st$Value)##
## Shapiro-Wilk normality test
##
## data: football_st$Value
## W = 0.37447, p-value < 2.2e-164.2 Autocorrelation
May not be very applicable here. But just for illustration……
library(car)## Loading required package: carData##
## Attaching package: 'car'## The following object is masked from 'package:dplyr':
##
## recodedurbinWatsonTest(value_simple) ## lag Autocorrelation D-W Statistic p-value
## 1 0.2167301 1.566536 0
## Alternative hypothesis: rho != 05. Multiple Linear Regression
Subset data for simplicity.
football_st_2 <- football_st[, c("Age", "Balance", "ShotPower", "Aggression",
"Positioning", "Composure", "Wage")]
head(football_st_2)## Age Balance ShotPower Aggression Positioning Composure Wage
## 1 29 32 78 75 78 70 1105
## 2 29 73 77 77 76 72 2138
## 3 26 60 78 67 72 83 3875
## 4 27 76 68 73 73 76 3661
## 5 26 64 73 49 75 74 2445
## 6 22 64 72 28 62 51 2216Convert to numeric.
library(dplyr)
football_st_2 <- football_st_2 %>% mutate_if(is.character, as.numeric)
str(football_st_2)## 'data.frame': 2152 obs. of 7 variables:
## $ Age : num 29 29 26 27 26 22 22 28 31 28 ...
## $ Balance : num 32 73 60 76 64 64 65 75 69 56 ...
## $ ShotPower : num 78 77 78 68 73 72 66 75 69 71 ...
## $ Aggression : num 75 77 67 73 49 28 30 36 68 59 ...
## $ Positioning: num 78 76 72 73 75 62 76 68 69 72 ...
## $ Composure : num 70 72 83 76 74 51 62 56 80 56 ...
## $ Wage : num 1105 2138 3875 3661 2445 ...A multiple regression model showing unstandardised estimates.
The predictors included in the model are: Age, Balance, ShotPower, Aggression, Positioning, and Composure.
names(football_st_2)## [1] "Age" "Balance" "ShotPower" "Aggression" "Positioning"
## [6] "Composure" "Wage"wage_model_st <- lm(Wage ~ Age + Balance + ShotPower +
Aggression + Positioning + Composure,
data = football_st_2)
summary(wage_model_st)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Aggression +
## Positioning + Composure, data = football_st_2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -31822 -8232 -2313 4754 350592
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -77073.40 4064.61 -18.962 < 2e-16 ***
## Age -1014.25 110.94 -9.143 < 2e-16 ***
## Balance 120.41 35.90 3.354 0.00081 ***
## ShotPower 498.07 74.43 6.692 2.81e-11 ***
## Aggression 15.96 32.29 0.494 0.62129
## Positioning 741.71 82.42 8.999 < 2e-16 ***
## Composure 424.72 71.66 5.927 3.58e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18840 on 2145 degrees of freedom
## Multiple R-squared: 0.2997, Adjusted R-squared: 0.2978
## F-statistic: 153 on 6 and 2145 DF, p-value: < 2.2e-16coef(wage_model_st)## (Intercept) Age Balance ShotPower Aggression Positioning
## -77073.39877 -1014.24567 120.40620 498.06517 15.95657 741.70804
## Composure
## 424.72405confint(wage_model_st, level = 0.95)## 2.5 % 97.5 %
## (Intercept) -85044.38590 -69102.41165
## Age -1231.79758 -796.69375
## Balance 50.00615 190.80626
## ShotPower 352.09956 644.03079
## Aggression -47.37581 79.28895
## Positioning 580.07796 903.33813
## Composure 284.19780 565.250315.1 Standardised estimates
A multiple regression model showing standardised estimates.
The predictors included in the model are: Age, Balance, ShotPower, Aggression, Positioning, and Composure.
library(lm.beta)## Warning: package 'lm.beta' was built under R version 4.4.2wage_model_st_std <- lm.beta::lm.beta(wage_model_st)
coef(wage_model_st_std)## (Intercept) Age Balance ShotPower Aggression Positioning
## NA -0.21358305 0.06178231 0.20182976 0.01126852 0.30316025
## Composure
## 0.19146721confint(wage_model_st_std)## 2.5 % 97.5 %
## (Intercept) NA NA
## Age -217.76550 217.33833
## Balance -70.33827 70.46184
## ShotPower -145.76378 146.16744
## Aggression -63.32111 63.34365
## Positioning -161.32692 161.93324
## Composure -140.33479 140.717725.2 Residuals
wage_model_st_residuals <- rstandard(wage_model_st)
head(wage_model_st_residuals)## 1 2 3 4 5 6
## -1.2711799 -1.4183035 -1.5151160 -1.1956035 -1.3820667 -0.5348701football_st_comb_2 <- cbind(football_st_2, wage_model_st_residuals)
head(football_st_comb_2)## Age Balance ShotPower Aggression Positioning Composure Wage
## 1 29 32 78 75 78 70 1105
## 2 29 73 77 77 76 72 2138
## 3 26 60 78 67 72 83 3875
## 4 27 76 68 73 73 76 3661
## 5 26 64 73 49 75 74 2445
## 6 22 64 72 28 62 51 2216
## wage_model_st_residuals
## 1 -1.2711799
## 2 -1.4183035
## 3 -1.5151160
## 4 -1.1956035
## 5 -1.3820667
## 6 -0.5348701ggplot(football_st_comb_2) + aes(x = Wage, y = wage_model_st_residuals) +
geom_point() + xlab("Wage") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Wage Prediction")
ggplot(football_st_comb_2) + aes(x = Age, y = wage_model_st_residuals) +
geom_point() + xlab("Age") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Age")
ggplot(football_st_comb_2) + aes(x = ShotPower, y = wage_model_st_residuals) +
geom_point() + xlab("Shot Power") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Shot Power")
ggplot(football_st_comb_2) + aes(x = Aggression, y = wage_model_st_residuals) +
geom_point() + xlab("Aggression") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Aggression")
ggplot(football_st_comb_2) + aes(x = Positioning, y = wage_model_st_residuals) +
geom_point() + xlab("Positioning") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Positionng")
ggplot(football_st_comb_2) + aes(x = Composure, y = wage_model_st_residuals) +
geom_point() + xlab("Composure") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Composure")
5.3 Model evaluation
5.3.1 Normality
library(ggplot2)
ggplot(football_st_2) + aes(x = Wage) +
geom_histogram() +
ylab("Count") +
ggtitle("Distribution of wage (strikers)")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
ggplot(football_st_2) + aes(x = Wage) +
geom_histogram() +
ylab("Count") +
scale_x_log10() +
ggtitle("Distribution of log(wage) (strikers)")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Using the Shapiro-Wilks test.
