Sifting the evidence --- what's wrong with significance tests?
physicaltherapyjournal.net
physicaltherapyjournal.net
If a study, significant at p=0.05, is replicated again at p=0.05, then the odds of that happening when there actually is no effect (assuming independence) are the product of the two (0.0025). Note: there are more rigorous ways to perform a meta-analysis than this...
I think p=0.05 is a good balance for noisy data. It's enough to publish, and publishing is frequently enough for others to replicate it. That's good science.
The conclusion shouldn't be that "we need to use a stricter test of significance (i.e. p<0.001)" Instead, I think it means the field is ripe for replicated studies and meta-analyses.
It can be hard on people's reputations when something is falsified, but let's face it: falsifiability is one of the best things going for science. It looks bad in the media, but it's really healthy for the field.
EDIT: I do want to mention that this paper is a really thoughtful analysis...
I think that must be approximately right, but I'm not sure it is precisely right. Imagine you do a very large number of studies and suppose the null hypothesis is in fact true. P will always be less than 1, maybe slightly less than one - or suppose that it is 1 occasionally. This is because P varies from 0 to 1. Now go ahead and multiply these Ps, and if you've done enough studies, the product is going to be very small even if all the individual Ps are close to 1 and even though the null hypothesis is true.
So I don't think a simple multiplication is exactly the right formula, though it does seem as though it must be about right.
I googled the question and found a formula for combining P of two independent studies, but it's not simple multiplication. You start with the Ps, find the corresponding Zs (I do not know what those are), then add them and divide by sqrt(2). This is your new Z, and then you take the corresponding P. Also, it requires that the Ps must be one-tailed, so it is not a fully general formula. I do not understand the Zs but my point is that if it were simply a matter of multiplying the Ps then why go to the trouble of adding Zs? I found it here:
http://books.google.com/books?id=nxOFMQYMIlgC&lpg=PA527&...
As for why it doesn't match exactly the familiar formula for combining independent probabilities (i.e. you simply multiply them), I think the answer lies in the nature of P. P is not really "the probability of that result" but "the probability of a result that is at least that extreme", and this subtly different meaning results in a different way in which the individual Ps must be combined.
As for the definition of P, that's something they address in the original article, and you are in agreement with the authors: P is _definitely_ not "the probability of that result." In its interpretation, P is the chance that you accept the null hypothesis when it is false (i.e. there is no effect, but your data randomly showed an effect).
For two-sided tests there's a small table at http://www.bmj.com/collections/statsbk/apptaba.dtl I submitted that booklet - which is a really good intro to statistics - at http://news.ycombinator.com/item?id=845275
So, if you look at the entire scientific landscape, let's pretend there is an experiment done where p=.05. The reality is that there is no effect, but in fact 20 of these experiments are done around the global at roughly the same time. At p=.05, we would expect 1 of those 20 experiments to support the null hypothesis /by random chance/.
So the author here says that the problem is that all 20 of those experiments don't get published, but instead only the 1 that happened to show an effect. That's misleading because it makes the result seem significant, when in fact it's just an artifact.
So then you say, but wait! It's okay, because now we can try to replicate those results so the bogus result from before will be found out, and we can all rest easy knowing that the brave men and women of statistics are vigilant.
But here's the problem: this one reported result leads to a wave of experiments trying to replicate the result, and according to the author there's a strong bias against publishing a paper that doesn't support the result. So let's say the finding was SUPER EXCITING (OMG), and 100 teams went out to replicate it. 5 of them should support the original result just by coincidence, even though they are all wrong.
Who publishes? Those 5! And maybe a couple others. So now, even though the work has been done (but not published) that pretty clearly demonstrates that there is no effect, we have in public view:
1 Study that shows an effect to p=.05 5 studies that support the findings of the first also to p=.05 2 studies that reject the original null hypothesis
6 studies supporting. 2 against. Ah, interesting. It's a small controversy, but "clearly" there's something here worth studying. Right?
Not so much. What about the other 112 studies?
This is a major discussion now in some fields, and there is a small movement to publish experiments that showed no effects.
It should be a bigger movement for at least two reasons:
- this will make meta-analyses stronger
- in the field I'm closest to, some scientists don't know their statistics well enough to detect an effect even if it's there. Publishing it will give someone else a chance to make something out of that data.
>* Confidence intervals for the main results should always be included, but 90% rather than 95% levels should be used
>* When there is a meaningful null hypothesis, the strength of evidence against it should be indexed by the P value. The smaller the P value, the stronger is the evidence
This sounded to me like "You shouldn't use P-values to decide, except when you do." (For non-mathy types, P values is 1-confidence, so they measure the same quantity).
I think some of this stems from their strong voiced preference for Bayesian statistics (there's something of a civil war among statisticians between bayesian and classical statistics). I've never been able to swallow the Bayesian theories; they're just too weird for my feeble mind to grasp.