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This is the insight that made statistics "click" for me many years ago: a statistical test answers one central question: what are the odds that the results you observed could have arisen by chance? If those odds are low, then you are justified in concluding that the results probably did not arise by chance, and so there must be some other explanation (usually, but not always, the causal hypothesis you are advancing).

Once consequence of this is that it is crucial that you advance your hypothesis before you collect (or at least look at) the data because the odds of something arising by chance change depending on whether you predict or postdict the results. Also, the more data you have, the more likely you are to find something in there that looks like a signal but is in fact just a coincidence. Many a day-trading fortune has been lost to this one mistake.

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Unfortunately no. Very much no, even though it's widely believed that that is a good definition/intuition (and used in many places).

It's the odds of having that results due to chance, if the null hypothesis is true[0]. That latter part might sound pedantic, but the whole point is that we don't know how likely the null hypothesis is. If I test wheather the sun has just died[1] and get a p-value of 0.01 it's still very likely that this result is due to change (surely more than 1%)! We need a prior probability (i.e. bayesian statistics) to calculate the probability that the result was due to chance, that is why that partial definition is incomplete and actually very misleading. This point is subtle, but very important to really understand p-values.

Another way to look at it is: if we knew the probability that the result was due to chance we could also just take 1-p and have to probability of there actually being some effect, a probability that hypothesis testing cannot give us.

There is one nice property that hypothesis testing does have (and why presumably it's so widely used): if the idea you are testing is wrong (which actually means "null hypothesis true") you will most likely (1-p) not find any positive results. This is good, this means that if the sun in fact did not die, and use 0.01 as your threshold, 99% of the experiments will conclude that there is no reason to believe the sun has died. So hypothesis testing does limit the number of false positive findings. The xkcd comic is a bit misleading it this regard, yes it does highlight the limitations of frequentist hypothesis testing, but the scenario depicted is a very unlikely one, in 99% of the cases there would have been a boring and reasonable "No, the sun hasn't died".

For an incredibly interesting article about the difficulty of concluding anything definitive from scientific results I highly recommend "The Control Group is out of Control" at slatestarcodex[2].

[0] To be even more pedantic you would have to add "equal or more extreme", and "under a given model", but "if the null hypothesis is true" is by far the most important piece often missing.

[1] https://xkcd.com/1132/

[2] http://slatestarcodex.com/2014/04/28/the-control-group-is-ou...

> It's the odds of having that results due to chance, if the null hypothesis is true[0].

Yes, that's right. I don't know why you think this is at odds with what I said. In fact, I clarified this myself a few hours ago in a sibling comment:

https://news.ycombinator.com/item?id=13026907

"what are the odds that the results you observed could have arisen by chance?"

If you say it like this it will very easily be misinterpreted. Once your results are in there are two cases: (1) either the null hypothesis is true and you got those results due to chance, or (2) the null hypothesis is false and there was some actual effect outside of the null hypothesis that helped you get the results.

Due to this it is very easy to interpret you statement as referring to the probability of (1).

Two two following definitions of p-values sound similar but are not:

[Correct] The probability of getting the results by chance if the null hypothesis is true P(Results|H0)

[Wrong] The probability that you got the results by chance and thus the null hypothesis was actually true P(H0|Results)

I'm not saying you didn't get it, but somebody reading what you wrote can very easily be fooled. And there are a lot of dead wrong definitions on the web[0][1][2][3].

[0] https://www.americannursetoday.com/the-p-value-what-it-reall...

[1] https://practice.sph.umich.edu/micphp/epicentral/p_value.php

[2] http://natajournals.org/doi/full/10.4085/1062-6050-51.1.04

[3] http://www.cdc.gov/des/consumers/research/understanding_scie...

The OP is referring to Fisher's p-value rather than the more common Neyman-Pierson method that you refer to. Fisher's method doesn't have the concept of the null hypothesis. The difference is fascinating and I do believe the Fisherian method is superior if you can't easily replicate.

The definition of p-value is the same independent of method, as far as I can tell the only real difference is that by Neyman–Pearson you just look at whether the p-value is below a threshold, and Fisher looks at p-value as "strength of evidence" valuable in itself. It's still not the probability that your result was due to chance, it's the probability that under the null hypothesis (and you will definitely need one) you would get that value (or more extreme) by chance.

