R Notebook

data <- read.table("NSSdata.txt")
pre <- data[,1]
post <- data[,2]

# Paired t-test
t.test(pre, post, paired=TRUE)

    Paired t-test

data:  pre and post
t = 10.058, df = 142, p-value < 2.2e-16
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.06545084 0.09747364
sample estimates:
mean of the differences 
             0.08146224 

# Two-sample t-test
t.test(pre, post)

    Welch Two Sample t-test

data:  pre and post
t = 7.3372, df = 264.99, p-value = 2.669e-12
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.05960153 0.10332295
sample estimates:
mean of x mean of y 
0.8180357 0.7365734 

# Welch test
t.test(pre, post, var.equal=FALSE)

    Welch Two Sample t-test

data:  pre and post
t = 7.3372, df = 264.99, p-value = 2.669e-12
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 0.05960153 0.10332295
sample estimates:
mean of x mean of y 
0.8180357 0.7365734 

# Plots
boxplot(pre, post, names=c("2019","2021"),
        main="NSS Satisfaction Comparison")

plot(pre, post,
     xlab="2019", ylab="2021",
     main="Scatter Plot NSS")
abline(0,1,col="red")

Add a new chunk by clicking the Insert Chunk button on the toolbar or by pressing Ctrl+Alt+I.

# Question 2(a)

volume <- c(0.7458,0.7458,0.7386,1.1875,1.3613,1.4265,1.1296,1.3179,
1.6365,1.4410,1.7524,1.5206,1.5496,1.5424,1.3831,1.6075,
2.4475,1.9841,1.8610,1.8030,2.4982,2.2954,2.6285,2.7734,
3.0847,4.0116,4.0333,4.2216,3.7292,3.6930,5.5757)

girth <- c(66.23,68.62,70.22,85.38,83.79,86.18,87.78,87.78,
88.57,89.37,90.17,90.97,90.97,93.36,95.76,102.94,
102.94,106.13,109.32,110.12,111.71,113.31,115.70,127.67,
130.07,138.05,139.64,142.84,143.63,143.63,164.38)

height <- c(21.0,19.5,18.9,21.6,24.3,24.9,19.8,22.5,
24.0,22.5,23.7,22.8,22.8,20.7,22.5,22.2,
25.5,25.8,21.3,19.2,23.4,24.0,22.2,21.6,
23.1,24.3,24.6,24.0,24.0,24.0,26.1)

data <- data.frame(volume,girth,height)

#Question 2(b)
model <- lm(volume ~ girth + height, data=data)
summary(model)

Call:
lm(formula = volume ~ girth + height, data = data)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.47061 -0.19120 -0.02143  0.15448  0.61193 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -4.241242   0.624123  -6.796 2.21e-07 ***
girth        0.042641   0.002389  17.845  < 2e-16 ***
height       0.084128   0.031302   2.688    0.012 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2807 on 28 degrees of freedom
Multiple R-squared:  0.9481,    Adjusted R-squared:  0.9444 
F-statistic: 255.8 on 2 and 28 DF,  p-value: < 2.2e-16

#Question 2(c)
new <- data.frame(
girth=c(65,90,115,140,165),
height=c(21,22,23,24,25)
)

predict(model, new)

        1         2         3         4         5 
0.2970858 1.4472299 2.5973739 3.7475180 4.8976620 

#Question 2(d)
predict(model, new, interval="prediction")

        fit        lwr       upr
1 0.2970858 -0.3115897 0.9057613
2 1.4472299  0.8592903 2.0351695
3 2.5973739  2.0118607 3.1828871
4 3.7475180  3.1459002 4.3491358
5 4.8976620  4.2628173 5.5325068

#Question 2(e)
plot(model)  # gives residual + QQ plots

#Question 3(a)

scores <- read.table("ScoreData.txt")

rating <- scores[,1]
product <- scores[,2]
region <- scores[,3]
regionmissing <- scores[,4]

rating <- rating[regionmissing < 1]
product <- product[regionmissing < 1]
region <- region[regionmissing < 1]

#Question 3(b)

product <- factor(product)
region <- factor(region)


#Question 3(c)

model <- lm(rating ~ region)
anova(model)

