2.6

Overview of Two-Way Tables
- Statistical analysis of two-way tables is considered the most widely used method in the subfield of statistics referred to as categorical data analysis.
- The discussion begins with a motivating example to illustrate the concept.

Definition
- Simpson’s Paradox refers to the phenomenon where the effects of lurking variables can strongly influence the perceived relationships between two categorical variables.
Application of Simpson's Paradox
- A significant example is provided to demonstrate this concept.

Context
- Air travelers generally prefer to arrive on time.
- Airlines collect data on on-time arrivals across their flights.
- The example discusses data collected for a year regarding flights directed to Prince George from two airlines: Air Canada and WestJet Air.
Data Reporting
- The table provided details a year’s worth of on-time and delayed arrivals for two airlines:
- Air Canada:
  - On time: 718
  - Delayed: 74
  - Total flights: 792
- WestJet Air:
  - On time: 5534
  - Delayed: 532
  - Total flights: 6066
Percentage of Delays
- WestJet Air's delay percentage:
- $rac{532}{6066} = 0.088$ , or 8.8%.
- Air Canada’s delay percentage:
- $rac{74}{792} = 0.093$ , or 9.3%.
Initial Impressions
- Analysis looking only at these percentages suggests that WestJet Air is preferred to minimize delays.

Analysis by Departure City
- The subsequent data table further breaks down delays by city of departure:
Vancouver Departures:
- Air Canada:
  - On time: 497
  - Delayed: 62
  - Total: 559
  - Delay percentage: $rac{62}{559} = 0.111$ , or 11.1%.
- WestJet Air:
  - On time: 694
  - Delayed: 117
  - Total: 811
  - Delay percentage: $rac{117}{811} = 0.144$ , or 14.4%.
Toronto Departures:
- Air Canada:
  - On time: 221
  - Delayed: 12
  - Total: 233
  - Delay percentage: $rac{12}{233} = 0.052$ , or 5.2%.
- WestJet Air:
  - On time: 4840
  - Delayed: 415
  - Total: 5255
  - Delay percentage: $rac{415}{5255} = 0.079$ , or 7.9%.
Conclusions from Departure Data
- For both Vancouver and Toronto, Air Canada shows a lower percentage of delays compared to WestJet Air, suggesting that in both instances, Air Canada is the better choice for on-time arrivals.

Understanding Data Aggregation
- The previous table is an example of a three-way table, as it incorporates the city of departure.
- The initial two-way airline data table was derived by aggregating data without considering the city variable, which consequently masked the effects of the lurking variable (city departure).

Definition Reconfirmed
- An association or comparison that holds for all several groups can reverse direction when the data are combined to form a single group; this is termed Simpson’s Paradox.
Simpson’s Paradox for Quantitative Variables
- An example involving two quantitative variables will be introduced, demonstrating how relationships can differ when viewed through varying lenses.

Illustrative Graphs
- Data points and their relationships (shown in graphical format across several slides) illustrate how relationships can vary, with the following described:
Estimated Regression Equations
- An example regression equation includes:
- $Y = 14.618 - 0.445X$
- Correlation coefficient: $r = -0.564$
Visual Representation
- Different scatter plots show various data groupings with their respective trends, including:
- An upper left grouping with an equation: $Y = 8.364 + 0.455X$ , and correlation $r = 0.455$ .
- A lower right grouping with an equation: $Y = 0.182 + 0.455X$ and correlation $r = 0.455$ .
Recap of Key Statistics
- Summary of regression data:
- All data set:
  - Intercept: 14.618
  - Slope: -0.445
  - Correlation: -0.564
- Upper Left data set:
  - Intercept: 8.364
  - Slope: 0.455
  - Correlation: 0.455
- Lower Right data set:
  - Intercept: 0.182
  - Slope: 0.455
  - Correlation: 0.455
Final Remarks on Data Relationships
- It is emphasized that the association observed within each of the groups can reverse when the data is aggregated into a single set, reinforcing the significance of consistent analysis across varying data groupings and the potential confounding effects of lurking variables.