> a statistical test answers one central question: what are the odds that the results you observed could have arisen by chance

Well, no, that'd be very interesting but unfortunately what a statistical test really says is the probability of the results you observed (or more extreme) given chance. P(data|model) and not P(model|data).

> what are the odds that the results you observed could have arisen by chance?

> the probability of the results you observed (or more extreme) given chance

I think he said the same thing as you. He didn't talk about "the odds that the results you observed have arisen by chance" but the odds that the results you have observed (the or more extreme is implicit) could have arisen by chance (i.e. the probability of happening by chance alone, given that only chance was involved).

It's actually P(data|null-hypothesis).

How about you guys are both right, sort of.

There are both Bayesian and Frequentist approaches in statistics!

They represent very different methods to statistics, but they are also quite similar. My apologies, I couldn't find one link that gave a good description of Bayesian vs. Frequentist. Here are a couple links to get started:

https://xkcd.com/1132/

http://jakevdp.github.io/blog/2014/03/11/frequentism-and-bay...

If anyone comes across a link that describes Bayesian and Frequentist clearly, please share it if you don't mind.

There are some cases when the difference between bayesian and frequentist statistics is just a matter of interpretation, but not in this case: the mathematics for a bayesian statistical test are different from the mathematics of a frequentist statistical test and they will produce different results, because the former accounts for prior probabilities and the latter does not. The article talks about frequentist tests, as does almost everyone else who mentions statistical tests without further specification.

Frequentists count their chickens after they've hatched. Bayesians bet the chickens that they will.

That cartoon is actually a pretty good illustration of why frequentists are wrong.

Do you mean probably wrong?

Either way I disagree. Both approaches have pros and cons depending on the type of analysis.

At the very least, just the presence of competing approaches in the field has pushed statisticians to have more rigour and do way more double checking than they might have, out of fear the other side actively looking to poke holes. It's easy to lie with statistics and the only people who can call statisticians out on their BS is other statisticians...Need more of this.

> Do you mean probably wrong?

Nope.

> Both approaches have pros and cons

What are the pros of the frequentist approach?

You can justify that you did everything "by the book" and you don't have to make up priors.

Now. You might read this as me saying that Bayesian statistics is all made up, and that's not what I'm saying. I'm saying that if you go Bayesian all the way, and your result is even remotely controversial, someone could easily challenge you by saying "this result would have been different with different priors, and why won't they just come out and say p < 0.01? What are they hiding?"

When you do frequentist statistics, the equivalent of priors were, instead, made up for you by long scientific tradition. They're not very good priors, as the relevant xkcd illustrates. But at least it's not you making them up.

In a more complex model, there's also the thing where you can't calculate the result exactly, you have to approximate it with Markov Chain Monte Carlo or something, leading to another way to doubt your results (did the MCMC converge correctly?)

So instead you do frequentist statistics. You use your favorite stats package and it spits out a nice comforting p-value that will satisfy the reviewers. You have tons of guidance about how to do things. It's not a great thing, but it is a definite advantage of using frequentist statistics.

In other words: the advantage of frequentism is that it gives you better plausible deniability when you get the wrong answer. Did I get that right?

No, and I don't know where you're getting your certainty that the answer is wrong from.

How about this: find me a published scientific paper with only Bayesian results in it, no frequentist statistics at all. Argue for its superiority all you want, but I don't think anyone does it. It would be seen as a stunt.

Don't know about a paper, but here's a book:

https://www.amazon.com/Proving-History-Bayess-Theorem-Histor...

> find me a published scientific paper with only Bayesian results in it

Trivial¹, surely?

> [B]ut I don't think anyone does it. It would be seen as a stunt.

What did you mean by "scientific"?

You'd have to hunt for a narrow reading s.t. the above holds — extant counterexamples aren't limited to any one branch.

Check it out² for yourself.

Further: cogsci's bayesian adoption is rapidly accelerating.

The replication crisis is brutalizing huge swathes of psych:

• loss of confidence in NHST is becoming near-total for many

• journals are purging in turn — e.g. BASP's p-value ban³⁴

• others are overhauling stats-in-psych entirely

J Math Psych alone has two recent special issues⁵⁶ on this.