Analysis of Variance Table

Response: rating
             Df  Sum Sq Mean Sq F value  Pr(>F)  
region        8    22.3  2.7847  2.3015 0.01837 *
Residuals 23472 28399.2  1.2099                  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#Question 3(d)

scores <- read.table("ScoreData.txt")

rating <- scores[,1]
region <- scores[,3]
regionmissing <- scores[,4]

rating <- rating[regionmissing < 1]
region <- region[regionmissing < 1]

region <- factor(region)

aggregate(rating ~ region, FUN = mean)

############################################################
# QUESTION 4 – FINAL R CODE
############################################################

#############################
# (a) Read EPL data file
#############################

EPL <- read.table("epl.txt", header = FALSE)

#############################
# (b) Convert matrix to score vector
#############################

scores <- c(EPL[1, ], EPL[2, ], EPL[3, ], EPL[4, ],
            EPL[5, ], EPL[6, ], EPL[7, ], EPL[8, ],
            EPL[9, ], EPL[10, ], EPL[11, ], EPL[12, ],
            EPL[13, ], EPL[14, ], EPL[15, ], EPL[16, ],
            EPL[17, ], EPL[18, ], EPL[19, ], EPL[20, ])

scores <- as.numeric(scores)

#############################
# (c) Team names
#############################

teamnames <- c("Arsenal","Villa","Brentford","Brighton","Burnley",
               "Chelsea","Palace","Everton","Leeds","Leicester",
               "Liverpool","City","United","Newcastle","Norwich",
               "Southampton","Tottenham","Watford","Westham","Wolves")

#############################
# (d) Create fixtures
#############################

teammix <- function(x, y){
  names <- vector("character", 38)
  for(i in 1:19){
    names[(2*i)-1] <- x
    names[2*i] <- y[i]
  }
  names
}

opponentmix <- function(x, y){
  names2 <- vector("character", 38)
  for(i in 1:19){
    names2[(2*i)-1] <- y[i]
    names2[2*i] <- x
  }
  names2
}

teammix2 <- function(a){
  teammix(teamnames[a], teamnames[-a])
}

teammix3 <- function(a){
  opponentmix(teamnames[a], teamnames[-a])
}

team <- c(teammix2(1), teammix2(2), teammix2(3), teammix2(4),
          teammix2(5), teammix2(6), teammix2(7), teammix2(8),
          teammix2(9), teammix2(10), teammix2(11), teammix2(12),
          teammix2(13), teammix2(14), teammix2(15), teammix2(16),
          teammix2(17), teammix2(18), teammix2(19), teammix2(20))

opponent <- c(teammix3(1), teammix3(2), teammix3(3), teammix3(4),
              teammix3(5), teammix3(6), teammix3(7), teammix3(8),
              teammix3(9), teammix3(10), teammix3(11), teammix3(12),
              teammix3(13), teammix3(14), teammix3(15), teammix3(16),
              teammix3(17), teammix3(18), teammix3(19), teammix3(20))

#############################
# (e) Home advantage variable
#############################

home <- rep(c(1,0), 380)

#############################
# (f) Response transformation
#############################

yvariable <- 1/(1 + scores)

#############################
# (g) Convert to factors
#############################

team <- factor(team)
opponent <- factor(opponent)

#############################
# (h) Fit null model
#############################

null.lm <- lm(yvariable ~ 1)

#############################
# (i) Fit full model
#############################

full.lm <- lm(yvariable ~ home + team + opponent)

#############################
# (j) Output results
#############################

summary(null.lm)

Call:
lm(formula = yvariable ~ 1)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.42795 -0.21962 -0.05295  0.44705  0.44705 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.55295    0.01071   51.64   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2952 on 759 degrees of freedom

summary(full.lm)

Call:
lm(formula = yvariable ~ home + team + opponent)

