Nevertheless, I suppose what you would suggest is just setting limits to start with and looking for a bump there instead. That's not a bad idea, and some (Gross, e.g.) have suggested setting limits even in discovery papers.
I don't see how this addresses any of the issues, it just makes it more expensive (ceteris paribus). As I already wrote:
"A more stringent significance level just means it costs more money or the data is much cheaper to collect. ...The way it works is simple, the null model acts as a strawman. The confidence in rejecting the null model is somehow translated into support for the researcher's favorite theory. How does changing the significance level address this problem?"
>"Nevertheless, I suppose what you would suggest is just setting limits to start with and looking for a bump there instead."
Exactly, and the more precise the limits, the more meaningful for the theory if its consistent with the data. At the extreme, theories that allow for any value at all (as I understand the case is for every possible observation under string theory) are not supported by the data no matter how it turns out.
Confidence in rejecting the null isn't "somehow translated into support for the researchers' favorite theory", it's translated into support for the existence of a signal, which makes sense since the null is the background-only hypothesis.
After we establish rudimentary support for a model by finding one or more such signals, we can then embark on the process of doing precision measurement to really test that model (as is being done for the SM).
If its only a statistical fluctuation, the p-value due to sequential sampling follows a markov process (random walk) between zero and one (there is no convergence). So starting from a significant result, the p-value would be just as likely to decrease and become "more significant" as increase and "lose significance". The significance cutoff is irrelevant to this behavior, although the random walk will spend less time in these extreme regions to begin with.
Here is an example of the random walk in R. It was my first attempt when going up to 100k samples, after n = 45 the p-value was ~5e-4 (~3.3 sigma), later at n = 36441 it went even more extreme in the other direction, corresponding to a p-value of ~1.9e-6 (~4.65 sigma). Here is the plot and code: https://image.ibb.co/hOHO08/p_markov_chain.png
set.seed(1234)
n = 1e5
a = rnorm(n)
b = rnorm(n)
p = sapply(10:length(a), function(i) t.test(a[1:i], b[1:i])$p.value)
plot(p, type = "l", panel.first = grid())
> min(p)
[1] 0.0005394751
> 1 -max(p)
[1] 1.908121e-06
>"_if_ what you're seeing is a statistical fluctuating"
You aren't you are seeing that, so everything that follows is irrelevant. The background model is imperfect (eg nobody believes in a standard model with no higgs boson, everyone knows it is wrong somehow). In this (realistic) case convergence to statistical significance is guaranteed with enough data. However, in the rare (non-existent?) case where the background model is thought to be perfect, it is a different story.
>"Confidence in rejecting the null isn't "somehow translated into support for the researchers' favorite theory", it's translated into support for the existence of a signal, which makes sense since the null is the background-only hypothesis."
This amounts to claiming that in most peoples minds (including the authors of the paper) the Higgs boson, gravitational waves, etc haven't been "detected with 5 sigma confidence"... How much would you want to bet on this? The null model could be rejected due to a loose cable (ftl neutrinos), but the p-value doesn't distinguish between these explanations. Perhaps there was "a loose cable" in the LHC experiments but nobody looked very hard since the deviation agreed with preconceived notions? The p-value doesn't help you here.
>"After we establish rudimentary support for a model by finding one or more such signals, we can then embark on the process of doing precision measurement to really test that model (as is being done for the SM)."
Why not just skip the first step? Instead collect data to check whatever models that exist. When none of them can explain it, modify or come up with new models until it fits. Then derive new predictions from those and collect new data to check them, repeat.
EDIT: I ran it again and got this: https://image.ibb.co/g9rGL8/p_markov_chain2.png
> min(p)
[1] 0.005265033
> 1-max(p)
[1] 1.453814e-05If you collect, say, 1000 samples of a normally distributed variable and find that the mean (for instance) is 2 sigma from the expected mean, then if you collect another 1000 samples and take the mean of all 2000 samples, you will probably find that the significance decreases, assuming the first result was a statistical fluctuation. This is just [regression to the mean](https://en.wikipedia.org/wiki/Regression_toward_the_mean).
What we don't do is dishonestly drip-feed data until we hit the significance we want, then stop. Data is added in fairly large chunks (generally an year's run at a time), and when a new dataset is added the analysis is developed with (at least the new) data blinded. The dataset is only unblinded when the analysis has been finalized.
You have found a deviation from your null model. That is it.
>"so it's right to claim a discovery."
No, if you knew the null model was flawed to begin with nothing has been learned. Also I wouldn't refer to something like "equipment was malfunctioning" as a discovery. Drawing any conclusions about the actual research hypothesis must be done outside the NHST framework, or else there is a fallacy at play (probably strawman).
>"What we don't do is dishonestly drip-feed data until we hit the significance we want, then stop. Data is added in fairly large chunks (generally an year's run at a time), and when a new dataset is added the analysis is developed with (at least the new) data blinded. The dataset is only unblinded when the analysis has been finalized."
Great, but even if you don't p-hack it still doesn't work unless it really makes sense to assume the null/background model is perfectly true. P-hacking is just more BS on top of an already BS procedure.
In summary: If there is anything wrong with the null/background model at all, the p-value will converge on zero, it is just a matter of collecting enough data. This will happen regardless of whether the theory of interest (research hypothesis) is accurate or not.
We don't know that the null is flawed to begin with. Not in a way that matters to the analysis anyway, otherwise you've picked the wrong null. No one runs analyses to discover the pion for the millionth time.
For BSM work, the way it generally works is this (ignoring limit setting):
- Pick a signature predicted by one or more BSM models (or even none), that either isn't predicted by the SM or is exceedingly rare in the SM. For example, you might predict a new particle that decays in some predetermined way.
- Define a signal region in some combination of variables where you expect to find a signal corresponding to this signature. This region is blinded, and you don't look at it until after fixing your entire procedure.
- Estimate the background in this signal region with a combination of Monte Carlo simulations and extrapolation from outside this region (data-driven backgrounds). This obviously has both statistical and systematic uncertainties, and a lot of the hard work is in getting this right. For instance, you might use independent Monte Carlo generators, or define control regions to check your estimation in.
- "Open the box" and look at the signal region data. If you see an excess over the estimated background, calculate the significance. Even here, if you were looking for a new particle, for instance, and the excess isn't a localized bump, this would be an indication that the background estimation may be flawed.
- If the significance is over 5 sigma, you have a discovery.
As you can see, you aren't using a null hypothesis that you already know is flawed. Your null is specifically "no signal exists". A positive deviation that isn't a statistical fluctuation is by definition a discovery of a signal.
Things like malfunctioning equipment go into the uncertainties if they occur in a way that the researchers have considered (which often means uncertainties are set conservatively if the equipment behaviour is poorly understood). If they occur in a way that no one considered, there's no statistical trickery that's every going to compensate for that. If we just went straight to setting limits, we would still see a deviation from what's expected there if equipment malfunctioned in a manner that faked a signal.
> In summary: If there is anything wrong with the null/background model at all, the p-value will converge on zero, it is just a matter of collecting enough data.
In summary: Assuming the background estimation is correct, the p-value will only converge to zero if a true signal exists. If the background estimation is wrong, then the background estimation is wrong and this is a problem with the background estimation, not hypothesis testing.
I'm by no means claiming the system is perfect, but it isn't systematically flawed the way you seem to be claiming.