Hello everyone. Below you will find some queries about the 8th chapter of CT4 on past papers. 1. April 2006 / Question B5 I can't understand why the last interval is t>=4+3/12. 2. April 2010 / Question 8 In the second sub-question (ii) we are asked for the survival function of the qualification. I can't understand why we didn't find F(t) and we found S(t) since the hazard is dj/nj. 3. September 2011 / Question 4 Could i make a table with intervals rather than points with the last interval being t>12? 4. September 2013 / Question 6 From what i understand the survival function relates to the event that the symptoms were disappeared so i cannot understand why in the sub-question (iii) the answer is 0.93777 and not 1-0.93777. In general, could you explain when the last interval is t>x and when t=x? where x is the last day/time etc. Thank you!
First of all, whether the last interval is t=x or t>=x depends on whether the data enables us to make inferences about the probabilities only up to time x, or after time x as well. For example, let's say I'm running a mortality study and I decide that I will run it for 100 days. If it at the end of 100 days, there are still some participants alive, then I can't know when they would go on to die. I have information only about the mortality up to time 100 days, so the final interval would be t=100. If however, I'm running a mortality study, and I continue until all lives have died, with the last death at time 125 days (say), then it appears from the data that no one survives for more than 125 days, so the final interval would be t>=125 (with a survival probability of 0 attached to it).
Looking at your other questions: 1. I think the final interval here should be 4.25<=t<=5, as we cannot make any comment after time 5 years (as we don't know when the remaining survivors at time 5 would go on to die). This looks like an oversight in the Examiners Report - the graph in (ii)(b) clearly stops at time 5. 2. S(t) is the survival function (which is what the question asks us to calculate). F(t) is the distribution function, and is equal to 1 - S(t). Here, the "hazard" is that of qualifying, so "survival" relates to remaining in the student population, and "death" relates to becoming a qualified actuary dj/nj relates to the hazard (or the probability of qualifying/dying at each tj). The Kaplan-Meier method uses factors 1 - dj/nj, which is the probability of not qualifying/dying, so gives the survival function directly. 3. Yes, I think you could label your table with intervals rather than points. So long as you state your answer for the probability of no re-growth by 9 months clearly you should be given the marks. I'd say the last interval was t>=12 (not t>12). 4. Here, the "hazard" is the skin condition disappearing. Participants in the study start in the "with condition" state (= alive, in the context of a mortality investigation) and might move to the "without condition" state (=dead). So the survival function relates to the probability of staying the in state of having the condition, and this is what is needed in (v).
For chapter 9 from what i understand is that proportional hazards model is used to express that the hazard for an individual is estimated from the data available and it is proportional to the general hazard. Is that right? I am a little confused with the utility of such models. Also, could you please explain to me what are the central exposed to risk and initial exposed to risk? i cannot understand their utility & definition. Thank you.
Yes, that's right. Proportional hazards models are useful because they enable us to quantify the effect of a particular variable on a hazard, eg they enable us to work out the effect of smoking on mortality. The central exposed to risk and initial exposed to risk are introduced in Chapter 10. These quantities are used in the calculation of the mortality statistics mu and q, respectively, which is what we're really interested in in most actuarial situations. Specifically, the central exposed to risk is the total time under observation for the lives in our population - the formula appears in Chapter 10.