Residuals:
     Min       1Q   Median       3Q      Max 
-0.51672 -0.19866 -0.07396  0.21017  0.71047 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)          0.598798   0.063464   9.435  < 2e-16 ***
home                -0.050457   0.019560  -2.580 0.010087 *  
teamBrentford        0.039282   0.061939   0.634 0.526143    
teamBrighton         0.073310   0.061939   1.184 0.236965    
teamBurnley          0.129144   0.061939   2.085 0.037419 *  
teamChelsea         -0.105918   0.061939  -1.710 0.087690 .  
teamCity            -0.151682   0.061939  -2.449 0.014567 *  
teamEverton          0.052523   0.061939   0.848 0.396728    
teamLeeds            0.064840   0.061939   1.047 0.295527    
teamLeicester       -0.058343   0.061939  -0.942 0.346534    
teamLiverpool       -0.193244   0.061939  -3.120 0.001881 ** 
teamNewcastle        0.020486   0.061939   0.331 0.740931    
teamNorwich          0.218252   0.061939   3.524 0.000452 ***
teamPalace           0.025231   0.061939   0.407 0.683863    
teamSouthampton      0.026379   0.061939   0.426 0.670317    
teamTottenham       -0.025741   0.061939  -0.416 0.677838    
teamUnited          -0.007917   0.061939  -0.128 0.898332    
teamVilla           -0.013912   0.061939  -0.225 0.822347    
teamWatford          0.150648   0.061939   2.432 0.015250 *  
teamWestham         -0.031782   0.061939  -0.513 0.608020    
teamWolves           0.115625   0.061939   1.867 0.062341 .  
opponentBrentford   -0.070301   0.061939  -1.135 0.256750    
opponentBrighton    -0.007106   0.061939  -0.115 0.908688    
opponentBurnley     -0.054606   0.061939  -0.882 0.378276    
opponentChelsea      0.078636   0.061939   1.270 0.204646    
opponentCity         0.142455   0.061939   2.300 0.021737 *  
opponentEverton     -0.093727   0.061939  -1.513 0.130663    
opponentLeeds       -0.151381   0.061939  -2.444 0.014763 *  
opponentLeicester   -0.100188   0.061939  -1.618 0.106199    
opponentLiverpool    0.140268   0.061939   2.265 0.023832 *  
opponentNewcastle   -0.077431   0.061939  -1.250 0.211664    
opponentNorwich     -0.153206   0.061939  -2.474 0.013609 *  
opponentPalace      -0.003935   0.061939  -0.064 0.949359    
opponentSouthampton -0.107133   0.061939  -1.730 0.084120 .  
opponentTottenham    0.052593   0.061939   0.849 0.396104    
opponentUnited      -0.067083   0.061939  -1.083 0.279145    
opponentVilla       -0.039329   0.061939  -0.635 0.525655    
opponentWatford     -0.162685   0.061939  -2.627 0.008809 ** 
opponentWestham     -0.070532   0.061939  -1.139 0.255189    
opponentWolves       0.005208   0.061939   0.084 0.933009    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.2696 on 720 degrees of freedom
Multiple R-squared:  0.2086,    Adjusted R-squared:  0.1658 
F-statistic: 4.867 on 39 and 720 DF,  p-value: < 2.2e-16

#############################
# (k) Compare models
#############################

anova(null.lm, full.lm)

Analysis of Variance Table

Model 1: yvariable ~ 1
Model 2: yvariable ~ home + team + opponent
  Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
1    759 66.135                                  
2    720 52.337 39    13.798 4.8673 < 2.2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

#############################
# (l) Diagnostics plots
#############################

par(mfrow=c(2,2))
plot(full.lm)

#############################
# (m) Team coefficients
#############################

coef(full.lm)

        (Intercept)                home       teamBrentford 
        0.598798037        -0.050457393         0.039282407 
       teamBrighton         teamBurnley         teamChelsea 
        0.073310185         0.129143519        -0.105917659 
           teamCity         teamEverton           teamLeeds 
       -0.151681548         0.052523148         0.064839616 
      teamLeicester       teamLiverpool       teamNewcastle 
       -0.058343254        -0.193244048         0.020486111 
        teamNorwich          teamPalace     teamSouthampton 
        0.218252315         0.025231481         0.026378968 
      teamTottenham          teamUnited           teamVilla 
       -0.025740741        -0.007916667        -0.013912037 
        teamWatford         teamWestham          teamWolves 
        0.150648148        -0.031782407         0.115625000 
  opponentBrentford    opponentBrighton     opponentBurnley 
       -0.070300926        -0.007106481        -0.054606481 
    opponentChelsea        opponentCity     opponentEverton 
        0.078635913         0.142455357        -0.093726852 
      opponentLeeds   opponentLeicester   opponentLiverpool 
       -0.151380622        -0.100188492         0.140267857 
  opponentNewcastle     opponentNorwich      opponentPalace 
       -0.077430556        -0.153206019        -0.003935185 
opponentSouthampton   opponentTottenham      opponentUnited 
       -0.107132937         0.052592593        -0.067083333 
      opponentVilla     opponentWatford     opponentWestham 
       -0.039328704        -0.162685185        -0.070532407 
     opponentWolves 
        0.005208333