MATH333: Assessment Week 18

Assessment Deadline: Week 19, Tuesday 5pm.

A Bernoulli regression assumes $Y_i\sim \mathrm{Bernoulli}({\mu }_i)$, $i=1,\mathrm{\dots },n$ with free parameters $(\alpha ,\beta )$ specified by $\mathrm{logit}({\mu }_i)=\alpha +\beta x_i$ where $x$ is a given covariate.

1. Explain what is the saturated model and null model. [2]

2. For the observed responses $y_1,\mathrm{\dots },y_n$, the fitted means are ${\widehat{\mu }}_1,\mathrm{\dots },{\widehat{\mu }}_n$. Find an expression for the residual deviance of the model in terms of the fitted means. [3]

3. The data file titanic.dat (accessible from MOODLE) contains information about 1046 passengers on RMS Titanic in 1912 and whether or not they survived the sinking of the ship (survived = 1 or died = 0).

   In R, fit a Bernoulli regression model to the survival of passengers with the linear predictor:

   $$\mathrm{logit}({\mu }_i)={\alpha }_0+{\alpha }_1{\mathrm{age}}_i$$

   1. What is the fitted probability of surviving, ${\widehat{\mu }}_i$, for a 20 year old passenger? [1]

   2. Evaluate a 95% confidence interval for the age co-efficient. Is age a good explanatory variable for predicting the survival of passengers? You may assume the regression coefficient estimator has a normal distribution with mean given by the true value of the regression coefficient and standard deviation approximated by the standard error from the R output. [2]

   Consider instead the following Bernoulli regression model where the linear predictor is based on the sex of the passengers:

   $$\mathrm{logit}({\mu }_i)={\beta }_1{\mathrm{female}}_i+{\beta }_2{\mathrm{male}}_i$$

   where ${\mathrm{female}}_i$ is an indicator variable that is $1$ if passenger $i$ is female or $0$ otherwise, and likewise for the indicator variable ${\mathrm{male}}_i$. Fit this regression model in R. Note that there is no intercept term.

   1. What is the fitted probability of surviving for a male passenger? [1]

   2. Is this model adequate in describing the variability in the data at the 5% significance level? [1]