Assignment 5
Sarim Ali Ansari
3 November 2018
Load data
bomdelbom <- read.csv("AirlinePricingData.csv")
attach(bomdelbom)
head(bomdelbom)## FlightNumber Airline DepartureCityCode ArrivalCityCode DepartureTime
## 1 9W 313 Jet DEL BOM 225
## 2 9W 339 Jet BOM DEL 300
## 3 SG 161 Spice Jet DEL BOM 350
## 4 6E 171 IndiGo DEL BOM 455
## 5 SG 160 Spice Jet BOM DEL 555
## 6 9W 762 Jet BOM DEL 605
## ArrivalTime Departure FlyingMinutes Aircraft PlaneModel Capacity
## 1 435 AM 130 Boeing 738 156
## 2 505 AM 125 Boeing 738 156
## 3 605 AM 135 Boeing 738 189
## 4 710 AM 135 Airbus A320 180
## 5 805 AM 130 Boeing 738 189
## 6 815 AM 130 Boeing 738 156
## SeatPitch SeatWidth DataCollectionDate DateDeparture IsWeekend Price
## 1 30 17 Sep 13 2018 Nov 6 2018 No 4051
## 2 30 17 Sep 15 2018 Nov 6 2018 No 11587
## 3 29 17 Sep 19 2018 Nov 6 2018 No 3977
## 4 30 18 Sep 8 2018 Nov 6 2018 No 4234
## 5 29 17 Sep 19 2018 Nov 6 2018 No 6837
## 6 30 17 Sep 15 2018 Nov 6 2018 No 6518
## AdvancedBookingDays IsDiwali DayBeforeDiwali DayAfterDiwali
## 1 54 1 1 0
## 2 52 1 1 0
## 3 48 1 1 0
## 4 59 1 1 0
## 5 48 1 1 0
## 6 52 1 1 0
## MetroDeparture MetroArrival MarketShare LoadFactor
## 1 1 1 15.4 83.32
## 2 1 1 15.4 83.32
## 3 1 1 13.2 94.06
## 4 1 1 39.6 87.20
## 5 1 1 13.2 94.06
## 6 1 1 15.4 83.321. Run linear-linear regression model
model1 <- Price ~ AdvancedBookingDays + Airline + Departure + IsWeekend + IsDiwali + DepartureCityCode + FlyingMinutes + SeatPitch + SeatWidth
fit1 <- lm(model1, data = bomdelbom)
summary(fit1)##
## Call:
## lm(formula = model1, data = bomdelbom)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2671.2 -1266.2 -456.4 517.4 11953.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4292.94 8897.87 -0.482 0.6298
## AdvancedBookingDays -87.70 12.47 -7.033 1.43e-11 ***
## AirlineIndiGo -577.17 778.64 -0.741 0.4591
## AirlineJet -120.75 436.69 -0.277 0.7823
## AirlineSpice Jet -1118.38 697.85 -1.603 0.1101
## DeparturePM -589.79 275.23 -2.143 0.0329 *
## IsWeekendYes -345.92 408.06 -0.848 0.3973
## IsDiwali 4346.80 568.14 7.651 2.90e-13 ***
## DepartureCityCodeDEL -1413.46 351.54 -4.021 7.38e-05 ***
## FlyingMinutes 38.97 29.27 1.331 0.1841
## SeatPitch -279.19 226.64 -1.232 0.2190
## SeatWidth 868.58 507.54 1.711 0.0881 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2079 on 293 degrees of freedom
## Multiple R-squared: 0.2695, Adjusted R-squared: 0.2421
## F-statistic: 9.828 on 11 and 293 DF, p-value: 3.604e-152. Run log-linear regression model
model2 <- log(Price) ~ AdvancedBookingDays + Airline + Departure + IsWeekend + IsDiwali + DepartureCityCode + FlyingMinutes + SeatPitch + SeatWidth
fit2 <- lm(model2, data = bomdelbom)
summary(fit2)##
## Call:
## lm(formula = model2, data = bomdelbom)
##
## Residuals:
## Min 1Q Median 3Q Max
## -0.57006 -0.19770 -0.05792 0.12935 1.24672
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 6.549474 1.243788 5.266 2.71e-07 ***
## AdvancedBookingDays -0.014639 0.001743 -8.399 1.97e-15 ***
## AirlineIndiGo -0.098622 0.108842 -0.906 0.3656
## AirlineJet 0.001113 0.061043 0.018 0.9855
## AirlineSpice Jet -0.127169 0.097548 -1.304 0.1934
## DeparturePM -0.055844 0.038473 -1.452 0.1477
## IsWeekendYes -0.036748 0.057041 -0.644 0.5199
## IsDiwali 0.744738 0.079418 9.377 < 2e-16 ***
## DepartureCityCodeDEL -0.264017 0.049140 -5.373 1.58e-07 ***
## FlyingMinutes 0.008717 0.004092 2.131 0.0340 *
## SeatPitch -0.032824 0.031681 -1.036 0.3010
## SeatWidth 0.122364 0.070947 1.725 0.0856 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.2906 on 293 degrees of freedom
## Multiple R-squared: 0.3671, Adjusted R-squared: 0.3433
## F-statistic: 15.45 on 11 and 293 DF, p-value: < 2.2e-163. Comparing the models
Adjusted R-squared for model1 = 24.21%
Adjusted R-squared for model2 = 34.33%
=> Log-linear model is a better fit
4a. Visual check for normality
plot(fit1, 2)
plot(fit2, 2)
# In both plots, all the points does not fall approximately along this reference line, so we cannot assume normality
4b. Normality test for linear-linear model
shapiro.test(fit1$fitted.values)##