H-0: normal distribution
H-1: distribution is different from a normal distribution.
shapiro.test(football_st_2$Wage)##
## Shapiro-Wilk normality test
##
## data: football_st_2$Wage
## W = 0.39056, p-value < 2.2e-165.3.2 Multicollinearity
How much the variance of an estimated regression coefficient increases if your predictors are correlated.
In other words, no 2 pairs of predicts should not be strongly correlated with each other.
If no factors are correlated, the VIFs will all be 1.
Rule of thumb: If VIF > 10, mullticollinearity is high.
library(car)
vif(wage_model_st)## Age Balance ShotPower Aggression Positioning Composure
## 1.671663 1.039327 2.786601 1.593244 3.476150 3.1964335.3.3 Autocorrelation
0 <= D-W <= 4.
Rule of thumb:
D-W = 2.0 means that there is no autocorrelation.
D-W < = means there is positive autocorrelation.
D-W > 2 means negative autocorrelation.
This applies in time series data; so not so applicable here.
durbinWatsonTest(wage_model_st)## lag Autocorrelation D-W Statistic p-value
## 1 0.5038085 0.9915208 0
## Alternative hypothesis: rho != 05.3.4 Heteroskedasticity
Perform a Breusch-Pagan Test to test for heteroskedasticity/homoskedasticity.
library(lmtest)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericbptest(wage_model_st)##
## studentized Breusch-Pagan test
##
## data: wage_model_st
## BP = 91.188, df = 6, p-value < 2.2e-165.3.5 Automatic evaluation
We can also automatically evaluate the model.
library(gvlma)
gvlma(wage_model_st)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Aggression +
## Positioning + Composure, data = football_st_2)
##
## Coefficients:
## (Intercept) Age Balance ShotPower Aggression Positioning
## -77073.40 -1014.25 120.41 498.07 15.96 741.71
## Composure
## 424.72
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st)
##
## Value p-value Decision
## Global Stat 1307104.5 0 Assumptions NOT satisfied!
## Skewness 26054.7 0 Assumptions NOT satisfied!
## Kurtosis 1280082.5 0 Assumptions NOT satisfied!
## Link Function 791.9 0 Assumptions NOT satisfied!
## Heteroscedasticity 175.5 0 Assumptions NOT satisfied!6. Stepwise Regression
Stepwise regression is a modification of the ordinary regression approach.
library(stats)
wage_model_st_step <- step(wage_model_st,
direction = "both")## Start: AIC=42374.94
## Wage ~ Age + Balance + ShotPower + Aggression + Positioning +
## Composure
##
## Df Sum of Sq RSS AIC
## - Aggression 1 8.6672e+07 7.6162e+11 42373
## <none> 7.6154e+11 42375
## - Balance 1 3.9939e+09 7.6553e+11 42384
## - Composure 1 1.2472e+10 7.7401e+11 42408
## - ShotPower 1 1.5897e+10 7.7743e+11 42417
## - Positioning 1 2.8752e+10 7.9029e+11 42453
## - Age 1 2.9676e+10 7.9121e+11 42455
##
## Step: AIC=42373.18
## Wage ~ Age + Balance + ShotPower + Positioning + Composure
##
## Df Sum of Sq RSS AIC
## <none> 7.6162e+11 42373
## + Aggression 1 8.6672e+07 7.6154e+11 42375
## - Balance 1 3.9197e+09 7.6554e+11 42382
## - Composure 1 1.2939e+10 7.7456e+11 42407
## - ShotPower 1 1.7279e+10 7.7890e+11 42419
## - Positioning 1 2.8770e+10 7.9039e+11 42451
## - Age 1 3.0373e+10 7.9200e+11 42455summary(wage_model_st_step)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Positioning +
## Composure, data = football_st_2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -31793 -8228 -2326 4830 350282
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -77250.10 4048.13 -19.083 < 2e-16 ***
## Age -1002.58 108.38 -9.251 < 2e-16 ***
## Balance 118.78 35.74 3.323 0.000904 ***
## ShotPower 506.25 72.55 6.978 3.98e-12 ***
## Positioning 741.93 82.40 9.004 < 2e-16 ***
## Composure 429.17 71.08 6.038 1.83e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18840 on 2146 degrees of freedom
## Multiple R-squared: 0.2997, Adjusted R-squared: 0.298
## F-statistic: 183.6 on 5 and 2146 DF, p-value: < 2.2e-16gvlma(wage_model_st_step)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Positioning +
## Composure, data = football_st_2)
##
## Coefficients:
## (Intercept) Age Balance ShotPower Positioning Composure
## -77250.1 -1002.6 118.8 506.2 741.9 429.2
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_step)
##
## Value p-value Decision
## Global Stat 1300530.2 0 Assumptions NOT satisfied!
## Skewness 25983.6 0 Assumptions NOT satisfied!
## Kurtosis 1273577.0 0 Assumptions NOT satisfied!
## Link Function 794.0 0 Assumptions NOT satisfied!
## Heteroscedasticity 175.5 0 Assumptions NOT satisfied!7. Data mining approach
Now, we will use the data mining approach.
7.1 Training validation split
Split the data into training and validation sets.
