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 \(n = 714\) 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:

• Sex: the sex of the passenger, limited to male and female.
• Pclass: The class of the ticket held by the passenger; 1 = 1st class, 2 = 2nd class, 3 = 3rd class.
• Age: The age of the passenger in years. Note: Decimals are for children under a year in age.

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

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

# Some Data cleaning
Titanic$Sex <- as.factor(Titanic$Sex)
Titanic$Pclass <- as.factor(Titanic$Pclass)
Titanic <- na.omit(Titanic)

Model 1: One Categorical X

The first question we are going to explore is the relationship between \(X\) = sex and \(Y\) = ticket cost. In other words, were ticket prices different for men versus women? We are going to use a regression model to explore the relationship between the two variables.

library(ggplot2)

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

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

model_sex <- lm( Fare ~ Sex, data = Titanic)

summary( model_sex )$coefficients 

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

Model 2: Multiple Regression Model 1

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? Recall that our goal in modeling is to try to capture patterns in \(Y\). Some passengers have low fares, some have high fares, and some are in the middle. What is different about the passengers that might relate to these differences in prices?? It makes sense that more than one thing might be related to differences in prices, so by adding in more than one \(X\), we can include more things that might be related to higher/lower prices.

In addition to \(X_1 =\) sex, we will include:

• Age: The age of the passenger in years. Note: Decimals are for children under a year in age.

ggplot( Titanic, aes( x =  Age, y = Fare)) + 
  geom_point()

model_sex_age <- lm( Fare ~ Sex + Age, data = Titanic)

plot_MRmodels <- function( numericX, categoricalX, Y, dataset){
  
  xlab = numericX
  ylab = Y
  groupLab <- categoricalX
  
  Y <- dataset[,which(colnames(dataset)==Y)]
  numericX <- dataset[,which(colnames(dataset)==numericX)]
  categoricalX <- dataset[,which(colnames(dataset)==categoricalX)]
  
    colorList <- c("#000000", "#E69F00" , "#56B4E9" ,"#009E73","#F0E442" , "#0072B2", "#D55E00", "#CC79A7")
  
  colorsneeded <- levels(categoricalX)
  colors <- rep(NA,nrow(dataset))
  for(i in 1:nrow(dataset)){
    for( j in 1:length(colorsneeded)){
     if(categoricalX[i] == colorsneeded[j]){
        colors[i] <- colorList[j]
      }
    }
  }
  
  # Build the model
  model <- lm( Y ~ numericX + categoricalX, data = dataset)
  
  # Create the lines
  ggplot( dataset, aes( x = numericX, y = Y, group = categoricalX)) + 
    geom_point(col = colors) +
    geom_abline(intercept = model$coefficients[1], slope = model$coefficients[2], color = colorList[1], lwd = 1) + 
    geom_abline(intercept = model$coefficients[1] + model$coefficients[3], slope = model$coefficients[2], color = colorList[2], lwd = 1) +  geom_abline(intercept = model$coefficients[1] + model$coefficients[4], slope = model$coefficients[2], color = colorList[3], lwd = 1) + labs(x = xlab, y = ylab)
    
}

plot_MRmodels("Age", "Sex", "Fare", dataset = 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 = passenger class, we can use the following:

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

Model 4: Multiple Regression Model 2

Now that we have explored a categorical X with 3 features, let’s build another multiple regression model. This time, we are going to build a model with \(X_1\) = passenger class and \(X_2\) = age.

When we had our model with age and sex, we graphed our results using:

plot_MRmodels("Age", "Sex", "Fare", dataset = Titanic)

The only change we have now is that instead of “Sex”, our categorical explanatory variable is “Pclass”.

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. 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!