Hi Akansha
The contribution to the partial likelihood when someone experiences the event of interest is:
hazard of person who experienced the event at the time of the event / sum of all hazards at the time of the event of those at risk just before the event
If multiple people experience the event at the same time, we use Breslow's approximation. In this case, the contribution to the partial likelihood is the product of the above ratios for each of the people experiencing the event.
This is covered in Section 4 of Chapter 8.
In this question, the hazard of an individual at time t is given by:
\( \lambda_0 (t) * exp(\beta_1 * X_1 + \beta_2 * X_2) \)
The second individual recovering was treated with medication on the second day after symptoms developed. This means we have X1 = 1 and X2 = 1. The hazard for this person is then \( \lambda_0 (t) * exp(\beta_1 + \beta_2) \).
For the denominator, we started with 600 individuals and one person recovered after 2 days of treatment. So just before the second recovery, there are 599 individuals at risk. 100 of these individuals have X1 = 1 and X2 = 0, 99 of them have X1 = 1 and X2 = 1 (as the first individual to recover had X1 = 1 and X2 = 1) and so on. Similar to the above, we can work out the hazards for each of these types of individuals. We can then take the sum of these for the denominator. Finally, when we take the above ratio, we note that all the hazards have the same baseline hazard in and so this cancels out.
Hope this helps!
Andy
