The Data

The RMS Titanic was a huge, luxury passenger liner designed and built in the early 20th century. Despite the fact that the ship was believed to be unsinkable, during her maiden voyage on April 15, 1912, the Titanic collided with an iceberg and sank.

We have information on \(n = 891\) passengers from the Titanic, and for today we are interested in modeling:

• Fare: How much each passenger’s ticket cost in US dollars.

We will build a few different models using these explanatory variables:

• Survived: An indicator for whether the passenger survived (1) or perished (0) during the disaster.
• Pclass: The class of the ticket held by the passenger; 1 = 1st class, 2 = 2nd class, 3 = 3rd class.
• Parch: number of parents/children the passenger had aboard the Titanic. Here, parent is defined as mother/father and child is defined as daughter, son, stepdaughter or stepson. NOTE: Some children traveled only with a nanny, therefore parch=0 for them. There were no parents aboard for these children.

To load the data, copy and paste the following into a code chunk and press play:

Titanic <- read.csv("https://www.dropbox.com/scl/fi/26vhy0o0n4zykjvpjgolo/Titanic.csv?rlkey=b4a6ewvjrgvd25gr42tuayoyh&st=yppg7u63&dl=1")

Titanic$Survived <- ifelse(Titanic$Survived == 1, "Yes", "No")
Titanic$Survived <- as.factor(Titanic$Survived)
Titanic$Pclass <- as.factor(Titanic$Pclass)

Model 1: One Categorical X

The first question we are going to explore is the relationship between \(X\) = survival and \(Y\) = ticket cost. In other words, can we find a relationship between whether or not someone survived the Titanic and how much they paid for their ticket? We are going to use a regression model to explore the relationship between the two variables.

library(ggplot2)

ggplot( Titanic, aes( x = Fare, y = Survived)) + geom_boxplot()

ggplot( Titanic, aes( x = Survived, y = Fare, y)) + geom_boxplot()

model_survived <- lm( Fare ~ Survived, data = Titanic)

summary( model_survived )$coefficients 

knitr::kable( summary( model_survived )$coefficients ) 

Model 2: Two Explanatory Variables

Now that we have explored a model with one explanatory variable, let’s add a second variable. When we add more than one explanatory variable to the model, the model is called a multiple regression (MR) model.

Why might we want to include multiple X variables in a model? One reason is that that putting two or more X variables together in the model allows us to build models that account for more sources of variation in \(Y\). It is reasonable to assume that more than one thing is related to \(Y\)!!

By including more explanatory variables, we may be able to improve our model fit (how well we are capturing the patterns in \(Y\)) and/or provide more accurate predictions for \(Y\). Now, this isn’t always true - we have to chose the \(X\) variables we add into the model wisely. Our goal is to start to explore models with multiple \(X\)s to see how we can work with them and evaluate them.

For now, in addition to \(X_1 =\) survival, we will include:

• Parch: number of parents/children the passenger had aboard the Titanic. Here, parent is defined as mother/father and child is defined as daughter, son, stepdaughter or stepson. NOTE: Some children traveled only with a nanny, therefore parch=0 for them. There were no parents aboard for these children.

ggplot( Titanic, aes( x =  Parch, y = Fare)) + 
  geom_point() + 
  stat_smooth(formula = y~x, method = "lm", se = FALSE)

model_survived_parch <- lm( Fare ~ Survived + Parch, data = Titanic)

Model 3: Categorical X with 3 levels

So far we have built (1) a model with a categorical X with 2 levels and (2) a model with a numeric X and a categorical X with 2 levels. However, categorical variables can have more than 2 levels. How does the model work in that case?

Consider the variable \(X\) = Pclass, where:

• Pclass: The class of the ticket held by the passenger; 1 = 1st class, 2 = 2nd class, 3 = 3rd class.

To build a regression model for Y = ticket price and X = class level, we can use the following:

model_class <- lm( Fare ~ Pclass , data = Titanic)

Model 4: Putting Them Together

We are going to build a model with \(X_1\) = passenger class and \(X_2\) = Parch (parent/child count).

Next Steps

As we move forward in the course, we will start to learn how we can use this idea of using multiple explanatory variables to build stronger models. This means we want to build models that are a better reflection of the variability in \(Y\) and the relationships between \(Y\) and other variables. We also need to learn how we can compare models, and how we might decide whether including more explanatory variables in the model is effective. All of that is coming up!