1. Read the data into R

airline.df <- read.csv(paste("SixAirlinesDataV2.csv", sep=""))
attach(airline.df)

Data Summary

2. Summary Statistics - mean, sd, median, min, max of variables

library(psych)
describe(airline.df)[,c(2,3,4,5,8,9)]
##                       n    mean      sd  median   min     max
## Airline*            458    3.01    1.65    2.00  1.00    6.00
## Aircraft*           458    1.67    0.47    2.00  1.00    2.00
## FlightDuration      458    7.58    3.54    7.79  1.25   14.66
## TravelMonth*        458    2.56    1.17    3.00  1.00    4.00
## IsInternational*    458    1.91    0.28    2.00  1.00    2.00
## SeatsEconomy        458  202.31   76.37  185.00 78.00  389.00
## SeatsPremium        458   33.65   13.26   36.00  8.00   66.00
## PitchEconomy        458   31.22    0.66   31.00 30.00   33.00
## PitchPremium        458   37.91    1.31   38.00 34.00   40.00
## WidthEconomy        458   17.84    0.56   18.00 17.00   19.00
## WidthPremium        458   19.47    1.10   19.00 17.00   21.00
## PriceEconomy        458 1327.08  988.27 1242.00 65.00 3593.00
## PricePremium        458 1845.26 1288.14 1737.00 86.00 7414.00
## PriceRelative       458    0.49    0.45    0.36  0.02    1.89
## SeatsTotal          458  235.96   85.29  227.00 98.00  441.00
## PitchDifference     458    6.69    1.76    7.00  2.00   10.00
## WidthDifference     458    1.63    1.19    1.00  0.00    4.00
## PercentPremiumSeats 458   14.65    4.84   13.21  4.71   24.69

Visualization

Premium Economy seats are more expensive than Economy seats, as expected

3. Plot of Premium Economy Ticket Prices versus Economy Ticket Prices

##  Price Premium versus Price Economy
plot(~PriceEconomy + PricePremium, main="Premium Economy Price vs. Economy Price")
abline(0,1)

  • The points are above the 45 degree line.
  • As expected, Premium Economy Airfares are higher than the corresponding Economy Airfares on the same flight.

Role of Difference in Pitch between Premium Economy and Economy Seats on the Pricing

4a. Distribution of the difference in the pitch of Premium Economy seats and the pitch of Economy seats

pitchDifferenceTable <- table(airline.df$PitchDifference)
pitchDifferenceTable
## 
##   2   3   6   7  10 
##  24  16 121 243  54
library(lattice)
histogram(~PitchDifference, data = airline.df,
 main = "Distribution of Pitch Difference", xlab="Difference in Pitch", col='gray' )

The difference in pitch of Premium Economy and Economy seats are {2, 3, 6, 7, 10} inches.

Result: The most frequently observed difference in pitch of seats in Premium Economy and Economy is 7 inches.

4b. Effect of Pitch Difference on the relative price of Economy and Premium Economy airfares

pd = aggregate(cbind(PriceEconomy,PricePremium, PriceRelative) ~ PitchDifference, 
                   data = airline.df, mean)
pd
##   PitchDifference PriceEconomy PricePremium PriceRelative
## 1               2     348.0000     377.3333    0.08708333
## 2               3     369.5625     398.7500    0.08125000
## 3               6    2008.6942    2333.7438    0.34082645
## 4               7    1388.1317    2155.4897    0.51888889
## 5              10     243.8519     435.6481    0.97074074

4c. Effect of Pitch Difference on the price of Economy and Premium Economy airfares

boxplot(PriceRelative~PitchDifference,data=airline.df, main="Relative Price Difference vs. Pitch", ylab="Pitch Difference", xlab="Relative Price b/w Economy and Premium Economy", horizontal=TRUE)

Result 1:

The relative difference in airfare (PriceRelative) between Premium Economy and Economy seats increases as the difference in their pitch (PitchDifference) increases.

