I’m going to have to object. We don’t use “false positive” and “false negative” as synonyms for Type I and Type II error because they’re not the same thing. The difference is at the heart of the misuse of p-values by so many researchers, and the root of the so-called replication crisis.
Type I error is the risk of falsely concluding that the quantities being compared are meaningfully different when they are not, in fact, meaningfully different. Type II error is the risk of falsely concluding that they are essentially equivalent when they are not, in fact, essentially equivalent. Both are conditional probabilities; you can only get a Type I error when the things are, in truth, essentially equivalent and you can only get a Type II error when they are, in truth, meaningfully different. We define Type I and Type II errors as part of the design of a trial. We cannot calculate the risk of a false positive or a false negative without knowing the probability that the two things are meaningfully different.
This may be a little easier to follow with an example:
Let’s say we have designed an RCT to compare two treatments with Type I error of 0.05 (95% confidence) and Type II error of 0.1 (90% power). Let’s also say that this is the first large phase 3 trial of a promising drug and we know from experience with thousands of similar trials in this context that the new drug will turn out to be meaningfully different from control around 10% of the time.
So, in 1000 trials of this sort, 100 trials will be comparing drugs which are meaningfully different and we will get a false negative for 10 of them (because we only have 90% power). 900 trials will be comparing drugs which are essentially equivalent and we will get a false positive for 45 of them (because we only have 95% confidence).
The false positive rate is 45/135 (33.3%), nowhere near the 5% Type I error we designed the trial with.
Statisticians are awful at naming things. But there is a reason we don’t give these error rates the nice, intuitive names you’d expect. Unfortunately we’re also awful at explaining things properly, so the misunderstanding has persisted anyway.
And this paper tries to rescue p-values from oblivion by calling for 0.005 to replace the usual 0.05 threshold for alpha: Redefine statistical significance.
I’m a spatial-visual person, so when presented with this problem as a teenager, I instead solved it spatially. If you stack squares like.
█.
██.
███.
…
To the hundredth row, you get a shape that is a half filled square that is 100x100. Except the diagonal is fully filled in, so you need to add another 50.
So the answer was 0.5x100x100 + 0.5x100. Easy to visualize, easy to solve. 5050.
There’s a similar problem in sports – I was a teaching assistant for our rural school’s gym class so this one also popped up for me as a teenager. If you have 100 teams and each team needs to play each other team once… You fill in a similar grid, with the teams on both the x and y axis. The diagonal gets removed in this scenario because a team cannot play itself. So the answer is 0.5x100x100 - 0.5x100. 4950. Anyone who has ever tried to plan any sort of tournament can probably solve this intuitively, but 25 years ago I though I was the smartest gym class teaching assistant ever ;)
This is why I never succeeded at math. Like why does this shit work?? How can people just take a problem and be like, nah I’m going to just throw numbers all over the place and reassemble them in all sorts of ways and get an answer somehow…
I can’t just memorize arbitrary nonsense that “just is” I need to know how it works or it never sticks and all the math I’ve ever been taught was just “memorize this arbitrary nonsense and regurgitate a specific formula for a specific application that we’ve spent 0 time explaining other than telling you to memorize it. You want proofs and you can’t get proofs until advanced college courses” well guess I’ll just never understand mathematical manipulation then…
I feel like 50/50 school failed me and I failed at math.
It’s not arbitrary. Really try to think about the problem at hand. The ‘why’ is quite apparent. Ask yourself why did they go with 99+1+98+2… in the first place? And why is that the same as 101+101…? What was the benefit of simplifying it to that? How did it save the student time?
You can deduce this yourself and literally no memorization is involved to figure this out. No formulas needed either.
Once you have the idea, seeing that it works if often easy. But coming up with ideas like that can be really hard, which is why gauss was the only one in his class who got it. There is no general method, you just have to think about stuff for a while, but you can get better with practice. And it feels really good when you prove something for yourself, even if it’s relatively straightforward. You can just try to prove some simple things yourself, if you want, the advanced college courses are just for proving really advanced stuff.
The rules underpinning math are axioms in the end, but they’re not completely arbitrary, because if you change them in most cases it just fucks everything up.
The axioms that were chosen were chosen for good reason, and the rules they result in (such as summation and multiplication being commutative so 3x4=4x3 and 3+4=4+3) allow more complex rules to be created.
There’s a lot of philosophy of math at the core of all this , but it’s not really true that this is all arbitrary.
The algorithm gets a little weird if you’re summing the numbers to an odd number, though since there will be a left over number you have to deal with . By calculating 2S it works exactly the same in either case.
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