Does smoking increase the risk of acne? Schafer et al. (2001) conducted a study involving 851 subjects whose smoking status was known. Each subject was independently assessed by dermatologists and rated on four characteristics associated with acne. For the purposes of the study, an individual is deemed to have acne at the time if at least one of the characteristics was present. Suppose a summary of the data is as given in the table below:



Schafer, T., Nienhaus, A., Vielue, D., Berger, J., and Ring, J. (2001): Epidemiology of acne in the general population: the risk of smoking. British Journal of Dermatology 145, 100-104.

Part a)
Find the risk ratio (of having acne if a non-smoker compared to being a smoker) from the data above, giving your answer to two decimal places.


Part b)
Find the corresponding odds ratio, to two decimal places.


Part c)
Suppose we fit a logistic regression model to the above data, of the form

for some parameters $\beta _{0}$ and $\beta _{1}$, and in which $Y=1$ indicates the presence of acne ($Y=0$ is absence), and $X_{1}=1$ indicates smoking ($X_{1}=0$ otherwise). What would be the estimate of the parameter $\beta _{1}$? Give your answer to two decimal places.


Part d)
Other data were gathered on each subject, including age and gender. Suppose the following logistic regression model is fitted to the data:



where $Y$ and $X_{1}$ are as above,



and $X_{3}$ is age (in years). Using the above model estimate the chance that a twenty-year-old female smoker has acne, to three decimal places.


Part e)
Estimate the multiplicative increase in the odds of acne for a smoking male compared to a non-smoking male of the same age, giving your answer to two decimal places.