Introduction

• There are many domestic flights departing from Melbourne Airport in day and night time.
• It has one common problem from unexpected factors which is a flight delayed.
• It happens frequently during a high-season holiday, heavy rain, storm, or it might happened from airlines or airports problem.
• If the departure flight was delayed, the arrival flights are likely to be delayed.
• However, it is not always depends on departure flight.
• Because flying through a storm or rain can make flight delayed too.

Introduction Cont.

• The purpose of this investigation is find the correlation between departures delayed and arrivals delayed that departing from Melbourne Airport only.

Problem Statement

• Is it possible for flight departing on time, but arriving late?
• Can I use the departures flight delayed to predict the arrivals flight delayed?
• There is the correlation between the departures flight delayed and the arrivals flight delayed.
• The Pearson correlation coefficient will be use in this investigation.
• It use for measure the strength of the linear relationship between two variables.

Data

• The data was recorded by the Bureau of Infrastructure, Transport and Regional Economics (BITRE).
• The domestic airlines (on time performance) dataset, downloaded from the data.gov.au website.
• The dataset contains over 60,000 rows of data.
• It contains data recording domestic flight time performance with Australia.
• The data was reported monthly by airlines since 2004.
• The scale of numeric variables was recorded by number of flights.

Domestic_Airlines <- read_csv("~/Downloads/domestic_time_performance.csv")

• The sample data will be chosen from departing via Melbourne only.

Melbourne_Airport <- Domestic_Airlines %>% filter(Departing_Port == "Melbourne")

Data Cont.

• Specifically, the Domestic_Airlines contains the following variables:
  • Route: Route description
  • Departing_Port: Departure city
  • Arriving_Port: Arrival city
  • Airline: Airline name
  • Month: 5-digits numbers
  • Sectors_Scheduled: Number of flight schedule in one sector
  • Sectors_Flown: Number of flight that has flown
  • Cancellations: Number of flight removed from service within 7 days of scheduled
  • Departures_On_Time: Number of departures on time
  • Arrivals_On_Time: Number of arrivals on time (arrived at the gate before 15 minutes)
  • Departures_Delayed: Number of departures Delayed (departs the gate before 15 minutes)
  • Arrivals_Delayed: Number of arrivals delayed
  • Year: Year’s data has been collected
  • Month_Num: Month’s data has been collected

Descriptive Statistics

• There was an issue in the dataset. It has the total rows of all airlines and all ports of departing and arriving.
• Therefore, the total rows need to filter it out.

Decsriptive Statistics Cont. (2)

Departures Delayed

Melbourne_Airport %>% summarise(Min = min(Departures_Delayed,na.rm = TRUE),
                                           Q1 = quantile(Departures_Delayed,probs = .25,na.rm = TRUE),
                                           Median = median(Departures_Delayed, na.rm = TRUE),
                                           Q3 = quantile(Departures_Delayed,probs = .75,na.rm = TRUE),
                                           Max = max(Departures_Delayed,na.rm = TRUE),
                                           Mean = mean(Departures_Delayed, na.rm = TRUE),
                                           SD = sd(Departures_Delayed, na.rm = TRUE),
                                           n = n(),
                                           Missing = sum(is.na(Departures_Delayed))) -> table1
knitr::kable(table1)

Decsriptive Statistics Cont. (3)

Arrivals Delayed

Melbourne_Airport %>% summarise(Min = min(Arrivals_Delayed,na.rm = TRUE),
                                           Q1 = quantile(Arrivals_Delayed,probs = .25,na.rm = TRUE),
                                           Median = median(Arrivals_Delayed, na.rm = TRUE),
                                           Q3 = quantile(Arrivals_Delayed,probs = .75,na.rm = TRUE),
                                           Max = max(Arrivals_Delayed,na.rm = TRUE),
                                           Mean = mean(Arrivals_Delayed, na.rm = TRUE),
                                           SD = sd(Arrivals_Delayed, na.rm = TRUE),
                                           n = n(),
                                           Missing = sum(is.na(Arrivals_Delayed))) -> table2
knitr::kable(table2)

Linear Regression

plot(log(Departures_Delayed) ~ log(Arrivals_Delayed), data = Melbourne_Airport)

- The plot above shows a strong positive of correlations.

