Saturday, March 9, 2013

Nervously approaching significance

We're deluged with claims that we should do this, that or the other thing because some study has a "statistically significant" result. But don't let this particular use of the word "significant" trip you up: when it's paired with "statistically", it doesn't mean it's necessarily important. Nor is it a magic number that means that something has been proven to work (or not to work).

The p-value on its own really tells you very little. It is one way of trying to tell whether the result is more or less likely to be "signal" than "noise".  If a study sample is very small, only a big difference might reach that level, while it is far easier in a bigger study.

But statistical significance is not a way to prove the "truth" of a claim or hypothesis. What's more, you don't even need the p-value, because other measures tell you everything the p-value can tell you, and more useful things besides.

This is roughly how the statistical test behind the p-value works. The test is based on the assumption that what the study is looking for is not true - but instead, that the "null hypothesis" is true. The statistical test estimates whether you would expect the result you got, or one further away from "null" than that result, if the hypothesis isn't true.

If the p-value is <0.05 (less than 5%), then the result is compatible with what you would get if the hypothesis actually is true. But it doesn't prove it is true. You can't conclude too much based on that alone. The threshold of 0.05 for statistical significance means the level for the test has been set at 95%. That is common practice, but still a bit arbitrary.

You can read more about statistical significance over here in my blog, Absolutely Maybe - and in Data Bingo! Oh no! and Does it work? here at Statistically Funny.

Always keep in mind that a statistically significant result is not necessarily significant in the sense of "important". It's "significant" only in the sense of signifying something. A sliver of a difference could reach statistical significance if a study is big enough. For example, if one group of people sleeps a tiny bit longer on average a night than another group of people, that could be statistically significant. But it wouldn't be enough for one group of people to feel more rested than the other.

This is why people will often say something was statistically significant, but clinically unimportant, or not clinically significant. Clinical significance is a value judgment, often implying a difference that would change the decision that a clinician or patient would make. Others speak of a minimal clinically important difference (MCID or MID). That can mean they are talking about the minimum difference a patient could detect - but there is a lot of confusion around these terms.

Researchers and medical journals are more likely to trumpet "statistically significant" trial results to get attention from doctors and journalists, for example. Those medical journal articles are a key part of marketing pharmaceuticals, too. Selling copies of articles to drug companies is a major part of the business of many (but not all) medical journals. 

And while I'm on the subject of medical journals, I need to declare my own relationship with one I've long admired: PLOS Medicine - an international open access journal. As well as being proud to have published there, I'm delighted to have recently joined their Editorial Board.

(This post was revised following Bruce Scott's comment below.)


  1. Yikes.

    When you say: "The probability that it is a fluke is less than 5% (0.05 or 5 out of a 100)."

    You've taken the view that the frequentist position is right.

    If you ask a Bayesian, they'd say that you can't make that statement without having an a priori estimate of the likelihood.

    You can argue that the frequentist position is the most useful to adopt. You can't argue that it is straightforwardly true or actually bears up to even the tiniest degree of scrutiny.

    Take the thought experiment:

    1) If someone tells me that their study shows that drinking a big glass of orange juice raises blood sugar (P<0.05), I'm happy to agree that the probability that this is just noise is less than 5%. (Much less, actually, and I'll wonder why the study needed to be done.)

    2) If someone tells me that their study shows that drinking a big glass of orange juice cures metastatic pancreatic cancer (P<0.05), it would be perverse to agree that the result is just noise is only 5%.

    Sorry for only commenting on the thing that bugged me. I found this site based on someone posting a link to a nice cartoon about multiple comparisons. Your batting average seems to be pretty good so far.

  2. Thanks for the comment - and glad you liked the Bonferroni one. Well, I'm not a Bayesian, it's true, but not entirely a frequentist either. I tried to make the explanation of the mathematical conception understandable, but I think I avoided saying it meant it must be true: I only spoke of probability, and linked back to another post explaining a certain proportion will always be wrong.

