MATH333: Assessment Week 19

Assessment Deadline: Week 20, Tuesday 5pm.

Consider the titanic data set that was introduced in the last assessment. Here, the event that $i$th passenger survived the sinking RMS Titanic is assumed to follow a Bernoulli distribution with some unknown probability ${\mu }_i$. The unknown probability is related to the linear predictor ${\eta }_i$ via the logit transform: i.e. $\mathrm{logit}({\mu }_i)={\eta }_i$. The data set titanic contains three explanatory variables:

• pclass – passenger class, either 1st, 2nd or 3rd.

• sex – the gender of the passenger, either male or female.

• age – the age of the passenger in years.

1. Let the indicator variable $s_i^M$ be $1$ if passenger $i$ is male or $0$ otherwise, and the indicator variable $s_i^F$ be $1$ if passenger $i$ is female or $0$ otherwise. An analysis proposes the following linear predictor:

   $${\eta }_i={\beta }_0+{\beta }_1{s}_i^M+{\beta }_2{s}_i^F.$$

   Is this a good linear predictor? Justify your answer. [1]

2. State the linear predictor that contains age and passenger class. Explain any notation that you introduce.

   [Hint: To get started let $𝐚$ denote the vector of ages, with the $i$th entry corresponding to the age of the $i$th person in the data set, and let let ${𝐜}^{1st}$ be an indicator vector that is $1$ in the $i$th position if the $i$th passenger is in 1st class, $0$ otherwise. Construct other vectors as required carefully defining them and use these to formulate the linear predictor]

   [2]

3. State the linear predictor for age and passenger class with an interaction term between these two explanatory variables. [1]

4. Explain what effect including an age and class interaction term has on the linear predictor. [1]

5. Using R, fit a generalised linear model to the Titanic survivor data where the linear predictor contains passenger class, the gender of the passenger and the interaction between these explanatory variables. Write down the R command you used to fit this model. [1]

6. Estimate the probability for a female 1st class passenger to survive the sinking of the Titanic. Calculate a 95% confidence interval. [2]

7. Estimate the probability for a male 2nd class passenger to survive the sinking of the Titanic. [2]