Correlation

\[r = \frac{L_{xy}}{\sqrt[]{L_{xx}-L_{yy}}}\]

cor(Melbourne_Airport$Departures_Delayed,Melbourne_Airport$Arrivals_Delayed, use="complete.obs")

## [1] 0.9575857

• Need a \(p\)-value or CI to test the null hypothesis.

Correlation Cont.

• Perform a full correlation analysis between departures delayed and arrivals delayed.

library(Hmisc)
bivariate<-as.matrix(dplyr::select(Melbourne_Airport, Departures_Delayed,Arrivals_Delayed))
rcorr(bivariate, type = "pearson")

##                    Departures_Delayed Arrivals_Delayed
## Departures_Delayed               1.00             0.96
## Arrivals_Delayed                 0.96             1.00
## 
## n
##                    Departures_Delayed Arrivals_Delayed
## Departures_Delayed               6483             6483
## Arrivals_Delayed                 6483             6487
## 
## P
##                    Departures_Delayed Arrivals_Delayed
## Departures_Delayed                     0              
## Arrivals_Delayed    0

Hypothesis Testing

• A hypothesis test for \(r\) has the followings statistical hypotheses:

\[H_{0}:r = 0 \] \[H_{A}:r \neq 0 \]

• The correlation between departures delayed and arrival delayed to be \(r\) = .96 and the \(p\)-value = 0, so \(p\) < 0.001.

Hypothesis Testing Cont.

• The \(p\)-value for \(r\) can be calculated by converting \(r\) to a \(t\) statistics: \[t=r\sqrt[]{\frac{n-2}{1-r^2}}=0.96\sqrt[]{\frac{6496-2}{1-0.96^2}}=276.293\]

• Therefore, a two-tailed \(p\)-value for \(r\) can be calculated using:

2*pt(q = 276.293, df = 6496 - 2, lower.tail=FALSE)

## [1] 0

Confidence Interval

• Testing \(H_{0}\) using a confidence interval approach.

\[r=z=\frac{1}{2}ln(\frac{1+r}{1-r})\]

0.5*(log((1+.96)/(1-.96)))

## [1] 1.94591

Confidence Interval Cont.

• A 95% CI for \(r\), [\(p_{1}, p_{2}\)]:

library(psychometric)
r=cor(Melbourne_Airport$Departures_Delayed,Melbourne_Airport$Arrivals_Delayed,use="complete.obs")
CIr(r = r, n = 6496, level = .95)

## [1] 0.9555183 0.9595589

• Therefore, \(r\)=0.96, 95% CI [0.956, 0.96]
• This confidence interval does not capture \(H_{0}\), so \(H_{0}\) was rejected.

Interpretation

• Correlation summary:
  • \(r\) = .96
  • 95% CI [0.956, 0.96]
  • \(p\) < .001
• Decision:
  • Reject \(H_{0}\)
• Conclusion
  • There was a statistically significant positive correlation between departures delayed and arrival delayed.

Discussion

• In this investigation, it aims to find the correlation between the departure flight delayed and the arrivals flight delayed.
• The plot shows a strong positive relationship.
• After performing the hypothesis test, \(p\)-value has less than 0.01 which means reject \(H_{0}\).
• Therefore, it was a statistically significant between departures and arrivals delayed.
• Strengths: It has enough sample data, because it was recorded over 10 years.
• Limitations: When the departures flight delayed, the arrivals flight normally delayed.
• Future investigations: Find the correlation between delayed flight during each month (Rainy season might make flight delayed).

• Overall, a Pearson’s correlation was calculated to measure the strength of association between departures delayed and arrivals delayed (r=0.96), and the correlation was statistically significant (\(p\) < 0.001).

• Re-state
  • Now, the arriving time can be predict after departing.

References

• Bureau of Infrastructure, Transport and Regional Economics. (2017). Domestic Airlines - On Time Performance. Retrieved from: https://data.gov.au/dataset/domestic-airline-on-time-performance
• Horne, T. (2013). A Flight with Jetstar and Tiger Airlines. Retrieved from https://think-tasmania.com/flight/