I mean:

> no frequentist stats at all

isn't even a strawman lately, let alone an absurdity.

In some fields, it's a battle-cry.

All that being said… re:

> Argue for its superiority all you want

I wouldn't even go that far. Don't give em that.

Probability interpretation fundies

• are all wrong,

• narrow minds and waste lifespans with the cultism, and

• should at least learn of the other interpretations.

Pitching probability as a 1v1 isn't merely wrong-prime⁷ — it's doublepluswrong″⁸.

It's pseudofundamentalism: fundies uphold foundations.

Flamewars predicated on ignorance of the same gotta go, no matter how fashionable they may be.

____________________________________________________________

[1] http://enwp.org/bayesian_game

[2] http://google.com/scholar?q=fully+bayesian

[3] http://doi.org/4z8 | BASP 37

[4] http://doi.org/34p | Nature re: BASP

[5] http://doi.org/btqx | J Math Psych 72

[6] http://doi.org/btqz | J Math Psych 74

[7] http://enwp.org/all_models_are_wrong

[8] http://enwp.org/not_even_wrong

The main advantage of the frequentist approach is that you can do the calculations much more easily. Bayesian statistics is great, but often the calculations are much more difficult because of your distribution of priors. You can make up simplified priors to ease the calculations, but then you run into some of the same problems as frequentist statistics.

Here is a simple example: lets say you flip a coin 10 times and get 8 heads, what is the probability that the coin is not fair? In frequentist statistics you only have to calculate the likelihood of the results for a null hypothesis, and then you use a p-value. While this approach is flawed, at least you can quickly do the calculation and get an approximate answer. In Bayesian statistics you have to specify the prior distribution, and calculate the likelihood of your results under every possible hypothesis. Correctly specifying this prior distribution and calculating the results is quite challenging - especially if you want to use a realistic prior (not just uniform). This is a pretty simple example, you can imagine how much more challenging this becomes in real-world problems. On the other hand, it is true that the frequentist approach doesn't really answer the question asked, so it is misleading (especially if you choose a p-value that isn't specific to the problem). If you choose p-values based on prior knowledge, than the differences between frequentist and bayesian are less extreme.

A man was walking down a city street when he saw another man wandering around a lamp post looking at the sidewalk. "What are you doing?" the first man asked. "Looking for my keys," said the second man. "Oh, did you lose them around here?" asked the first man. "No," the second man replied, "but the light is better here."

Often times getting close to an answer that is approximately correct is better than trying to find the perfect solution. Most real world problems, especially in analyzing data, don't have perfect answers. For example approximations are made all the time in physics, because without these approximations the calculations can't be done. Knowing when to make approximations and what approximations to make is an essential skill for analyzing data.

> Bayesian statistics is great, but often the calculations are much more difficult because of your distribution of priors.

This is often a feature, forcing you to actually look at the complexity head on before you sweep it under a rug.

Simple maths that can be done without a computer. Useful in the past - not so useful today.

All models are wrong. That much is entailed by 'models'.

There is no Zuul.

The model is the null hypothesis here... I don't think the parent comment was wrong w.r.t. this.

No, there's a difference. The model is something like, "This drug attaches itself selectively to cancer cells and kills them." The null hypothesis is, "This drug has no effect." So you conduct a double-blind study, measure the effect of the drug on cancer cells, collect some data and compute that P(data|null-hypothesis) is 1%. It it not the case that there is a 99% chance that your model is correct and that the drug does in fact attach itself to cancer cells and kill them. Because there are other possible models, e.g.: the drug redirects your Chi and channels it into your chakras which kills the cancer cells. Statistics alone cannot tell you which of those two models is correct.

>"Statistics alone cannot tell you which of those two models is correct."

Statistics can tell you whether a model is consistent with the data. But you need to deduce the null hypothesis from your model rather than use the default "no difference" (of course, sometimes no difference is deduced from a real model, but not often, in that case: great!).

In fact, that is the proper use of statistics. I would guess >99.99% of current usage is incorrect (ie pseudoscience) and amounts to a waste of time at best. The usual usage turns scientific reasoning on its head, and has lead to a (literally for most people) unbelievable amount of trouble.

