Categorical Variables, Two-Way Tables, and Proportions: Titanic Survival Case Study

Research Context and Variable Classification

• Primary Investigation Objective:

  • The study investigates whether a passenger's class of travel served as a predictor of survival status during the sinking of the Titanic.
  • The analytical focus is placed on the relationship between two specific categorical variables: class of travel and survival status.

• Variable Identification:

  • Explanatory Variable: Class of travel.
  • The categories within the explanatory variable are First Class, Second Class, and Third Class.
  • Standard two-way table design convention specifies that the explanatory variable labels the rows of the contingency table.
  • Response Variable: Survival status.
  • The categories within the response variable are Survived (Yes) and Did Not Survive (No).
  • Standard two-way table design convention specifies that the response variable labels the columns of the contingency table.

Construction and Calculation of the Two-Way Table

• Summary Data Inputs:

  • First Class: Exactly $200$ passengers survived, and $123$ passengers did not survive.
  • Second Class: There were $277$ total passengers, $57\%$ of whom did not survive.
  • Third Class: A total of $181$ out of $709$ passengers survived.

• Mathematical Derivations for Cell Frequencies:

  • First Class Totals:
  • Survived ($\text{Yes}$): $200$
  • Did Not Survive ($\text{No}$): $123$
  • Total First Class Passengers: $200 + 123 = 323$
  • Second Class Cell Values:
  • The proportion of non-survivors is given as $57\%$ of $277$.
  • Set up the proportion formula:       $\frac{\text{part}}{\text{whole}} = \frac{\%}{100}$$\frac{x}{277} = \frac{57}{100}$$(277)(57) = 100(x)$$15789 = 100x$$x = 157.89$
  • Because $x$ represents a discrete count of human passengers, it is rounded to the nearest whole number: $x = 158$.
  • Calculating the number of survivors in Second Class:       $\text{Total} - \text{Did Not Survive} = 277 - 158 = 119$
  • Third Class Cell Values:
  • Survived ($\text{Yes}$): $181$
  • Total Third Class Passengers: $709$
  • Calculating non-survivors in Third Class:       $\text{Total} - \text{Survived} = 709 - 181 = 528$
  • Marginal and Grand Totals:
  • Total Survived ($\text{Yes}$): $200 + 119 + 181 = 500$
  • Total Did Not Survive ($\text{No}$): $123 + 158 + 528 = 809$
  • Grand Total of Passengers: $323 + 277 + 709 = 1309$ (equivalently confirmed by $500 + 809 = 1309$).

• Complete Contingency Table:

  • First Class: $200$ Survived ($\text{Yes}$), $123$ Did Not Survive ($\text{No}$), Total = $323$.
  • Second Class: $119$ Survived ($\text{Yes}$), $158$ Did Not Survive ($\text{No}$), Total = $277$.
  • Third Class: $181$ Survived ($\text{Yes}$), $528$ Did Not Survive ($\text{No}$), Total = $709$.
  • Marginal Column Totals: Total Survived ($\text{Yes}$) = $500$, Total Did Not Survive ($\text{No}$) = $809$, Grand Total = $1309$.

Limitations of Comparing Raw Counts

• Comparison of Raw Frequencies:

  • When comparing raw survivor counts between Second Class and Third Class passengers, Third Class recorded more survivors in raw numbers ($181$ survivors) than Second Class ($119$ survivors).

• Methodological Invalidation of Raw Counts:

  • Raw counts do not provide a meaningful or useful basis for comparison between groups of differing sizes.
  • The total number of passengers in Third Class ($709$) was substantially larger than the total number of passengers in Second Class ($277$).
  • Because group baselines vary substantially, relative proportions or rates must be calculated to evaluate whether an association exists between passenger class and survival status.

Population Proportions and Group Comparisons

• Population Parameter Framework:

  • The Titanic data set is treated as an entire population rather than a sample.
  • Consequently, the population parameter symbol $P$ is utilized for survival proportions rather than the sample statistic symbol $\hat{p}$.

• Proportion Calculations by Class:

  • First Class Survival Proportion ($P_1$):$P_1 = \frac{200}{323} \approx 0.6191...$
  • Rounded to three decimal places: $P_1 = 0.619$ (or $61.9\%$).
  • Second Class Survival Proportion ($P_2$):$P_2 = \frac{119}{277} \approx 0.4296...$
  • Rounded to three decimal places: $P_2 = 0.430$ (or $43.0\%$).
  • Third Class Survival Proportion ($P_3$):$P_3 = \frac{181}{709} \approx 0.2552...$
  • Rounded to three decimal places: $P_3 = 0.255$ (or $25.5\%$).

• Proportional Comparison Between Second and Third Class:

  • Comparing the rates reveals that $P_2 = 0.430$ is significantly higher than $P_3 = 0.255$.
  • Although Third Class had more survivors in absolute count ($181$ vs. $119$), Second Class passengers survived at a substantially higher proportion ($43.0\%$ vs. $25.5\%$).

Difference of Proportions Analysis

• Parameter Definitions and Notation:

  • Let $P_1$ represent the proportion of all First Class passengers who survived: $P_1 = 0.619$.
  • Let $P_3$ represent the proportion of all Third Class passengers who survived: $P_3 = 0.255$.
  • The direction of subtraction must be clearly specified through subscript notation to define the groups being contrasted.

• Calculation:$P_1 - P_3 = 0.619 - 0.255 = 0.364$

• Substantive Interpretation:

  • The difference in survival proportions between First Class and Third Class is $0.364$ (or $36.4\%$ percentage points).
  • A First Class passenger was $36.4\%$ percentage points more likely to survive the sinking than a Third Class passenger.

Visualizing Association: Segmented and Grouped Bar Charts

• Segmented Bar Chart Characteristics:

  • Displays stacked absolute frequencies representing survival status categories ($0 = \text{did not survive}$, $1 = \text{survived}$) across the three passenger classes on the horizontal axis.
  • Bar heights reflect the total passenger capacity of each class ($323$ in Class $1$, $277$ in Class $2$, and $709$ in Class $3$).
  • Visual inspection illustrates that while Class $3$ dominates total passenger volume, the shaded segment denoting survivors ($1$) constitutes a noticeably smaller fraction of the Third Class column relative to the First Class column.

• Grouped Bar Chart Characteristics:

  • Places separate bars side-by-side for non-survivors ($0$) and survivors ($1$) within each travel class.
  • Class 1: The survivor bar ($200$) exceeds the non-survivor bar ($123$).
  • Class 2: The non-survivor bar ($158$) exceeds the survivor bar ($119$).
  • Class 3: The non-survivor bar ($528$) vastly exceeds the survivor bar ($181$).

• Evidence of Association:

  • The graphical representations establish a strong association between class of travel and survival status.
  • Survival rates show a consistent, monotonic decline as class rank decreases from First Class ($61.9\%$) to Second Class ($43.0\%$) to Third Class ($25.5\%$).
  • The proportion of survivors was dramatically higher in First Class compared to Third Class, confirming that travel class was a strong predictor of survival.