Role of Difference in Seat Width of Premium Economy and Economy Seats

5a. Distribution of the difference in the width of Premium Economy seats and the width of Economy seats

widthDifferenceTable <- table(airline.df$WidthDifference)
widthDifferenceTable
## 
##   0   1   2   3   4 
##  40 264  32  68  54
library(lattice)
histogram(~WidthDifference, data = airline.df,
 main = "Distribution of Difference in Seat Width", xlab="Difference in Seat Width", col='gray' )

The difference in seat width of Premium Economy and Economy seats are {0, 1, 2, 3, 4} inches. Result: The most frequently observed difference in width of seats in Premium Economy and Economy is 1 inch.

5b. Effect of Seat WidthDifference on the price of Economy and Premium Economy airfares

aggregate(cbind(PriceEconomy,PricePremium, PriceRelative) ~ WidthDifference, 
                   data = airline.df, mean)
##   WidthDifference PriceEconomy PricePremium PriceRelative
## 1               0     356.6250     385.9000     0.0847500
## 2               1    1428.4053    1966.0795     0.4184091
## 3               2    2884.7500    3197.4375     0.2296875
## 4               3    1631.7206    2717.7059     0.7282353
## 5               4     243.8519     435.6481     0.9707407

Result: The relative difference in airfare (PriceRelative) between Premium Economy and Economy seats increases as the difference in their seat width (WidthDifference) increases.

5c. Effect of Seat WidthDifference on the price of Economy and Premium Economy airfares

boxplot(PriceRelative~WidthDifference,data=airline.df, main="Relative Price Difference vs. Seat Width", ylab="Seat Width Difference", xlab="Relative Price b/w Economy and Premium Economy", horizontal=TRUE)

Combined role of Differences in Pitch and Seat Width between Premium Economy and Economy Seats on Pricing

6a.

pitchWidthTable <- xtabs(~WidthDifference + PitchDifference, data=airline.df)
ftable(pitchWidthTable) # print table
##                 PitchDifference   2   3   6   7  10
## WidthDifference                                    
## 0                                24  16   0   0   0
## 1                                 0   0  89 175   0
## 2                                 0   0  32   0   0
## 3                                 0   0   0  68   0
## 4                                 0   0   0   0  54

6b. Mosaic Plot comparing the Pitch Difference and Width Difference between Premium Economy and Economy seats

library(vcd)
## Loading required package: grid
mosaic(pitchWidthTable, shade=TRUE, legend=TRUE, main=" Dist. of Diff. in Pitch and Seat Width")

Effect of Plane Capacity (Total Seats=Economy Seats + Premium Economy Seats)

Note: We ignore Business Class and First Class

7a.

t1 = aggregate(cbind(PriceEconomy,PricePremium, PriceRelative) ~ SeatsTotal, 
                   data = airline.df, mean)
t1
##    SeatsTotal PriceEconomy PricePremium PriceRelative
## 1          98     291.7500     306.7500    0.06125000
## 2         138     299.4000     316.4000    0.06200000
## 3         140     184.7037     377.8889    1.15370370
## 4         144     365.7692     402.6154    0.10230769
## 5         156     328.4000     348.6000    0.07200000
## 6         158     458.0000     497.0000    0.09000000
## 7         160     415.5000     459.5000    0.10750000
## 8         162    1511.8627    2154.0000    0.42254902
## 9         166    1040.4444    1459.8333    0.40722222
## 10        168    2194.6857    2448.4286    0.27914286
## 11        170     257.0526     461.1579    0.95263158
## 12        198    3341.0000    3450.6667    0.03000000
## 13        200     479.2500     522.0000    0.09500000
## 14        212    1035.4167    1361.8750    0.34958333
## 15        227    1625.5000    2910.8333    0.60416667
## 16        228    3144.3636    3398.8182    0.08590909
## 17        233    1509.7273    2552.2121    0.82121212
## 18        240     772.5000    1710.0000    1.32000000
## 19        271    1638.2222    2658.1111    0.61111111
## 20        279    1666.2195    2439.9268    0.50414634
## 21        299    1633.8750    2765.3750    0.73500000
## 22        358     631.8654     965.8077    0.32211538
## 23        367      74.0000      97.0000    0.31000000
## 24        369     597.5000    1057.0000    0.80000000
## 25        427    2822.5714    3198.0714    0.20928571
## 26        441    2113.0000    3601.5000    0.74000000
xyplot(PriceRelative ~ SeatsTotal, data = airline.df
       ,type = c("p", "g"),
       xlab = "Total Seats (Economy + Premium Economy Seats)", ylab = "Rel. Price Difference"
       )