Set the seed using our favourite number :-)
set.seed(666)Create the indices for the split This samples the row indices to split the data into training and validation.
train_index <- sample(1:nrow(football_st_2), 0.7 * nrow(football_st_2))
valid_index <- setdiff(1:nrow(football_st_2), train_index)Using the indices, create the training and validation sets This is similar in principle to splitting a data frame by row.
train_df_st <- football_st_2[train_index, ]
valid_df_st <- football_st_2[valid_index, ]It is a good habit to check after splitting.
nrow(train_df_st)## [1] 1506nrow(valid_df_st)## [1] 6467.2 Training
Training the model on the training set.
wage_model_st_2 <- lm(Wage ~ Age + Balance + ShotPower +
Aggression + Positioning + Composure,
data = train_df_st)
summary(wage_model_st_2)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Aggression +
## Positioning + Composure, data = train_df_st)
##
## Residuals:
## Min 1Q Median 3Q Max
## -32861 -8569 -2336 5182 347609
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -81327.50 5253.63 -15.480 < 2e-16 ***
## Age -1032.61 146.38 -7.054 2.64e-12 ***
## Balance 131.37 46.63 2.817 0.00491 **
## ShotPower 514.89 98.00 5.254 1.70e-07 ***
## Aggression 13.64 41.73 0.327 0.74380
## Positioning 692.34 107.84 6.420 1.82e-10 ***
## Composure 533.27 93.18 5.723 1.26e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 20410 on 1499 degrees of freedom
## Multiple R-squared: 0.2877, Adjusted R-squared: 0.2848
## F-statistic: 100.9 on 6 and 1499 DF, p-value: < 2.2e-167.3 Model evaluation
Predict the outcome (i.e. wage) of the training and validation sets using the model from the training set. Compare the errors between the training and validation sets.
library(forecast)## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo##
## Attaching package: 'forecast'## The following object is masked from 'package:ggpubr':
##
## gghistogramwage_model_st_2_pred_train <- predict(wage_model_st_2,
train_df_st)
accuracy(wage_model_st_2_pred_train, train_df_st$Wage)## ME RMSE MAE MPE MAPE
## Test set -3.02254e-10 20363.72 9804.857 -30.84404 131.6738wage_model_st_2_pred_valid <- predict(wage_model_st_2,
valid_df_st)
accuracy(wage_model_st_2_pred_valid, valid_df_st$Wage)## ME RMSE MAE MPE MAPE
## Test set -910.8093 14653.34 9444.861 -32.5431 130.3103max(train_df_st$Wage) - min(train_df_st$Wage)## [1] 406504sd(train_df_st$Wage)## [1] 24135.81max(valid_df_st$Wage) - min(valid_df_st$Wage)## [1] 205030sd(valid_df_st$Wage)## [1] 18074.14library(car)
vif(wage_model_st_2)## Age Balance ShotPower Aggression Positioning Composure
## 1.690727 1.037080 2.900017 1.602035 3.573232 3.185606library(lmtest)
bptest(wage_model_st_2)##
## studentized Breusch-Pagan test
##
## data: wage_model_st_2
## BP = 72.484, df = 6, p-value = 1.264e-137.4 Predicting
Predict new players
new <- read.csv("new.csv", header = TRUE)
wage_model_st_2_pred_new <- predict(wage_model_st_2,
newdata = new, interval = "confidence")
wage_model_st_2_pred_new## fit lwr upr
## 1 21523.43 18689.82 24357.04
## 2 23759.40 20030.25 27488.55
## 3 21465.21 19657.65 23272.778. Categorical Predictors
Subset to include categorical variable: preferred foot
football_st_3 <- football_st[, c("Preferred Foot", "Positioning", "Composure", "Wage")]
head(football_st_3)## Preferred Foot Positioning Composure Wage
## 1 Right 78 70 1105
## 2 Left 76 72 2138
## 3 Right 72 83 3875
## 4 Right 73 76 3661
## 5 Right 75 74 2445
## 6 Right 62 51 2216names(football_st_3)[1] <- "Preferred_Foot"
football_st_3$Positioning <- as.numeric(football_st_3$Positioning)
football_st_3$Composure <- as.numeric(football_st_3$Composure)8.1 Traditional Statistics
wage_model_st_cat <- lm(Wage ~ factor(Preferred_Foot) + Positioning + Composure, data = football_st_3)confint(wage_model_st_cat, level = 0.95)## 2.5 % 97.5 %
## (Intercept) -71769.7525 -59180.1997
## factor(Preferred_Foot)Right -3482.4352 1307.4754
## Positioning 668.9041 964.3812
## Composure 304.0428 572.1551Residuals.
wage_model_st_cat_stdresiduals <- rstandard(wage_model_st_cat)
head(wage_model_st_cat_stdresiduals)## 1 2 3 4 5 6
## -1.3769868 -1.3424571 -1.2770311 -1.1703986 -1.2718790 -0.2163923football_st_3_cat <- cbind(football_st_3, wage_model_st_cat_stdresiduals)
head(football_st_3_cat)## Preferred_Foot Positioning Composure Wage wage_model_st_cat_stdresiduals
## 1 Right 78 70 1105 -1.3769868
## 2 Left 76 72 2138 -1.3424571
## 3 Right 72 83 3875 -1.2770311
## 4 Right 73 76 3661 -1.1703986
## 5 Right 75 74 2445 -1.2718790
## 6 Right 62 51 2216 -0.2163923ggplot(football_st_3_cat) + aes(x = Wage, y = wage_model_st_cat_stdresiduals) +
geom_point() + xlab("Wage") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Wage")
Positioning
ggplot(football_st_3_cat) + aes(x = Positioning, y = wage_model_st_cat_stdresiduals) +
geom_point() + xlab("Positioning") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Positioning")
Composure
ggplot(football_st_3_cat) + aes(x = Composure, y = wage_model_st_cat_stdresiduals) +
geom_point() + xlab("Composure") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Composure")
ggplot(football_st_3_cat) + aes(x = Preferred_Foot, y = wage_model_st_cat_stdresiduals) +
geom_point() + xlab("Preferred Foot") + ylab("Standarised Residuals") +
ggtitle("Standarised Residual Plot, Preferred Foot")
8.2 Model Evaluation
ggplot(football_st_3_cat) + aes(x = Wage) +
geom_histogram() +
ylab("Count") +
ggtitle("Distribution of Wage")## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Using the Shapiro-Wilks test.
