The problem with p-values
aeon.co
aeon.co
Sometimes I fail to follow the distinctions made in certain strains of classical statistics. How is his "conundrum" different than this one? "Roll a 6 sided die whose result we cannot see. Take the proposition that the top shows a 6. It either does or it doesn’t, so it’s hard to see how we could pick a probability for this statement."
What matters to a scientific observer is how often you’ll be wrong if you claim that an effect is real, rather than being merely random.
I think "scientific observer" may mean statistician here.
For the scientist, what should matter is the probability that the claimed effect is real -- period. That is, unlike the statistician, the scientist isn't (shouldn't be) allowed to blame "modeling error" when it turns out that the measurements are biased, the samples are correlated, or the effects are nonlinear. False assumptions that "randomness" is the only (or main) danger can lead to unrealistic error bars and unwarranted confidence in the effectiveness of flawed models.
In case of the six-sided die, we have a good physical model and years of memory to know that there should be a 1/6 chance of any number coming up, so it's easy to estimate that prior. Similarly, when doing a disease screening test, we have data on how common diseases are in the general population that can be easily used.
The problem comes when using Bayesian methods on unsolved scientific problems. Suppose you don't know if the Earth goes around the Sun or the other way around, it's the early Renaissance and you've gathered some data that could indicate a probability of one or the other. What's the prior odds that one theory is correct? You have no idea, that's why you're investigating! The worry when using Bayes's theorem to replace deductive p-value methods is that the prior probabilities may just be made up out of baseless intuition, and skew the final calculation. (It can still be used effectively, you just have to get a little fancy.)
What's the probability of the earth going around some object bigger than it?
What's the probability of the earth going around something bigger than it that isn't the sun?
The answer to all 3 cannot all be 50/50.
I agree, but in the case of a die, "estimating the priors" is the same thing as "knowing the probability of a 6". So you are saying (correctly) that once we have a model that we believe in, we can assign a probability because we believe we know the probability in advance.
I was pointing more to the author's emphasis on "either it does or it doesn't". One view is that probability requires replication, and we can only speak of probabilities in the long run of many trials. The other view (Bayesian to my limited understanding) is that probability can also be used to measure appropriate degree of belief in a proposition.
While it's hard to quantify, and dependent on one's assumptions, I do think it's possible to speak of "the probability that the earth revolves around the sun", much as I think it's meaningful to talk about "the probability of anthropogenic warming" even though we only have a single Earth to study.
I don't really understand how classical statistics disallows discussion of the first while allowing discussion of the second. Maybe it doesn't?
That's what the author seems to mean by "either it does or it doesn't," though it isn't worded that well. Even if the die has already been rolled, we know that it could have been a 4 or a 5, and have good knowledge of the probabilities of those events. This is more like looking at a picture of the top of a die and seeing a six: what are the odds that it's a standard die, vs. a die with sixes on each face? I don't think it's meaningful to talk about that, or the sun going around the Earth, without using a Bayesian prior.
Right, using Bayes makes the difference not a "fundamental" one, but just a practical one of coming up with the Bayesian prior. Even if it would be hard to establish a consensus on the most appropriate complete set of factors determining the Bayesian prior, there are clearly some examples of meaningful inputs. Like, for example, if you somehow had information about how many of the researcher's previous hypotheses on the subject failed to obtain a "p-value of significance".
But perhaps more practically, you could consider things like the (Kolmogorov) complexity of the hypothesis. Since the number of "low complexity" hypotheses are finite, they are less susceptible to "p-value mining"[1]. The challenge being deciding which inputs to use to evaluate the complexity.
This seems to me like an area where machine learning should be applicable. Rather than lament about the impracticality of determining an appropriate, tractable set of (quantifiable) criteria for determining a Bayesian prior, why not just include every potentially relevant piece of information and let Deep Thought[2] figure out which are actually relevant? So what we really need is unified data about all published results that have been confirmed and discredited.
[1] obligatory xkcd: https://www.xkcd.com/882/
[2] for the youngsters: https://en.wikipedia.org/wiki/List_of_minor_The_Hitchhiker%2...
My only issue with this kind of calculations is that their assumptions dramatically changes the results. He says that 10% of the drug tests are going to be effective. This might apply to an specific area, but not to others. Here is a Nature publication that changes the most important parameters, power and % of programs/drugs/tests that trully have an impact http://www.nature.com/nmeth/journal/v10/n12/full/nmeth.2738.....
If you assume power of 80% (not crazy in rigorous studies) and 50% of programs having an effect (not crazy in many areas) then you get that 94% of the programs that you stated that had impact, were trully effective.
You're making another assumption: that the power calculations are actually rigorous and the effect size (not "having an effect", but "having an effect of at least the size used to calculate power at a given alpha") is sufficient. If you've ever designed a clinical trial or written up the statistics for a grant, you know damned well that these numbers are cooked 6 ways from Sunday.
Colquhoun is a sharp guy (I have debated various fine points with him on several occasions and he has convinced me with reproducible examples that I was wrong in my beliefs). It would be nice if the name of the article was "It's time for science to abandon the term statistically significant", as in the URL, because that's the real point here.
"Signficance" can only be judged in context. Post all your data and then we'll see whether you can be believed. But that notion of transparency is absolutely terrifying to the sorts of senior scientists who control most funding and peer review at the moment. It's OK, though; the rest of us put up preprints knocking down the most egregious lies and those who care about the truth can have it (for free, no paywall).
I've got plenty of Cell, Nature, and NEJM papers on my CV; direct support on grants where I am PI, Co-I, or KP is over $5M; and I still think the situation is fucked up. I'm not bitter because I feel left out; I'm bitter because I fear that the bad money is driving out the good, as it always does.
