Note that 1/Y is generally not well defined when Y’s range contains 0, and it’s messy when it approaches it
I agree. Just one note, I think a distribution for Y which encompasses 0 cannot be correct, because it would lead to infinities, which I am happy to reject. Can you give some examples in which Y (i.e. a distribution in the denominator) is defined such that it could not be zero, but you still found messiness?
when both X and Y contains both positive and negative parts
For this case, one can get point estimates from:
E(X) = P(X > 0)*E(X | X > 0) + P(X < 0)*E(X | X < 0).
E(1/Y) = P(Y > 0)*E(1/Y | Y > 0) + P(Y < 0)*E(1/Y | Y < 0).
I think that in practice you (almost) always want to calculate lives/$, not $/life
Yes, I prefer to calculate the cost-effectiveness in terms of benefits per unit cost. This way, the expected cost-effectiveness can be multiplied by the cost to obtain the expected benefits. In contrast, the cost cannot be divided by the expected cost per unit benefit to obtain the expected benefits.
Another advantage of benefits per unit cost is that they always increase with the goodness of the intervention, whereas the cost per unit benefit has a more confusing relationship (when it can be both positive and negative).
the cost is practically never zero
Yes, I do not think the cost can be zero. Even if the monetary cost is zero, there are always time costs.
Hi Nuño,
Nice points!
I agree. Just one note, I think a distribution for Y which encompasses 0 cannot be correct, because it would lead to infinities, which I am happy to reject. Can you give some examples in which Y (i.e. a distribution in the denominator) is defined such that it could not be zero, but you still found messiness?
For this case, one can get point estimates from:
E(X) = P(X > 0)*E(X | X > 0) + P(X < 0)*E(X | X < 0).
E(1/Y) = P(Y > 0)*E(1/Y | Y > 0) + P(Y < 0)*E(1/Y | Y < 0).
This may not buy you enough. E.g., sometimes you may want to calculate the $/life saved, where life saved is a distribution which could be 0.
I think that in practice you (almost) always want to calculate lives/$, not $/life, and the cost is practically never zero
Hi Lorenzo,
Yes, I prefer to calculate the cost-effectiveness in terms of benefits per unit cost. This way, the expected cost-effectiveness can be multiplied by the cost to obtain the expected benefits. In contrast, the cost cannot be divided by the expected cost per unit benefit to obtain the expected benefits.
Another advantage of benefits per unit cost is that they always increase with the goodness of the intervention, whereas the cost per unit benefit has a more confusing relationship (when it can be both positive and negative).
Yes, I do not think the cost can be zero. Even if the monetary cost is zero, there are always time costs.