H-0: normal distribution.
H-alt: distribution is different from a normal distribution.
shapiro.test(football_st_3_cat$Wage)##
## Shapiro-Wilk normality test
##
## data: football_st_3_cat$Wage
## W = 0.39056, p-value < 2.2e-16Multicollinearity
vif(wage_model_st_cat)## factor(Preferred_Foot) Positioning Composure
## 1.002720 2.738872 2.743181Homoscedasticity.
bptest(wage_model_st_cat)##
## studentized Breusch-Pagan test
##
## data: wage_model_st_cat
## BP = 82.465, df = 3, p-value < 2.2e-16gvlma(wage_model_st_cat)##
## Call:
## lm(formula = Wage ~ factor(Preferred_Foot) + Positioning + Composure,
## data = football_st_3)
##
## Coefficients:
## (Intercept) factor(Preferred_Foot)Right
## -65475.0 -1087.5
## Positioning Composure
## 816.6 438.1
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_cat)
##
## Value p-value Decision
## Global Stat 1208344.4 0 Assumptions NOT satisfied!
## Skewness 25297.7 0 Assumptions NOT satisfied!
## Kurtosis 1182302.6 0 Assumptions NOT satisfied!
## Link Function 600.3 0 Assumptions NOT satisfied!
## Heteroscedasticity 143.8 0 Assumptions NOT satisfied!8.3 Data mining approach
set.seed(666)
train_index_3 <- sample(1:nrow(football_st_3), 0.7 *
nrow(football_st_3))
valid_index_3 <- setdiff(1:nrow(football_st_3), train_index)
train_df_st_3 <- football_st_3[train_index_3, ]
valid_df_st_3 <- football_st_3[valid_index_3, ]wage_model_st_cat_2 <- lm(Wage ~ factor(Preferred_Foot) + Positioning +
Composure, data = train_df_st_3)
summary(wage_model_st_cat_2)##
## Call:
## lm(formula = Wage ~ factor(Preferred_Foot) + Positioning + Composure,
## data = train_df_st_3)
##
## Residuals:
## Min 1Q Median 3Q Max
## -33599 -8588 -2271 5035 352343
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -68270.03 4142.64 -16.480 < 2e-16 ***
## factor(Preferred_Foot)Right -2040.14 1572.46 -1.297 0.195
## Positioning 787.78 97.32 8.095 1.17e-15 ***
## Composure 534.07 89.17 5.990 2.63e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 20920 on 1502 degrees of freedom
## Multiple R-squared: 0.2501, Adjusted R-squared: 0.2486
## F-statistic: 167 on 3 and 1502 DF, p-value: < 2.2e-16wage_model_st_cat_2_pred_train <- predict(wage_model_st_cat_2,
train_df_st_3)
accuracy(wage_model_st_cat_2_pred_train, train_df_st_3$Wage)## ME RMSE MAE MPE MAPE
## Test set -1.589124e-10 20893.75 9883.551 -37.84999 129.0229wage_model_st_cat_2_pred_valid <- predict(wage_model_st_cat_2,
valid_df_st_3)
accuracy(wage_model_st_cat_2_pred_valid, valid_df_st_3$Wage)## ME RMSE MAE MPE MAPE
## Test set -844.4194 15387.14 9700.628 -41.79669 130.5894sd(train_df_st_3$Wage)## [1] 24135.81sd(valid_df_st_3$Wage)## [1] 18074.14vif(wage_model_st_cat_2)## factor(Preferred_Foot) Positioning Composure
## 1.004517 2.769896 2.776543bptest(wage_model_st_cat_2)##
## studentized Breusch-Pagan test
##
## data: wage_model_st_cat_2
## BP = 64.695, df = 3, p-value = 5.829e-14new2 <- read.csv("new2.csv")
new2## Preferred.Foot Positioning Composure
## 1 Right 64 56
## 2 Right 65 47new2$Preferred.Foot <- as.factor(new2$Preferred.Foot)
names(new2)## [1] "Preferred.Foot" "Positioning" "Composure"names(new2)[1] <- "Preferred_Foot"
names(new2)## [1] "Preferred_Foot" "Positioning" "Composure"wage_model_st_cat_2_pred_new <- predict(wage_model_st_cat_2,
newdata = new2, interval = "confidence")
wage_model_st_cat_2_pred_new## fit lwr upr
## 1 10016.014 8776.276 11255.75
## 2 5997.149 3512.208 8482.099. Non-Linear Regression
Sometimes, a relationship may not be linear. In this case, we can specify a non-linear relationship in the model.
9.1 Traditional statistics
We start with the traditional statistics approach and evaluate.
The non-linear relationship is expressed in the model specification.
names(football_st_2)## [1] "Age" "Balance" "ShotPower" "Aggression" "Positioning"
## [6] "Composure" "Wage"wage_model_st_nl <- lm(Wage ~ Age + Balance + ShotPower +
Aggression + Positioning * Composure,
data = football_st_2)
summary(wage_model_st_nl)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Aggression +
## Positioning * Composure, data = football_st_2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -58380 -5245 80 4644 267683
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 297675.783 13442.584 22.144 <2e-16 ***
