Calculate expected survival time for two stage survival

Question

In a book about Biostatistics, I found this example to calculate expected value:

Consider the following hypothetical example of a lung cancer study in which all patients start in phase 1, transition into phase 2, and die at the end of phase 2. Unfortunately, but inevitably, all people die. Biostatistics is often concerned with studying approaches that could prolong or improve life. We assume the length of phase 1 is random and is well modeled by an exponential distribution with mean of five years. Similarly, the length of phase 2 is random and can be modeled by a Gamma distribution with parameters α = 5 and β = 4. Suppose that a new drug that can be administered at the beginning of phase 1 increases 3 times the length of phase 1 and 1.5 times the length of phase 2. Consider a person who today is healthy, is diagnosed with phase 1 lung cancer in 2 years, and is immediately administered the new treatment. We would like to calculate the expected value of the survival time for this person. Denote by X the time from entering in phase 1 to entering phase 2 and by Y the time from entering phase 2 to death without taking treatment. Thus, the total survival time is 2 + 3X + 1.5Y and the expected total survival time, in years, is E(2+3X +1.5Y) = 2+3E(X)+1.5E(Y) = 2+3×3+1.5×5/4 = 12.875 .

What I don't understand is why in the last equation E[x] is set to 3 while in my opinion it should be 5 because the length of phase 1 (for patients not taking the drug) has the exponential distribution with mean of five years?

I'm also a bit confused with this sentence:

Consider a person who today is healthy, is diagnosed with phase 1 lung cancer in 2 years, and is immediately administered the new treatment

Does that mean that the person will enter the phase 1 in the next two years while today they are healthy? It sounds kind of strange to me how medical advancements can help predict a future disease for someone's today healthy two years from now.

Answer 1

There is one way that this might have been thought by the authors to make sense. But I think that you are correct.

Their argument might be if the diagnosis of Phase 1 came 2 years after Phase 1 actually started. Then the argument might be that expected remaining length of Phase 1 is $5-2=3$ years.

The problem with that argument is the memoryless nature of the assumed exponential survival function. For an exponential survival function, the mean residual life conditional upon survival to any time after 0 is always equal to the expected survival from time 0. That supports your argument that the remaining expected duration of Phase 1 for that individual should be 5 years, and that the extension of Phase 1 by the drug would be 3 times that. It's possible that the authors had some other model for the effect of the drug in mind that would support their argument, but that's not clear from what you quote.

It looks like the book you cited is essentially self-published, and thus might not have undergone the editorial review of a traditionally published text. This seems to be an error, perhaps made without properly thinking through the implications, that might have been caught during a traditional editorial process.

In this case, it would make sense to address your question directly to the authors. If they provide a compelling argument to support what you quote, please provide that as another answer to this question. (It's OK to provide and accept your own answer to your question on this site.)