In suicide studies, alcohol use is found to be an important predictor of suicide ideation. Suppose the following logistic model is used to model the effect:

\[\begin{equation*}\operatorname{logit}[\operatorname{Pr}(\text { has suicide ideation })]=\beta_{0}+\beta_{1} * \operatorname{Drink} \tag{4.51}\end{equation*}\]

where Drink is the daily alcohol usage in drinks.

(a) If we know that the odds ratio of having suicide ideation between a subject who drinks two drinks daily with a subject who drinks one drink daily is 2 , compute \(\beta_{1}\).

(b) Drink' is a measure of alcohol use under a new scale where two drinks are considered as one unit of drink. Thus, Drink \({ }^{\prime}=\frac{1}{2}\) Drink. If the same logistic model is fitted, \(\operatorname{logit}[\operatorname{Pr}(\) has suicide ideation \()]=\beta_{0}^{\prime}+\beta_{1}^{\prime} * \operatorname{Drink}^{\prime}\). How are \(\beta_{1}\) and \(\beta_{1}^{\prime}\) related to each other?

(c) If data are applied to test whether alcohol use is a predictor of suicide ideation, does it matter which scale is used to measure the alcohol use?