## Age -789.963 94.502 -8.359 <2e-16 ***
## Balance 57.694 30.555 1.888 0.0591 .
## ShotPower 642.408 63.389 10.134 <2e-16 ***
## Aggression 19.805 27.418 0.722 0.4702
## Positioning -5016.022 211.523 -23.714 <2e-16 ***
## Composure -6150.054 235.919 -26.069 <2e-16 ***
## Positioning:Composure 96.301 3.339 28.844 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 16000 on 2144 degrees of freedom
## Multiple R-squared: 0.4955, Adjusted R-squared: 0.4939
## F-statistic: 300.8 on 7 and 2144 DF, p-value: < 2.2e-16vif(wage_model_st_nl)## there are higher-order terms (interactions) in this model
## consider setting type = 'predictor'; see ?vif## Age Balance ShotPower
## 1.683057 1.044616 2.804077
## Aggression Positioning Composure
## 1.593281 31.765761 48.069231
## Positioning:Composure
## 127.119996durbinWatsonTest(wage_model_st_nl)## lag Autocorrelation D-W Statistic p-value
## 1 0.2531554 1.491911 0
## Alternative hypothesis: rho != 09.2 Traditional statistics stepwise
Perform a stepwise regression with a non-linear relationship and evaluate
wage_model_st_nl_step <- step(wage_model_st_nl,
direction = "both")## Start: AIC=41671.29
## Wage ~ Age + Balance + ShotPower + Aggression + Positioning *
## Composure
##
## Df Sum of Sq RSS AIC
## - Aggression 1 1.3352e+08 5.4877e+11 41670
## <none> 5.4863e+11 41671
## - Balance 1 9.1234e+08 5.4955e+11 41673
## - Age 1 1.7881e+10 5.6652e+11 41738
## - ShotPower 1 2.6282e+10 5.7492e+11 41770
## - Positioning:Composure 1 2.1290e+11 7.6154e+11 42375
##
## Step: AIC=41669.81
## Wage ~ Age + Balance + ShotPower + Positioning + Composure +
## Positioning:Composure
##
## Df Sum of Sq RSS AIC
## <none> 5.4877e+11 41670
## - Balance 1 8.5698e+08 5.4963e+11 41671
## + Aggression 1 1.3352e+08 5.4863e+11 41671
## - Age 1 1.8041e+10 5.6681e+11 41737
## - ShotPower 1 2.8516e+10 5.7728e+11 41777
## - Positioning:Composure 1 2.1286e+11 7.6162e+11 42373summary(wage_model_st_nl_step)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Positioning +
## Composure + Positioning:Composure, data = football_st_2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -58507 -5205 67 4579 267488
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 297410.796 13436.079 22.135 <2e-16 ***
## Age -775.517 92.352 -8.397 <2e-16 ***
## Balance 55.684 30.424 1.830 0.0674 .
## ShotPower 652.547 61.808 10.558 <2e-16 ***
## Positioning -5015.048 211.495 -23.712 <2e-16 ***
## Composure -6143.738 235.730 -26.063 <2e-16 ***
## Positioning:Composure 96.289 3.338 28.844 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 15990 on 2145 degrees of freedom
## Multiple R-squared: 0.4954, Adjusted R-squared: 0.494
## F-statistic: 351 on 6 and 2145 DF, p-value: < 2.2e-16vif(wage_model_st_nl_step)## there are higher-order terms (interactions) in this model
## consider setting type = 'predictor'; see ?vif## Age Balance ShotPower
## 1.607679 1.035950 2.666586
## Positioning Composure Positioning:Composure
## 31.764471 48.003192 127.116986durbinWatsonTest(wage_model_st_nl_step)## lag Autocorrelation D-W Statistic p-value
## 1 0.2522843 1.493672 0
## Alternative hypothesis: rho != 09.3 Data mining approach
A data mining approach with the non-linear relationship.
wage_model_st_nl_2 <- lm(Wage ~ Age + Balance + ShotPower + Aggression +
Positioning * Composure,
data = train_df_st)
summary(wage_model_st_nl_2)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Aggression +
## Positioning * Composure, data = train_df_st)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66085 -5447 302 4870 260260
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 342265.457 17203.875 19.895 < 2e-16 ***
## Age -758.277 122.802 -6.175 8.52e-10 ***
## Balance 79.001 39.022 2.025 0.0431 *
## ShotPower 699.312 82.219 8.505 < 2e-16 ***
## Aggression 13.470 34.875 0.386 0.6994
## Positioning -5818.796 271.105 -21.463 < 2e-16 ***
## Composure -6961.508 304.442 -22.866 < 2e-16 ***
## Positioning:Composure 109.127 4.285 25.465 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17060 on 1498 degrees of freedom
## Multiple R-squared: 0.5029, Adjusted R-squared: 0.5006
## F-statistic: 216.5 on 7 and 1498 DF, p-value: < 2.2e-16Predict the training and validation sets using the non-linear model. Check the accyracu.
wage_model_st_nl_2_pred_train <- predict(wage_model_st_nl_2,
train_df_st)
accuracy(wage_model_st_nl_2_pred_train, train_df_st$Wage)## ME RMSE MAE MPE MAPE
## Test set 1.966265e-09 17011.8 8516.679 -27.10041 105.2229wage_model_st_nl_2_pred_valid <- predict(wage_model_st_nl_2,
valid_df_st)
accuracy(wage_model_st_nl_2_pred_valid, valid_df_st$Wage)## ME RMSE MAE MPE MAPE
## Test set -855.2409 13504.82 8758.766 -32.40985 113.7297Predict the wages of new players using the non-linear model.
wage_model_st_nl_2_pred_new <- predict(wage_model_st_nl_2,
newdata = new, interval = "confidence")
wage_model_st_nl_2_pred_new## fit lwr upr
## 1 14285.88 11853.154 16718.61
## 2 12719.88 9489.583 15950.18
## 3 17129.30 15582.278 18676.32gvlma(wage_model_st_nl_2)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Aggression +
## Positioning * Composure, data = train_df_st)
##
## Coefficients:
## (Intercept) Age Balance
## 342265.46 -758.28 79.00
## ShotPower Aggression Positioning
## 699.31 13.47 -5818.80
## Composure Positioning:Composure
## -6961.51 109.13
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_nl_2)
##
## Value p-value Decision
## Global Stat 435462.84 0.0000000 Assumptions NOT satisfied!
## Skewness 9663.54 0.0000000 Assumptions NOT satisfied!
## Kurtosis 425239.16 0.0000000 Assumptions NOT satisfied!
## Link Function 545.56 0.0000000 Assumptions NOT satisfied!
## Heteroscedasticity 14.58 0.0001342 Assumptions NOT satisfied!9.4 Data mining approach using stepwise
A data mining approach using a stepwise regression and non-linear relationship.