Percentage of Premium Economy Seats

These are defined as the ratio of the Premium Economy Seats to the sum of the Premium Economy and Economy Seats.

7b.

boxplot(airline.df$PercentPremiumSeats, data=airline.df, main="Percentage of Premium Economy Seats", 
    xlab="Percentage of Premium Economy Seats in Plane", ylab="", horizontal=TRUE )

7c.

xyplot(PriceRelative ~ PercentPremiumSeats, data = airline.df
       ,type = c("p", "g"),
       xlab = "Percentage of Premium Economy Seats in Plane", ylab = "Rel. Price Difference"
       )

Correlations

8a.

# Scatterplot Matrices from the car Package
library(car)
## 
## Attaching package: 'car'
## The following object is masked from 'package:psych':
## 
##     logit
scatterplotMatrix(~PricePremium+PriceEconomy+PitchDifference+WidthDifference, data=airline.df,
    main="Premium Economy vs. Economy Airfares")

scatterplotMatrix(~PricePremium+PriceEconomy+SeatsTotal+PercentPremiumSeats, data=airline.df,
    main="Premium Economy vs. Economy Airfares")

Correlation Matrix

8b.

library(Hmisc)
## Loading required package: survival
## Loading required package: Formula
## Loading required package: ggplot2
## 
## Attaching package: 'ggplot2'
## The following objects are masked from 'package:psych':
## 
##     %+%, alpha
## 
## Attaching package: 'Hmisc'
## The following object is masked from 'package:psych':
## 
##     describe
## The following objects are masked from 'package:base':
## 
##     format.pval, units
colairlines <- c("PricePremium","PriceEconomy","PitchDifference","WidthDifference")
corMatrix <- rcorr(as.matrix(airline.df[,colairlines]))
corMatrix
##                 PricePremium PriceEconomy PitchDifference WidthDifference
## PricePremium            1.00         0.90           -0.02           -0.01
## PriceEconomy            0.90         1.00           -0.10           -0.08
## PitchDifference        -0.02        -0.10            1.00            0.76
## WidthDifference        -0.01        -0.08            0.76            1.00
## 
## n= 458 
## 
## 
## P
##                 PricePremium PriceEconomy PitchDifference WidthDifference
## PricePremium                 0.0000       0.6998          0.8059         
## PriceEconomy    0.0000                    0.0332          0.0708         
## PitchDifference 0.6998       0.0332                       0.0000         
## WidthDifference 0.8059       0.0708       0.0000
colairlines2 <- c("PricePremium","PriceEconomy","SeatsTotal","PercentPremiumSeats")
corMatrix2 <- rcorr(as.matrix(airline.df[,colairlines2]))
corMatrix2
##                     PricePremium PriceEconomy SeatsTotal
## PricePremium                1.00         0.90       0.19
## PriceEconomy                0.90         1.00       0.13
## SeatsTotal                  0.19         0.13       1.00
## PercentPremiumSeats         0.12         0.07      -0.22
##                     PercentPremiumSeats
## PricePremium                       0.12
## PriceEconomy                       0.07
## SeatsTotal                        -0.22
## PercentPremiumSeats                1.00
## 
## n= 458 
## 
## 
## P
##                     PricePremium PriceEconomy SeatsTotal
## PricePremium                     0.0000       0.0000    
## PriceEconomy        0.0000                    0.0045    
## SeatsTotal          0.0000       0.0045                 
## PercentPremiumSeats 0.0127       0.1628       0.0000    
##                     PercentPremiumSeats
## PricePremium        0.0127             
## PriceEconomy        0.1628             
## SeatsTotal          0.0000             
## PercentPremiumSeats
  • The upper matrix is the Correlation Matrix
  • The lower matrix gives the Pearson Rank Test