    The frequentist heuristic is going to blow up errors, for sure. But so will the Bayesian one, if the priors are based on a paradigm that comes unraveled. Thanks for adding the orange juice example!

    1. How about:

      For p<0.05
      "If the null hypothesis is true, then we would expect a study of this design to produce a false-positive result 5% of the time."

      It isn't as catchy or concise, but it is more true, isn't it? It might be that the attempt to make the explanation more understandable has meant that you've lost too much of the meaning. Although you might need to further define "null hypothesis" and "false positive", so the explanation gets longer and longer (and potentially off-putting to those trying to read the blog). Brevity isn't my strength.

      I don't think this explanation of p value would bother a Bayesian. (Even though they'd probably rather talk in likelihood ratios.)

      It's the "because it means that the relationship is highly unlikely to be a coincidence. The probability that it is a fluke is less than 5%" part that rankles, obviously.

      If a study conducted on 100C homeopathic treatment for influenza were reported out as positive because p<0.05, then I'd say that the relationship is extremely likely to be coincidence. (Or if we are less kind, bias or fraud.) The probability of it being a fluke is much, much greater than 5%. It approaches 100%, since the pretest probability is so vanishingly small.

    2. Thanks, Bruce: that nails more precisely what's grating. Very helpful. How about this?

      "...because it means the chance that this is a fluke is less than 5% (0.05 or 5 out of 100)."

    3. I still don't think this is true. The chance of it being a fluke could be much greater than 5% or considerably less than 5% depending on the pretest probability. In my orange juice example, it simply isn't the case that you SHOULD think that the study of orange juice as a cure for pancreatic cancer that was reported out as p<0.05 is a fluke less than 5% of the time. You would be on firm ground to believe that it was a fluke very nearly 100% of the time.

      My formulation was neutral in regards to the question of the importance of considering pretest probability. I happen to think it is very important to consider pretest probability. (So does Ioannidis, who you link to on multiple occasions.) It seems you feel it is less important than I do. So be it. It's an active debate with reasonable arguments about why one approach is superior to the other in practice. However, it IS possible to give a definition of p-value that doesn't require you to take a position on the importance of pretest probability.

      You do this by explaining it in terms of the expected false positive rate given the study design (and the assumption of no bias).

      "Given a study of this design, with an assumption of no bias, a p-value threshold of <0.05 would result in a false positive result 5% of the time."

      Doesn't this capture what we need to say about p-value?

      You can't take a position on whether or not the result is a "fluke" less than 5% of the time unless you take a position on whether or not it is important to consider pretest probability.

    4. Still haven't thought of a good way to resolve this - other than to do a Bayesian cartoon/explanation and then cross-link. I don't think there's really such a thing as "no bias", and statistical significance is used in such a variety of ways. Pretest probability is just too big an issue to tackle in a sentence. So watch this space.....

    5. OK, Bruce, I think I found a way to go about it: be great if you could let me know what you think now. I haven't gone into pretest probability because it would overload this post: I will do that in another post and then cross-link. But I've tried to be less "frequentist"/simplistic without getting too complex about it. I'd be grateful for feedback.

  3. Hello Hilda "Statistical significance is reached when a "p" value is less than 5% "
    5% is merely convention at best or habit at worst. It is entirely arbitrary. While 5% seems to be used in medicine, in others, a much lower value is used (eg in the hunt for fundamental particles like the Higgs Boson). I would encourage the use of such cut-off only in the design of trials. In the methodology report 5% (or whatever) as the alpha, but not use the term "statistical significance" for a result with p<0.05, rather just report the p value.


    1. G'day, John! Yes, it's arbitrary in some ways, although the choice was a genuine attempt to find a meaningful cut-off point. See the link included in this post: Don't's just a standard deviation

      Actually, for the reporting of trial results, the p value is discouraged by several journals. The confidence levels are more valuable. Click here for the CONSORT Statement.