wage_model_st_nl_2_step <- step(wage_model_st_nl_2,
direction = "both")## Start: AIC=29357.89
## Wage ~ Age + Balance + ShotPower + Aggression + Positioning *
## Composure
##
## Df Sum of Sq RSS AIC
## - Aggression 1 4.3404e+07 4.3588e+11 29356
## <none> 4.3584e+11 29358
## - Balance 1 1.1925e+09 4.3703e+11 29360
## - Age 1 1.1093e+10 4.4693e+11 29394
## - ShotPower 1 2.1048e+10 4.5689e+11 29427
## - Positioning:Composure 1 1.8867e+11 6.2451e+11 29898
##
## Step: AIC=29356.04
## Wage ~ Age + Balance + ShotPower + Positioning + Composure +
## Positioning:Composure
##
## Df Sum of Sq RSS AIC
## <none> 4.3588e+11 29356
## + Aggression 1 4.3404e+07 4.3584e+11 29358
## - Balance 1 1.1606e+09 4.3704e+11 29358
## - Age 1 1.1307e+10 4.4719e+11 29393
## - ShotPower 1 2.2847e+10 4.5873e+11 29431
## - Positioning:Composure 1 1.8867e+11 6.2455e+11 29896summary(wage_model_st_nl_2_step)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Positioning +
## Composure + Positioning:Composure, data = train_df_st)
##
## Residuals:
## Min 1Q Median 3Q Max
## -66274 -5469 260 4921 260121
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 342109.131 17194.232 19.897 < 2e-16 ***
## Age -748.254 119.996 -6.236 5.84e-10 ***
## Balance 77.598 38.841 1.998 0.0459 *
## ShotPower 706.988 79.759 8.864 < 2e-16 ***
## Positioning -5818.926 271.028 -21.470 < 2e-16 ***
## Composure -6958.216 304.237 -22.871 < 2e-16 ***
## Positioning:Composure 109.128 4.284 25.472 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 17050 on 1499 degrees of freedom
## Multiple R-squared: 0.5028, Adjusted R-squared: 0.5008
## F-statistic: 252.7 on 6 and 1499 DF, p-value: < 2.2e-16Predict the training and validation sets using the stepwise, non-linear model. Check the accuracy.
wage_model_st_nl_2_step_pred_train <- predict(wage_model_st_nl_2_step,
train_df_st)
accuracy(wage_model_st_nl_2_step_pred_train, train_df_st$Wage)## ME RMSE MAE MPE MAPE
## Test set 1.982657e-09 17012.65 8519.261 -27.09032 105.1927wage_model_st_nl_2_step_pred_valid <- predict(wage_model_st_nl_2_step,
valid_df_st)
accuracy(wage_model_st_nl_2_step_pred_valid, valid_df_st$Wage)## ME RMSE MAE MPE MAPE
## Test set -852.6068 13509.38 8763.627 -32.34321 113.7652Predict the wages of new players using the stepwise non-linear model.
wage_model_st_nl_2_step_pred_new <- predict(wage_model_st_nl_2_step,
newdata = new, interval = "confidence")
wage_model_st_nl_2_step_pred_new## fit lwr upr
## 1 14529.97 12437.349 16622.59
## 2 12274.73 9968.014 14581.45
## 3 17001.40 15597.748 18405.04gvlma(wage_model_st_nl_2_step)##
## Call:
## lm(formula = Wage ~ Age + Balance + ShotPower + Positioning +
## Composure + Positioning:Composure, data = train_df_st)
##
## Coefficients:
## (Intercept) Age Balance
## 342109.1 -748.3 77.6
## ShotPower Positioning Composure
## 707.0 -5818.9 -6958.2
## Positioning:Composure
## 109.1
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_nl_2_step)
##
## Value p-value Decision
## Global Stat 433134.34 0.0000000 Assumptions NOT satisfied!
## Skewness 9632.18 0.0000000 Assumptions NOT satisfied!
## Kurtosis 422942.94 0.0000000 Assumptions NOT satisfied!
## Link Function 544.53 0.0000000 Assumptions NOT satisfied!
## Heteroscedasticity 14.69 0.0001267 Assumptions NOT satisfied!10. Log Variables
Sometimes, the data need to be transformed. A common transformation is the log transformation.
10.1 Traditional statistics
A traditional statistics approach using a log transformation.
Here, the predictors are transformed using a log function.
wage_model_st_log <- lm(log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
log(Aggression) + log(Positioning) + log(Composure),
data = football_st_2)
summary(wage_model_st_log)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Aggression) + log(Positioning) + log(Composure), data = football_st_2)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.58521 -0.38112 -0.03033 0.36168 2.32404
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.49560 0.49568 -19.157 < 2e-16 ***
## log(Age) -0.79291 0.08748 -9.064 < 2e-16 ***
## log(Balance) 0.23309 0.06514 3.578 0.000354 ***
## log(ShotPower) 1.66883 0.14615 11.418 < 2e-16 ***
## log(Aggression) 0.06081 0.04912 1.238 0.215839
## log(Positioning) 1.94852 0.15876 12.273 < 2e-16 ***
## log(Composure) 1.16000 0.12750 9.098 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5898 on 2145 degrees of freedom
## Multiple R-squared: 0.4989, Adjusted R-squared: 0.4975
## F-statistic: 355.9 on 6 and 2145 DF, p-value: < 2.2e-16gvlma(wage_model_st_log)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Aggression) + log(Positioning) + log(Composure), data = football_st_2)
##
## Coefficients:
## (Intercept) log(Age) log(Balance) log(ShotPower)
## -9.49560 -0.79291 0.23309 1.66883
## log(Aggression) log(Positioning) log(Composure)
## 0.06081 1.94852 1.16000
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_log)
##
## Value p-value Decision
## Global Stat 561.540 0.000e+00 Assumptions NOT satisfied!
## Skewness 2.261 1.326e-01 Assumptions acceptable.
## Kurtosis 66.615 3.331e-16 Assumptions NOT satisfied!
## Link Function 481.799 0.000e+00 Assumptions NOT satisfied!
## Heteroscedasticity 10.865 9.802e-04 Assumptions NOT satisfied!10.2 Data mining with log
We can also use a data mining approach with the log transformation.
