O(D:¬D), read as the odds of dementia to no dementia, is the odds ratio that D is true compared to the odds ratio that D is false. O(D:¬D)=3:1 means that it’s 3 times as likely that somebody has dementia than that they don’t. It doesn’t say anything about the magnitude of the probability, so it could be small, like 3% and 1%, or big, like 90% and 30%.
P(D or ¬D) = 1 (with P=1 one either has dementia or doesn’t have it) and P(D and ¬D) = 0 (probability of having dementia and not having it is 0), so if O(D:¬D)=3:1 then P(D) = 75% and P(¬D) = 25%.
I mean in your examples.. if :P(D) = 3% and P(¬D) = 1% then what happens in other 96+% of cases (when patient neither has dementia nor doesn’t have it)? If P(D) = 90% and P(¬D) = 30% what is the state of the 20+% of patients who both have dementia and don’t have it?
You’re completely right here. I meant odds of 3:1 in general, as opposed to when they’re a complement. (Also, 90 + 30 is more than 100%.) I’ll edit it.
It’s only 75% and 25% when the sum of probabilities is 100%, but O(red car:green car) can be 3:1 when 60% of cars are red and 20% are green, or when 3% of cars are red and 1% are green. The remainder are different colours.
I kept on reading and wanted to check your numbers further (concrete math I could do in my head seems correct but I wanted to check moar) but I got lost in my tiredness and spreadseets. If you’re interested in feedback on the math you’re doing.. smaller steps are easier to verify. For example when you give the formula for P(D|+) in order to verify it I have to check the formula, value of each conditional probability (including figuring out formula for each of those), and the result at the same time.
It would be much easier to verify if you wrote down the intermediate steps (possibly simplifying verification from 30 minutes of spredsheet munching to a few in-head multiplications).
I’m pretty sure you got math wrong here:
P(D or ¬D) = 1 (with P=1 one either has dementia or doesn’t have it) and P(D and ¬D) = 0 (probability of having dementia and not having it is 0), so if O(D:¬D)=3:1 then P(D) = 75% and P(¬D) = 25%.
I mean in your examples.. if :P(D) = 3% and P(¬D) = 1% then what happens in other 96+% of cases (when patient neither has dementia nor doesn’t have it)? If P(D) = 90% and P(¬D) = 30% what is the state of the 20+% of patients who both have dementia and don’t have it?
You’re completely right here. I meant odds of 3:1 in general, as opposed to when they’re a complement. (Also, 90 + 30 is more than 100%.) I’ll edit it.
It’s only 75% and 25% when the sum of probabilities is 100%, but O(red car:green car) can be 3:1 when 60% of cars are red and 20% are green, or when 3% of cars are red and 1% are green. The remainder are different colours.
I kept on reading and wanted to check your numbers further (concrete math I could do in my head seems correct but I wanted to check moar) but I got lost in my tiredness and spreadseets. If you’re interested in feedback on the math you’re doing.. smaller steps are easier to verify. For example when you give the formula for P(D|+) in order to verify it I have to check the formula, value of each conditional probability (including figuring out formula for each of those), and the result at the same time.
It would be much easier to verify if you wrote down the intermediate steps (possibly simplifying verification from 30 minutes of spredsheet munching to a few in-head multiplications).
I’ll keep that in mind for next time. Thanks!