Corrgram

8c.

library(Hmisc)
library(car)
library(corrgram)
colairlines <- c("PricePremium","PriceEconomy","PitchDifference","WidthDifference","SeatsTotal","PercentPremiumSeats")
corrgram(airline.df[,colairlines], order=TRUE,
         main="Premium Economy vs. Economy Airfares",
         lower.panel=panel.pts, upper.panel=panel.pie,
         diag.panel=panel.minmax, text.panel=panel.txt)

T-Tests

9a. t-test of PricePremium, PriceEconomy

t.test(PricePremium,PriceEconomy)
## 
##  Welch Two Sample t-test
## 
## data:  PricePremium and PriceEconomy
## t = 6.8304, df = 856.56, p-value = 1.605e-11
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  369.2793 667.0831
## sample estimates:
## mean of x mean of y 
##  1845.258  1327.076
  • Since the p-value is less than 0.05, we fail to reject the null hypothesis that they are equal

9b. t-test of PitchPremium, PitchEconomy

t.test(PitchPremium, PitchEconomy)
## 
##  Welch Two Sample t-test
## 
## data:  PitchPremium and PitchEconomy
## t = 97.482, df = 671.02, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  6.553067 6.822479
## sample estimates:
## mean of x mean of y 
##  37.90611  31.21834

9c. Perfroming t-test on WidthPremium, WidthEconomy

t.test(WidthPremium, WidthEconomy)
## 
##  Welch Two Sample t-test
## 
## data:  WidthPremium and WidthEconomy
## t = 28.4, df = 678.24, p-value < 2.2e-16
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  1.520276 1.746100
## sample estimates:
## mean of x mean of y 
##  19.47162  17.83843

Regression

10. Formulating multivariate linear regression model to fit price with respect to width and pitch

Model 1

In this model we try regressing Price Premium on ALL the remaining columns.

Model1 <- PricePremium ~ PriceEconomy + PitchDifference + WidthDifference + PercentPremiumSeats + SeatsTotal + IsInternational + TravelMonth + FlightDuration + Aircraft
fit1 <- lm(Model1, data = airline.df)
summary(fit1)
## 
## Call:
## lm(formula = Model1, data = airline.df)
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -977.2 -246.3  -47.9  135.2 3419.7 
## 
## Coefficients:
##                                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                  -1.211e+03  1.755e+02  -6.898 1.82e-11 ***
## PriceEconomy                  1.064e+00  3.114e-02  34.175  < 2e-16 ***
## PitchDifference               8.510e+01  3.913e+01   2.175 0.030163 *  
## WidthDifference               1.240e+02  3.438e+01   3.607 0.000345 ***
## PercentPremiumSeats           3.177e+01  5.250e+00   6.052 3.04e-09 ***
## SeatsTotal                    1.925e+00  3.360e-01   5.729 1.87e-08 ***
## IsInternationalInternational -7.537e+02  2.135e+02  -3.530 0.000458 ***
## TravelMonthJul               -3.441e+01  7.074e+01  -0.486 0.626904    
## TravelMonthOct                2.692e+01  6.036e+01   0.446 0.655795    
## TravelMonthSep               -2.097e+00  6.015e+01  -0.035 0.972203    
## FlightDuration                8.455e+01  8.809e+00   9.598  < 2e-16 ***
## AircraftBoeing               -2.082e+00  5.651e+01  -0.037 0.970625    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 480.7 on 446 degrees of freedom
## Multiple R-squared:  0.8641, Adjusted R-squared:  0.8607 
## F-statistic: 257.7 on 11 and 446 DF,  p-value: < 2.2e-16