wage_model_st_log_2 <- lm(log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
log(Aggression) + log(Positioning) + log(Composure),
data = train_df_st)
summary(wage_model_st_log_2)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Aggression) + log(Positioning) + log(Composure), data = train_df_st)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.55406 -0.38093 -0.03289 0.35960 2.32610
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.67383 0.59662 -16.214 < 2e-16 ***
## log(Age) -0.69453 0.10718 -6.480 1.24e-10 ***
## log(Balance) 0.28514 0.07870 3.623 0.000301 ***
## log(ShotPower) 1.49775 0.17843 8.394 < 2e-16 ***
## log(Aggression) 0.07195 0.05938 1.212 0.225846
## log(Positioning) 1.87362 0.19243 9.737 < 2e-16 ***
## log(Composure) 1.31511 0.15334 8.577 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5945 on 1499 degrees of freedom
## Multiple R-squared: 0.4985, Adjusted R-squared: 0.4965
## F-statistic: 248.3 on 6 and 1499 DF, p-value: < 2.2e-16Predict the training and validation sets. Check the accuracy.
wage_model_st_log_2_pred_train <- predict(wage_model_st_log_2,
train_df_st)
train_df_st$logWage <- log(train_df_st$Wage)
accuracy(wage_model_st_log_2_pred_train, train_df_st$logWage)## ME RMSE MAE MPE MAPE
## Test set 4.312485e-14 0.5931541 0.4587753 -0.4038503 5.089574wage_model_st_log_2_pred_valid <- predict(wage_model_st_log_2,
valid_df_st)valid_df_st$logWage <- log(valid_df_st$Wage)
accuracy(wage_model_st_log_2_pred_valid, valid_df_st$logWage)## ME RMSE MAE MPE MAPE
## Test set 0.0002915404 0.5808893 0.4554191 -0.3779982 5.050841gvlma(wage_model_st_log_2)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Aggression) + log(Positioning) + log(Composure), data = train_df_st)
##
## Coefficients:
## (Intercept) log(Age) log(Balance) log(ShotPower)
## -9.67383 -0.69453 0.28514 1.49775
## log(Aggression) log(Positioning) log(Composure)
## 0.07195 1.87362 1.31511
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_log_2)
##
## Value p-value Decision
## Global Stat 394.158 0.000e+00 Assumptions NOT satisfied!
## Skewness 1.842 1.747e-01 Assumptions acceptable.
## Kurtosis 54.044 1.960e-13 Assumptions NOT satisfied!
## Link Function 336.747 0.000e+00 Assumptions NOT satisfied!
## Heteroscedasticity 1.524 2.170e-01 Assumptions acceptable.Predict new records
new3 <- read.csv("new3.csv", header = TRUE)
new3## X Age Balance ShotPower Aggression Positioning Composure
## 1 1 25 66 69 55 72 71
## 2 2 26 58 76 75 66 66
## 3 3 19 80 67 33 43 52wage_model_st_log_2_pred_new3 <- predict(wage_model_st_log_2,
newdata = new3, interval = "confidence")
wage_model_st_log_2_pred_new3## fit lwr upr
## 1 9.533908 9.484214 9.583602
## 2 9.377799 9.315354 9.440244
## 3 8.323197 8.170023 8.476371Results as a data frame (if desired).
wage_model_st_log_2_pred_new3_df <- as.data.frame(wage_model_st_log_2_pred_new3)
wage_model_st_log_2_pred_new3_df_value <- exp(1)^wage_model_st_log_2_pred_new3_df
wage_model_st_log_2_pred_new3_df_value## fit lwr upr
## 1 13820.495 13150.482 14524.645
## 2 11822.968 11107.261 12584.791
## 3 4118.303 3533.423 4799.99710.3 Stepwise data mining with log
A stepwise regression using data mining and log transformations.
wage_model_st_log_2_step <- step(wage_model_st_log_2,
direction = "both")## Start: AIC=-1559.17
## log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) + log(Aggression) +
## log(Positioning) + log(Composure)
##
## Df Sum of Sq RSS AIC
## - log(Aggression) 1 0.519 530.38 -1559.7
## <none> 529.86 -1559.2
## - log(Balance) 1 4.640 534.50 -1548.0
## - log(Age) 1 14.843 544.70 -1519.6
## - log(ShotPower) 1 24.907 554.77 -1492.0
## - log(Composure) 1 26.002 555.86 -1489.0
## - log(Positioning) 1 33.510 563.37 -1468.8
##
## Step: AIC=-1559.7
## log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) + log(Positioning) +
## log(Composure)
##
## Df Sum of Sq RSS AIC
## <none> 530.38 -1559.7
## + log(Aggression) 1 0.519 529.86 -1559.2
## - log(Balance) 1 4.393 534.77 -1549.3
## - log(Age) 1 14.336 544.71 -1521.5
## - log(Composure) 1 27.350 557.73 -1486.0
## - log(ShotPower) 1 27.771 558.15 -1484.8
## - log(Positioning) 1 33.605 563.98 -1469.2summary(wage_model_st_log_2_step)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Positioning) + log(Composure), data = train_df_st)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.5474 -0.3754 -0.0318 0.3602 2.3170
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -9.74502 0.59382 -16.411 < 2e-16 ***
## log(Age) -0.66674 0.10471 -6.367 2.55e-10 ***
## log(Balance) 0.27624 0.07837 3.525 0.000436 ***
## log(ShotPower) 1.54434 0.17426 8.862 < 2e-16 ***
## log(Positioning) 1.87617 0.19245 9.749 < 2e-16 ***
## log(Composure) 1.33827 0.15216 8.795 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.5946 on 1500 degrees of freedom
## Multiple R-squared: 0.498, Adjusted R-squared: 0.4963
## F-statistic: 297.6 on 5 and 1500 DF, p-value: < 2.2e-16gvlma(wage_model_st_log_2_step)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Positioning) + log(Composure), data = train_df_st)
##
## Coefficients:
## (Intercept) log(Age) log(Balance) log(ShotPower)
## -9.7450 -0.6667 0.2762 1.5443
## log(Positioning) log(Composure)
## 1.8762 1.3383
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_log_2_step)
##
## Value p-value Decision
## Global Stat 393.169 0.000e+00 Assumptions NOT satisfied!
## Skewness 2.178 1.400e-01 Assumptions acceptable.
## Kurtosis 52.586 4.117e-13 Assumptions NOT satisfied!
## Link Function 336.860 0.000e+00 Assumptions NOT satisfied!
## Heteroscedasticity 1.545 2.139e-01 Assumptions acceptable.Predict the training and validation sets. Check the accuracy.
