Titanic Survival
Titanic Survival
Ahmed Elhefnawy
Ahmed Elhefnawy
I found an interesting historical data set about Titanic, the famous British luxury passenger liner that sank in \(1912\). The data is available in the package carData under the name TitanicSurvival. You can see this source for more historical information about the Titanic. But for now, let’s have a closer look at our data.
Our data has \(1309\) observations of \(1309\) passengers on four variables: Whether each passenger survived or not, the passenger gender, the age, and the passenger class: first class, second class, or third class.
#load the package containing the data.
library(carData)
#load the data itself.
data("TitanicSurvival")
#have a look!
TitanicSurvival[1:15,] survived sex age passengerClass
Allen, Miss. Elisabeth Walton yes female 29.0000 1st
Allison, Master. Hudson Trevor yes male 0.9167 1st
Allison, Miss. Helen Loraine no female 2.0000 1st
Allison, Mr. Hudson Joshua Crei no male 30.0000 1st
Allison, Mrs. Hudson J C (Bessi no female 25.0000 1st
Anderson, Mr. Harry yes male 48.0000 1st
Andrews, Miss. Kornelia Theodos yes female 63.0000 1st
Andrews, Mr. Thomas Jr no male 39.0000 1st
Appleton, Mrs. Edward Dale (Cha yes female 53.0000 1st
Artagaveytia, Mr. Ramon no male 71.0000 1st
Astor, Col. John Jacob no male 47.0000 1st
Astor, Mrs. John Jacob (Madelei yes female 18.0000 1st
Aubart, Mme. Leontine Pauline yes female 24.0000 1st
Barber, Miss. Ellen Nellie yes female 26.0000 1st
Barkworth, Mr. Algernon Henry W yes male 80.0000 1st#create a summary for our variables.
summary(TitanicSurvival) survived sex age passengerClass
no :809 female:466 Min. : 0.1667 1st:323
yes:500 male :843 1st Qu.:21.0000 2nd:277
Median :28.0000 3rd:709
Mean :29.8811
3rd Qu.:39.0000
Max. :80.0000
NA's :263 The third class passengers were more than the first class and the second class passengers combined, and more than double the first-class passengers.
Let’s see the survived passengers by sex and by class.
library(ggplot2)
ggplot(data=TitanicSurvival,mapping=aes(x=survived))+geom_bar(aes(fill=sex))
ggplot(data=TitanicSurvival,mapping=aes(x=survived))+geom_bar(aes(fill=passengerClass))
#passengers' gender by class.
table(TitanicSurvival$passengerClass,TitanicSurvival$sex)
female male
1st 144 179
2nd 106 171
3rd 216 493#boxplot of the passengers' age.
boxplot(TitanicSurvival$age,horizontal = T,xlab="Passengers' Age")
abline(v=mean(TitanicSurvival$age,na.rm = T))
#passengers' age by class.
ggplot(data=TitanicSurvival,mapping=aes(x=age,fill=passengerClass))+geom_bar()
As you can tell, most of the second and third class passengers were younger than \(40\).
#passengers' age by survival proportion.
ggplot(data=TitanicSurvival,mapping=aes(x=age,fill=survived))+geom_bar()
The Probability of survival:
We want to see if the probability of survival differs by the class. Let \(S_i\) denotes the survival of passenger \(i\), and \(C_i\) denotes the class of passenger \(i\). Then, \[\begin{equation} \mathrm P(S_i \mid C_i) = \frac{\mathrm P (S_i \cap C_i)}{\mathrm P(C_i)} \end{equation}\]
The probability of survival conditional on having, for example, a first-class ticket, is the ratio between the probability that a first-class passenger survives and the probability of a passenger \(i\) to choose a first-class ticket.
#calculate the conditional probability of survival on having a first class ticket. You can use also this code to get the same result: nrow(dplyr::filter(TitanicSurvival,survived=="yes",passengerClass=="1st"))/nrow(dplyr::filter(TitanicSurvival,passengerClass=="1st"))
mean(TitanicSurvival$survived[TitanicSurvival$passengerClass=="1st"]=="yes")[1] 0.619195#calculate the conditional probability of survival on having a second class ticket.
mean(TitanicSurvival$survived[TitanicSurvival$passengerClass=="2nd"]=="yes")[1] 0.4296029#calculate the conditional probability of survival on having a third class ticket.
mean(TitanicSurvival$survived[TitanicSurvival$passengerClass=="3rd"]=="yes")[1] 0.2552891#calculate the probability of survival. You can also use this code to get the same result: nrow(dplyr::filter(TitanicSurvival,survived=="yes"))/nrow(TitanicSurvival)
mean(TitanicSurvival$survived=="yes")[1] 0.381971We might also use \(\chi^2\) test to know the magnitude of the p-value.
#run chi-squared test.
chisq.test(TitanicSurvival$survived,TitanicSurvival$passengerClass)
Pearson's Chi-squared test
data: TitanicSurvival$survived and TitanicSurvival$passengerClass
X-squared = 127.86, df = 2, p-value < 2.2e-16The p-value is almost equal to \(0\), implying that the probability of survival and the ticket class are not independent, at all levels.
To get a hint about the association between both variables, we might calculate the Contingency Coefficient and Cramer’s V:
#first have a look on our two variables.
table(TitanicSurvival$survived,TitanicSurvival$passengerClass)
1st 2nd 3rd
no 123 158 528
yes 200 119 181Contingency Coefficient \(=\sqrt{\frac {127.86}{1309+127.86}} \approx 0.298\).
Cramer’s V \(= \sqrt{\frac {127.86}{1309}} \approx 0.313\).
We didn’t consider the number of rows/columns in the calculation of Cramer’s V since the \(\min{(2-1,3-1)}=1\).
In \(R\), we can calculate Contingency Coefficient and Cramer’s V as well.
library(DescTools)
#calculate Contingency Coefficient.
ContCoef(TitanicSurvival$survived,TitanicSurvival$passengerClass)[1] 0.2983038#calculate Cramer's V.
CramerV(TitanicSurvival$survived,TitanicSurvival$passengerClass)[1] 0.3125332The two variables are not independent, as we said before, but the association is not very strong either.
Finally, I hope you had fun!