## Shapiro-Wilk normality test
##
## data: fit1$fitted.values
## W = 0.96562, p-value = 1.224e-06From the output, the p-value < 0.05 implying that the distribution of the data are significantly different from normal distribution. In other words, we cannot assume the normality
4b. Normality test for log-linear model
shapiro.test(fit2$fitted.values)##
## Shapiro-Wilk normality test
##
## data: fit2$fitted.values
## W = 0.96081, p-value = 2.589e-07From the output, the p-value < 0.05 implying that the distribution of the data are significantly different from normal distribution. In other words, we cannot assume the normality
5. Visual check for linearity
plot(fit1, 1)
plot(fit2, 1)
# Line is not horizontal for either model
6b. Running linear-linear regression on transformed data
model3 <- PriceNew ~ AdvancedBookingDays + Airline + Departure + IsWeekend + IsDiwali + DepartureCityCode + FlyingMinutes + SeatPitch + SeatWidth
fit3 <- lm(model1, data = bomdelbom)
summary(fit3)##
## Call:
## lm(formula = model1, data = bomdelbom)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2671.2 -1266.2 -456.4 517.4 11953.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4292.94 8897.87 -0.482 0.6298
## AdvancedBookingDays -87.70 12.47 -7.033 1.43e-11 ***
## AirlineIndiGo -577.17 778.64 -0.741 0.4591
## AirlineJet -120.75 436.69 -0.277 0.7823
## AirlineSpice Jet -1118.38 697.85 -1.603 0.1101
## DeparturePM -589.79 275.23 -2.143 0.0329 *
## IsWeekendYes -345.92 408.06 -0.848 0.3973
## IsDiwali 4346.80 568.14 7.651 2.90e-13 ***
## DepartureCityCodeDEL -1413.46 351.54 -4.021 7.38e-05 ***
## FlyingMinutes 38.97 29.27 1.331 0.1841
## SeatPitch -279.19 226.64 -1.232 0.2190
## SeatWidth 868.58 507.54 1.711 0.0881 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2079 on 293 degrees of freedom
## Multiple R-squared: 0.2695, Adjusted R-squared: 0.2421
## F-statistic: 9.828 on 11 and 293 DF, p-value: 3.604e-156c. Running log-linear regression on transformed data
model4 <- log(PriceNew) ~ AdvancedBookingDays + Airline + Departure + IsWeekend + IsDiwali + DepartureCityCode + FlyingMinutes + SeatPitch + SeatWidth
fit4 <- lm(model1, data = bomdelbom)
summary(fit4)##
## Call:
## lm(formula = model1, data = bomdelbom)
##
## Residuals:
## Min 1Q Median 3Q Max
## -2671.2 -1266.2 -456.4 517.4 11953.9
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) -4292.94 8897.87 -0.482 0.6298
## AdvancedBookingDays -87.70 12.47 -7.033 1.43e-11 ***
## AirlineIndiGo -577.17 778.64 -0.741 0.4591
## AirlineJet -120.75 436.69 -0.277 0.7823
## AirlineSpice Jet -1118.38 697.85 -1.603 0.1101
## DeparturePM -589.79 275.23 -2.143 0.0329 *
## IsWeekendYes -345.92 408.06 -0.848 0.3973
## IsDiwali 4346.80 568.14 7.651 2.90e-13 ***
## DepartureCityCodeDEL -1413.46 351.54 -4.021 7.38e-05 ***
## FlyingMinutes 38.97 29.27 1.331 0.1841
## SeatPitch -279.19 226.64 -1.232 0.2190
## SeatWidth 868.58 507.54 1.711 0.0881 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 2079 on 293 degrees of freedom
## Multiple R-squared: 0.2695, Adjusted R-squared: 0.2421
## F-statistic: 9.828 on 11 and 293 DF, p-value: 3.604e-157a. Visual check for normality for transformed model
plot(fit3, 2)
plot(fit4, 2) # In both plots, all the points does not fall approximately along this reference line, so we cannot assume normality
7b. Normality test for transformed linear-linear model
shapiro.test(fit3$fitted.values)##
## Shapiro-Wilk normality test
##
## data: fit3$fitted.values
## W = 0.96562, p-value = 1.224e-06From the output, the p-value < 0.05 implying that the distribution of the data are significantly different from normal distribution. In other words, we cannot assume the normality
7b. Normality test for transformed log-linear model
shapiro.test(fit4$fitted.values)##
## Shapiro-Wilk normality test
##
## data: fit4$fitted.values
## W = 0.96562, p-value = 1.224e-06From the output, the p-value < 0.05 implying that the distribution of the data are significantly different from normal distribution. In other words, we cannot assume the normality
7c. Visual check for linearityfor transformed model
plot(fit3, 1)
plot(fit4, 1) # Line is not horizontal for either model