wage_model_st_log_2_step_pred_train <- predict(wage_model_st_log_2_step, train_df_st)
accuracy(wage_model_st_log_2_step_pred_train, train_df_st$logWage)## ME RMSE MAE MPE MAPE
## Test set 4.4382e-14 0.5934445 0.4588034 -0.4041652 5.089808wage_model_st_log_2_step_pred_valid <- predict(wage_model_st_log_2_step, valid_df_st)
accuracy(wage_model_st_log_2_step_pred_valid, valid_df_st$logWage)## ME RMSE MAE MPE MAPE
## Test set 0.0007094787 0.580847 0.4553575 -0.3734178 5.049277gvlma(wage_model_st_log_2_step)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + log(Balance) + log(ShotPower) +
## log(Positioning) + log(Composure), data = train_df_st)
##
## Coefficients:
## (Intercept) log(Age) log(Balance) log(ShotPower)
## -9.7450 -0.6667 0.2762 1.5443
## log(Positioning) log(Composure)
## 1.8762 1.3383
##
##
## ASSESSMENT OF THE LINEAR MODEL ASSUMPTIONS
## USING THE GLOBAL TEST ON 4 DEGREES-OF-FREEDOM:
## Level of Significance = 0.05
##
## Call:
## gvlma(x = wage_model_st_log_2_step)
##
## Value p-value Decision
## Global Stat 393.169 0.000e+00 Assumptions NOT satisfied!
## Skewness 2.178 1.400e-01 Assumptions acceptable.
## Kurtosis 52.586 4.117e-13 Assumptions NOT satisfied!
## Link Function 336.860 0.000e+00 Assumptions NOT satisfied!
## Heteroscedasticity 1.545 2.139e-01 Assumptions acceptable.Predict new records using the stepwise log model
wage_model_st_log_2_step_pred_new3 <- predict(wage_model_st_log_2_step,
newdata = new3, interval = "confidence")
wage_model_st_log_2_step_pred_new3## fit lwr upr
## 1 9.533442 9.483746 9.583138
## 2 9.359849 9.304570 9.415129
## 3 8.340250 8.189562 8.490939wage_model_st_log_2_step_pred_new3_df <- as.data.frame(wage_model_st_log_2_step_pred_new3)
wage_model_st_log_2_step_pred_new3_df_value <- exp(1)^wage_model_st_log_2_step_pred_new3_df
wage_model_st_log_2_step_pred_new3_df_value## fit lwr upr
## 1 13814.060 13144.332 14517.911
## 2 11612.636 10988.116 12272.652
## 3 4189.138 3603.142 4870.43711. Combined
We can combine different settings.
wage_model_st_log_3 <- lm(log(Wage) ~ log(Age) + I(Positioning * Composure),
data = train_df_st)
summary(wage_model_st_log_3)##
## Call:
## lm(formula = log(Wage) ~ log(Age) + I(Positioning * Composure),
## data = train_df_st)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2.73661 -0.35786 -0.00011 0.35532 2.05304
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 8.666e+00 2.793e-01 31.03 < 2e-16 ***
## log(Age) -6.139e-01 9.807e-02 -6.26 5.02e-10 ***
## I(Positioning * Composure) 5.760e-04 1.599e-05 36.02 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.577 on 1503 degrees of freedom
## Multiple R-squared: 0.5264, Adjusted R-squared: 0.5258
## F-statistic: 835.2 on 2 and 1503 DF, p-value: < 2.2e-16Predict the training and validation sets. Check the accuracy.
wage_model_st_log_3_pred_train <- predict(wage_model_st_log_3,
train_df_st)
train_df_st$logWage <- log(train_df_st$Wage)
accuracy(wage_model_st_log_3_pred_train, train_df_st$logWage)## ME RMSE MAE MPE MAPE
## Test set 3.847045e-14 0.5764289 0.4423697 -0.388299 4.908134wage_model_st_log_3_pred_valid <- predict(wage_model_st_log_3,
valid_df_st)valid_df_st$logWage <- log(valid_df_st$Wage)
accuracy(wage_model_st_log_3_pred_valid, valid_df_st$logWage)## ME RMSE MAE MPE MAPE
## Test set 0.002845158 0.5841898 0.4527094 -0.3440159 5.01038vif(wage_model_st_log_3)## log(Age) I(Positioning * Composure)
## 1.568088 1.568088Predict new records
new3 <- read.csv("new3.csv", header = TRUE)
new3## X Age Balance ShotPower Aggression Positioning Composure
## 1 1 25 66 69 55 72 71
## 2 2 26 58 76 75 66 66
## 3 3 19 80 67 33 43 52wage_model_st_log_3_pred_new3 <- predict(wage_model_st_log_3,
newdata = new3, interval = "confidence")
wage_model_st_log_3_pred_new3## fit lwr upr
## 1 9.634869 9.590987 9.678752
## 2 9.175323 9.143823 9.206824
## 3 8.146727 8.092877 8.200578Results as a data frame (if desired).
wage_model_st_log_3_pred_new3_df <- as.data.frame(wage_model_st_log_3_pred_new3)
wage_model_st_log_3_pred_new3_df_value <- exp(1)^wage_model_st_log_3_pred_new3_df
wage_model_st_log_3_pred_new3_df_value## fit lwr upr
## 1 15288.703 14632.298 15974.554
## 2 9655.891 9356.468 9964.895
## 3 3452.064 3271.086 3643.054