This was pointed out most aptly by Paul Meehl long, long ago: http://www.fisme.science.uu.nl/staff/christianb/downloads/me...

> Statistics can tell you whether a model is consistent with the data.

Yes, that's true, but it badly misses the point. The power of statistics is to tell you when a model (the null hypothesis) is (most likely) inconsistent with the data so that you can confidently rule it out. Any finite data set is consistent with an infinite number of models, so knowing that a model and the data are consistent tells you absolutely nothing about whether or not that model has any relationship with reality (which, at the risk of stating the obvious, is what science actually cares about). This is the reason that rejecting the null hypothesis is considered a positive result.

Except such a data set is also inconsistent with an infinite number of models, so ruling one out via rejecting a null also provides practically no information value and moves us no closer to understanding. /devils advocate

In practical terms, we're not interested in true models, but useful ones, so the description of a model's consistency with observed data is often the more useful metric in practice than rejecting nulls :/ especially in applications where you can't set up repeated experiments.

OK, I realise its more nuanced than that too, but given how many papers and practitioners seem incapable of understanding that evidence against the null it's not explicit evidence for an arbitrary alternative, practically and consequentially I don't think that's how we should be working...

> Except such a data set is also inconsistent with an infinite number of models, so ruling one out via rejecting a null also provides practically no information value and moves us no closer to understanding.

No, that's not true, because experiments are not done in a (figurative) vacuum. They are done in the context of an explanatory theory that has already gone through a vigorous filter and shown to be consistent with the all prior experimental data and has better explanatory power than all of its competitors. It is only when more than one theory survives this filter that an experiment is done, and the experiment is designed specifically to distinguish between the surviving theories.

So while it is true that an experiment allows you to eliminate an infinite number of theories, it's irrelevant, because by the time the experiment is done nearly all of those theories have already been eliminated anyway.

>"The power of statistics is to tell you when a model (the null hypothesis) is (most likely) inconsistent with the data so that you can confidently rule it out"

If you know whether your model is consistent with the data, you know whether it is inconsistent... I think you are talking about some other issue than I am.

The point about deducing the null hypothesis from your explanatory model is that the null hypothesis is precise. In that case you will get a strong test of the model, and it will get stronger as more data gets collected. Using a default null and and vague alternative is the exact opposite. (check the Meehl 1967 paper I linked earlier in this thread).

> If you know whether your model is consistent with the data, you know whether it is inconsistent...

No, that's not true. And in fact I got it wrong earlier when I agreed with you that statistics can tell you when data is consistent with a model. They can't. At best they can tell you whether the data are not inconsistent. That sounds like the same thing, but it isn't. It's like the distinction between "not guilty" and "actually innocent." At best (or at worst depending on how you look at it) a statistical test can tell you, "This theory cannot be confidently ruled out on the basis of that data under the following assumptions..."

> I think you are talking about some other issue than I am.

That is quite possible.

[UPDATE:] BTW, I read the Meehl paper, and I completely agree with what he says. So you and I may be in "violent agreement" here.

I think this is just an argument over semantics, the "null hypothesis" is a perfectly valid model using the definition of the statistics community [0] (i.e. a collection of probability distributions over some sample space).

[0] https://en.wikipedia.org/wiki/Statistical_model

The word "model" is being used here in two mutually incompatible ways. Yes, the null hypothesis is a model, but it is not an explanatory model. A scientific hypothesis has to meet two tests to be considered a valid theory. It has to be consistent with the data, and it has to have explanatory power. The theory that cancer drugs work by aligning a patient's chi with their chakras is rejected not because it is inconsistent with the data (it's not) but because it lacks the explanatory power of alternative theories based on molecular biology.

The null hypothesis never has explanatory power. The null hypothesis is always a statement of the form, "The explanatory hypothesis under test is wrong for some unknown reason." This is why rejecting the null hypothesis, i.e. showing that the data are (with high probability) inconsistent with the null hypothesis, is considered a positive result.

Yup, and the null hypothesis is a reduced model. Different words, same thing.

I have no idea what you mean by "reduced" but they are not the same thing. The difference in their explanatory power. A (scientific) hypothesis explains things. A null hypothesis simply says, "This hypothesis is wrong" but doesn't say why. It's a crucial distinction.

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