Next, we want to find the best fit model. We will use regsubsets() from library leaps

Find the best predictors

library(leaps)
## Warning: package 'leaps' was built under R version 3.4.3
leap1 <- regsubsets(Model1, data = airline.df, nbest=1)
# summary(leap1)
plot(leap1, scale="adjr2")

The best fit model excludes TravelMonth and Aircraft type (Boeing or AirBus). Therefore, in our next model, we rerun the regression, excluding these variables.

Model 2 – Revise Model 1, by excluding TravelMonth and Aircraft from the list of regressors

Model2 <- PricePremium ~ PriceEconomy + PitchDifference + WidthDifference + PercentPremiumSeats + SeatsTotal + FlightDuration + IsInternational 
fit2 <- lm(Model2, data = airline.df)
summary(fit2)
## 
## Call:
## lm(formula = Model2, data = airline.df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1010.0  -258.4   -49.9   133.6  3416.7 
## 
## Coefficients:
##                                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)                  -1.213e+03  1.695e+02  -7.156 3.40e-12 ***
## PriceEconomy                  1.063e+00  3.077e-02  34.537  < 2e-16 ***
## PitchDifference               8.421e+01  3.656e+01   2.303 0.021722 *  
## WidthDifference               1.224e+02  3.373e+01   3.629 0.000318 ***
## PercentPremiumSeats           3.190e+01  5.220e+00   6.112 2.14e-09 ***
## SeatsTotal                    1.920e+00  3.241e-01   5.922 6.31e-09 ***
## FlightDuration                8.459e+01  8.507e+00   9.943  < 2e-16 ***
## IsInternationalInternational -7.412e+02  2.001e+02  -3.704 0.000238 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 479 on 450 degrees of freedom
## Multiple R-squared:  0.8638, Adjusted R-squared:  0.8617 
## F-statistic: 407.9 on 7 and 450 DF,  p-value: < 2.2e-16

Results

We measure the difference in quality between Premium Economy and Economy in two ways: The difference in pitch (PitchDifference) measures the additional legroom in Premium Economy seats, compared to Economy seats. The difference in width (WidthDifference) measures the additional width of the seat in Premium Economy, compared to Economy.

Effect of Quality Difference:

The airfare for Premium Economy, relative to the Economy airfare: 1) increases as the difference in pitch between Premium Economy and Economy seats increases 2) increases as the difference in seat width between Premium Economy and Economy seats increases.

Effect of the Supply of Premium Economy seats:

The airfare for Premium Economy, relative to the Economy airfare: 1) increases as the total number of seats in the airplane increases 2) increases as the percentage of Premium Economy seats in the plane increases

Visualize the Beta coefficiencts and their confidence intervals from Model 2

library(coefplot)
coefplot(fit2, intercept= FALSE, outerCI=1.96,coefficients=c("PriceEconomy","PitchDifference", "WidthDifference", "PercentPremiumSeats", "SeatsTotal", "FlightDuration"))

Model 2 fits better than Model 1, as indicated by Adjusted R Squared and AIC values

# the Adjusted R Squared for Model 2 is less than Model 1
summary(fit1)$adj.r.squared
## [1] 0.8607235
summary(fit2)$adj.r.squared
## [1] 0.861724
# the AIC for Model 2 is less than Model 1
AIC(fit1)
## [1] 6970.166
AIC(fit2)
## [1] 6962.954

Thus, Model 2 is our ‘best’ ordinary least squares model. Model 2 predicts the price of the premium economy seat PricePremium, as a function of the following explanatory variables: “PriceEconomy”,“PitchDifference”, “WidthDifference”, “PercentPremiumSeats”, “SeatsTotal”, “FlightDuration”