Hi, I have a couple of questions from chapter 5. Example given after Question 5.9. The example introduces the idea of integrating by the integrating factor method. In this case, all terms are multiplied by e ^0.1t. That is clear, however I do not fully understand what happens with term 0.11 e ^ 0.11t * p01(t) after integration it disappears. why? Another question I have is in solution to problem 5.17 for the generator matrix. Where does the 3/400 and 1/400 come from? I don't see how this is calculated.... This is my first time posting a question here and I hope it will get me some answers before I discover them in a while....
For the integrating factor question: After multiplying through by the integrating factor of e^0.11t we have the equation: e^0.11t * d/dt [p01(t)] + 0.11e^0.11t*p01(t) = 0.01e^0.11t The left hand side of this equation is equal to: d/dt [p01(t)*e^0.11t] ie the exact derivative of p01(t)*e^0.11t. You can check this is true by carrying out the differentiation using the product rule. The next step just integrates the equation, giving p01(t)*e^0.11t on the left hand side as the integration cancels out the differentiation. So no terms get lost, it's just part of the integrating factor method.
For the solution to 5.17: I assume you've worked out the other numbers in the generator matrix. So focussing on the ones you mention: The overall rate of leaving the A state is 1/20 (as the average waiting time while the assessment is completed is 20 minutes). When the assessment is complete, the process moves to state H with probability 0.8, state M with probability 0.15, and state S with probability 0.05. The rate of going from state A to state M is equal to the overall rate of leaving state A (1/20) multiplied by the proportion of time the process moves to state M on leaving state A (0.15). 0.15*(1/20) is 3/400. Similarly, the rate of going from state A to state S is (1/20)*0.05. I hope this helps!
One more question - Q&A bank 3.14 Here we are calculating a variance for Kaplan Meier. Everything is clear for me except the fact that the solution table does not take into account any censoring mechanism, which typically is denoted as cj. I assumed that dismissal are calculated as dj, but anything else that causes leaving e.g. deaths in this case or leaving on own account should be taken into account as cj in order to decrease the total nj for each time interval. Why the solution is skipping cj, thus nj is higher than it should be???
Hi Guys, Can someone explain how to work out the integrating factor in this question. I understand why it makes sense. Thanks N