Take away the monetary incentive, the Journal Impact Factor BS, the "prestige" of hoodwinking 3-5 referees with a pile of STAP or single-sample comparisons (had to warn a student about this when he pulled some data from a Nature paper that a respectable MSKCC scientist gooned for...) and let's see what's left...
"Take the proposition that the Earth goes round the Sun. It either does or it doesn’t, so it’s hard to see how we could pick a probability for this statement."
Bayes Factor, the Bayesian alternative to a NHST, is quite a bit different than simply creating the Bayesian equivalent of a t-test. Bayes factor asks "How many times better is my Hypothesis at explaining the data than an alternate Hypothesis". So the Bayesian statement would first pit one model of the Earth's orbit against another. The Bayesian statement of the question of the Earth's orbit would be:
"How much more likely is the astronomical data we've observed to have happened given that the Earth revolves around the sun than it is if the sun revolved around the Earth."
For a more concrete example let's suppose that we have a coin. I think the coin has only heads and you think it is a fair coin, with a 50/50 chance of getting heads or tails. We observe three heads in a row. My hypothesis says that the probability of getting 3 heads in a row given a trick coin is 1. Your hypothesis says that the probability of getting 3 heads in a row given a fair coin is 0.5 x 0.5 x 0.5 = 0.125. My hypothesis explains the data 1/0.125 = 8 times better than your hypothesis. Now suppose the next flip is a tail. The probability of HHHT in my model is 0 and yours is 0.5 x 0.5 x 0.5 x 0.5 = 0.0625. You're hypothesis explains the model infinitely better than mine!
Now we can say that our new Hypothesis is that the coin is fair. Suppose another friend comes along and claims that they thought the coin had a 75% chance of getting heads and only a 25% chance of tails. We flip the coin 5 more times and get HHTTH. Your hypothesis says 0.5^5 = 0.03125, and the friend's says 0.75^3 x 0.25^2 = 0.0263... You're hypothesis explains the data only 1.2 times better than theirs. Clearly, we need more data to feel really confident in one hypothesis over the other.
If you want an even longer example, I wrote a post awhile back about "Bayesian Reasoning in the Twilight Zone" that goes into more detail (including priors)[0]
[0] https://www.countbayesie.com/blog/2016/3/16/bayesian-reasoni...
The problem of induction is real and unavoidable in the general case, but there is no such thing as "the probability that there is a real effect". Either there is a "real effect" or there is not.
It might be possible to find a "probability" that you observed a "real effect".
>The problem of induction was solved, in principle, by the Reverend Thomas Bayes in the middle of the 18th century.
The problem of induction is fundamentally unsolvable (hence, "problem"). The article just states that it was solved and never mentions it again. Is it a widely held view that induction was solved by Bayes? Does anyone know where I can read more detailed claims about how people believe Bayes solved induction?
To name just a few of the NHST/p-value flaws:
1- I'm interested in P(H1 | data) but I get P(data | Ho). Contrary to popular belief P(data|Ho) != P(Ho|data). Let alone that conclusions about P(H1|data) can be drawn.
2- it is vulnerable to wrong interpretations. * No, a 95% confidence interval (a,b) does NOT mean there is a 95% chance that Mu is in (a,b). * No, p=0.04 does not mean they is a 96% chance that H1 is true.
3- the p-value depends on the intentions of the scientist. If you end your experiment after 80 observations, as planned, your p-value is different from that of an experiment that ended unplanned after 80 observations. So the same data have different evidential power, influenced by results you did not see in experiments you did not do. This is very unsatisfactory.
4- the idea of "an effect that exists or does not exist", based on some arbitrary threshold. The reality is, in many cases, uncertainty and variation. In group A I see effects of medicine A, with lots of variation between persons. In group B I see varying effects of medicine B. Then I introduce uncertainty by drawing random samples from A and B. Let's day I used those samples to make an inference: is, on average, medicine A better than medicine B ? Matras like "there is an effect, or there isn't" are not very helpful. Statistics should be about quantifying uncertainty rather than give false yes/no statements.
5. Basing decisions and knowledge on the data only makes t vulnerable to outliers, unlucky samples and so on. And why should you NOT use information, when it's there ?
The American Statistical Association actually put out a statement this year on the issue of p-value. You can find the whole article here http://dx.doi.org/10.1080/00031305.2016.1154108 but here are the main points:
1) P-values can indicate how incompatible the data are with a specified statistical model.
2) P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone.
3) Scientific conclusions and business or policy decisions should not be based only on whether a p-value passes a specific threshold.
4) Proper inference requires full reporting and transparency.
5) A p-value, or statistical significance, does not measure the size of an effect or the importance of a result.
6) By itself, a p-value does not provide a good measure of evidence regarding a model or hypothesis.
As an additional curiosity, the group of writers was not completely unconflicted coming up those definitions and the article contains a number of supplemental articles by the individual authors to clarify/dispute some of the points made.
[edit] fixed formatting
Perhaps this is nice fur Quality Asurance in factories. Where I do repeated measurements and where I want a simple YES or NO.
But science usually asks: "what can I learn from this specific data? I don't do 100 samples and I'm not interested in being "not to far off moist of the time". I want a best estimation based on this specific sample".
NHST does not give that answer. Bayes does.
Statistics and probability are difficult subjects. They are also not intuitive subjects for many people.
Besides the publish or perish thing, I would guess many authors of these unreliable/non-replicable biomedicine papers were focused on biomedicine during grad school - while they learned statistics it was a secondary subject.
A solution seems to be more statistics training for academics doing studies or requiring a trained statistician on the team/reviewing a study prior to publication. I don't see either happening to be honest, and this problem will likely continue.