6 ms·

The problem with p-values

58 points by vmuhonen 10y ago

cuchoi 10y ago
I think the article is a very good explanation of the application of Bayes theorem to p-values. 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.html 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.
- apathy 10y ago
  > If you assume power of 80% (not crazy in rigorous studies) and 50% of programs having an effect... 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...
nkurz 10y ago
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. 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.
- bbctol 10y ago
  Well, it's less of a conundrum in the case of the die because it's easier to estimate the priors. That's the power and difficulty of using Bayesian reasoning; you need some estimate of how likely something is to be true before you perform the statistical test. 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.)
  - ramidarigaz 10y ago
Homunculiheaded 10y ago
This statement is a misunderstanding of the Bayesian approach: "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-reasoning-in-the-twilight-zone https://www.countbayesie.com/blog/2016/3/16/bayesian-reasoni...
- fuckingmoron 10y ago
  Will you stop it. Bayes factors are computed as a ratio of posterior model likelihoods, a likelihood ratio, which is not so different from a t-test (which as you'll recall from stats 101 is the likelihood ratio test for two Gaussians with finite samples). The difference, then, is in using the raw ratio rather than calculating its probability, and (crucially) whether you use a uniform or something more informative to generate the prior distribution that is updated with the data to yield the posterior. Frequentist == flat, weak prior (usually a dumb idea). There is no reason you couldn't use a Beta for a prior for a p-value distribution (flat would then be Beta(1,1) aka uniform) and generate a posterior p-value probability distribution based on that (take the integral of the PDF from the posterior mode on up to 1 => p-value). Not unlike a t-test! Pierre-Simone Laplace himself (who Bayes' theorem should rightly have been named after) used the "sun rising tomorrow" example to contrast naive with subjective treatments of probability as belief. (If you're a strict frequentist, a uniform prior is called for; but everyone will laugh at you because obviously the prior is a lot more pointy than that.) http://lesswrong.com/lw/774/a_history_of_bayes_theorem/ http://lesswrong.com/lw/774/a_history_of_bayes_theorem/ is a nice treatment. All this NHST bullshit came later. The important point here is how you derive the posterior likelihood, and the "sun rising tomorrow" example (whence Colquhoun derived his) neatly makes that point. So while you could generate a "Bayesian p-value" with exactly the same tools, everyone would laugh at you for throwing away all that valuable information in the posterior (how pointy is it? how much more conclusive than the prior?) and THAT (I claim) is the real difference here.
jsprogrammer 10y ago
>The problem is that the p-value gives the right answer to the wrong question. What we really want to know is not the probability of the observations given a hypothesis about the existence of a real effect, but rather the probability that there is a real effect – that the hypothesis is true – given the observations. And that is a problem of induction. 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?
- bayeslives 10y ago
  Bayes did not "solve induction". If we define induction as telling which specific model generated this data: that is not possible. Countless models could in theory generate our data. What we need is some restriction. Like a prior. And when was the last time you started a research project without any idea what to expect? And if you did, wouldn't it be wiser to do some literature study, experts interviews etc. before starting experiments? Modeling the state of the art, pre-experiment, seems like a clever move anyway. 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 ?
qwrusz 10y ago
It's ironic the author starts off saying they need to get more rigorous in their science and statistics. Then goes on to write a relatively short and not very rigorous overview of Bayesian statistics. 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.

58 points by vmuhonen 10y ago

I think the article is a very good explanation of the application of Bayes theorem to p-values. 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.html 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.

apathy 10y ago
> If you assume power of 80% (not crazy in rigorous studies) and 50% of programs having an effect... 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...

nkurz 10y ago

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. 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.

bbctol 10y ago
Well, it's less of a conundrum in the case of the die because it's easier to estimate the priors. That's the power and difficulty of using Bayesian reasoning; you need some estimate of how likely something is to be true before you perform the statistical test. 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.)
- ramidarigaz 10y ago

Homunculiheaded 10y ago

This statement is a misunderstanding of the Bayesian approach: "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-reasoning-in-the-twilight-zone https://www.countbayesie.com/blog/2016/3/16/bayesian-reasoni...

fuckingmoron 10y ago
Will you stop it. Bayes factors are computed as a ratio of posterior model likelihoods, a likelihood ratio, which is not so different from a t-test (which as you'll recall from stats 101 is the likelihood ratio test for two Gaussians with finite samples). The difference, then, is in using the raw ratio rather than calculating its probability, and (crucially) whether you use a uniform or something more informative to generate the prior distribution that is updated with the data to yield the posterior. Frequentist == flat, weak prior (usually a dumb idea). There is no reason you couldn't use a Beta for a prior for a p-value distribution (flat would then be Beta(1,1) aka uniform) and generate a posterior p-value probability distribution based on that (take the integral of the PDF from the posterior mode on up to 1 => p-value). Not unlike a t-test! Pierre-Simone Laplace himself (who Bayes' theorem should rightly have been named after) used the "sun rising tomorrow" example to contrast naive with subjective treatments of probability as belief. (If you're a strict frequentist, a uniform prior is called for; but everyone will laugh at you because obviously the prior is a lot more pointy than that.) http://lesswrong.com/lw/774/a_history_of_bayes_theorem/ http://lesswrong.com/lw/774/a_history_of_bayes_theorem/ is a nice treatment. All this NHST bullshit came later. The important point here is how you derive the posterior likelihood, and the "sun rising tomorrow" example (whence Colquhoun derived his) neatly makes that point. So while you could generate a "Bayesian p-value" with exactly the same tools, everyone would laugh at you for throwing away all that valuable information in the posterior (how pointy is it? how much more conclusive than the prior?) and THAT (I claim) is the real difference here.

jsprogrammer 10y ago

>The problem is that the p-value gives the right answer to the wrong question. What we really want to know is not the probability of the observations given a hypothesis about the existence of a real effect, but rather the probability that there is a real effect – that the hypothesis is true – given the observations. And that is a problem of induction. 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?

bayeslives 10y ago
Bayes did not "solve induction". If we define induction as telling which specific model generated this data: that is not possible. Countless models could in theory generate our data. What we need is some restriction. Like a prior. And when was the last time you started a research project without any idea what to expect? And if you did, wouldn't it be wiser to do some literature study, experts interviews etc. before starting experiments? Modeling the state of the art, pre-experiment, seems like a clever move anyway. 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 ?

qwrusz 10y ago

It's ironic the author starts off saying they need to get more rigorous in their science and statistics. Then goes on to write a relatively short and not very rigorous overview of Bayesian statistics. 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.