A blog on statistics, methods, and open science. Understanding 20% of statistics will improve 80% of your inferences.

Thursday, May 29, 2014

The probability of p-values as a function of the statistical power of a test



I used to be really happy with any p-value smaller than .05, and very disappointed when p-values turned out to be higher than .05. Looking back, I realize I was suffering from a bi-polar p-value disorder. Nowadays, I interpret p-values more evenly. Instead of a polar division between p-values above and below the .05 significance level, I use a gradual interpretation of p-values. As a consequence, I'm no longer very convinced something is going on by p-values between .02 and .05. Let me explain.

In my previous blogpost, I explained how p-values can be calibrated to provide best-case posterior probabilities that the H0 was true. High p-values leave quite something to be desired, with a p = .05 yielding a best-case scenario with a 71% probability that H1 is true (assuming H0 and H1 are a-priori equally likely). Here, I want to move beyond best case scenario’s. Instead of only looking at p-values, we are going to look at the likelihood that a p-value represents a true effect, given the power of a statistical test.

This blog post is still based on the paper by Sellke, Bayarri, & Berger, 2001. The power of a statistical test that yields a specific p-value is determined by the size of the effect, the significance level, and the size of the sample. The more observations, and the larger the effect size, the higher the statistical power. The higher the statistical power, the higher the likelihood of observing a small (e.g., p = .01) compared to a high (e.g., p = .04) p-value, assuming there is a true effect in the population. We can see this in the figure below. The top and bottom halves of the figure display the same information, but the scale showing the percentage of expected p-values differs (from 0-100 in the top, from 0-10 in the bottom, where the percentages for p-values between .00 and .01 are cut off at .1). As the top pane illustrates, the probability of observing a p-value between 0.00 and 0.01 is more than twice as large if a test has 80% power, compared to when the test has only 50% power. In an extremely high powered experiment (e.g., 99% power) the p-value will be smaller than .01 in approximately 96% of the tests, and between 0.01 and 0.05 in only 3.5% of the tests.



In general, the higher the statistical power of a test, the less likely it is to observe relatively high p-values (e.g., p > .02). As can be seen in the lower pane in the figure, in extremely high powered statistical tests (i.e., 99% power), the probability of observing a p-value between .02 and .03 is less than 1%. If there is no real effect in the population, and the power of the statistical test is 0% (i.e., there is no chance to observe a real effect), p-values are uniformly distributed. This means that every p-value is equally likely to be observed, and thus that 1% of the p-values will fall within the .02 and .03 interval. As a consequence, when a test with extremely high statistical power returns a p = .024, this outcome is more likely when the null hypothesis is true, than when the alternative hypothesis is true (the bars for a p-value between .02 and .03 is higher when power = 0%, than when power = 99%). In other words, a statistical difference at the p < .05 level is surprising, assuming the null-hypothesis is true, but should still be interpreted as support for the null-hypothesis (we also explain this in Lakens & Evers (2014)

The fact that with increasing sample size, a result can at the same time be a statistical difference with p < .05, while also being stronger support for the null-hypothesis than for the alternative hypothesis, is known as Lindley’s paradox. This isn’t a true paradox – things just get more interesting to people if you call them a paradox. There are just two different questions that are asked. First, the probability of the data, assuming the null-hypothesis is true, or Pr(D|H0), is very low. Second, the probability of the alternative hypothesis, is lower than the probability of the null-hypothesis, given the data, or Pr(H1|D)<Pr(H0|D). Although it is often interpreted by advocates of Bayesian statistics as a demonstration of the ‘illogical status of significance testing’ (Rouder, Morey, Verhagen, Province, & Wagemakers, in press), it also an illustration of the consequences of using improper priors in Bayesian statistics (Robert, 2013).

An extension of these ideas is now more widely known in psychology as p-curve analysis (Simonsohn, Nelson, & Simmons, 2014, see www.p-curve.com). However, you can apply this logic (with care!) when subjectively evaluating single studies as well. In a well-powered study (with power = 80%) the odds of a statistical difference yielding a p-value smaller than .01 compared to a statistical difference between .01 and .05 is approximately 3 to 1. In general, the lower the p-value, the more the result supports the alternative hypothesis (but don't interpret p-values directly as support for H0 or H1, and always consider the prior probability of H0). Nevertheless, 'sensible p-values are related to weights of evidence' (Good, 1992), and the lower the p-value the better. A p-value for a true effect can be higher than .03, but it's relatively unlikely to happen a lot across  multiple studies, especially when sample sizes are large. In small sample sizes, there is a lot of variation in the data, and a relatively high percentage of higher p-values is possible (see the figure for 50% power). Remember that if studies only have 50% power, there should also be 50% non-significant findings.

The statistical reality explained above also means that in high-powered studies (e.g., with a power of .99, for example when you collect 400 participants (divided over 2 conditions in an independent t-test) and the effect size is d=.43), setting the significance level to .05 is not very logical. After all, p-values > .02 are not even more likely under the alternative hypothesis, than under the null-hypothesis. Unlike my previous blog, where subjective priors were involved, this blog post is focused on a the objective probability of observing p-values under the null hypothesis and the alternative hypothesis, as a function of power. It means that we need to stop using fixed significance levels of α = .05 for all our statistical tests, especially now that we are starting to collect larger samples. As Good (1992) remarks:
The real objection to p-values is not that they usually are utter nonsense, but rather that they can be highly misleading, especially if the value of N is not also taken into account and is large.’
How we can decide which significance level we should use, depending on our sample size, is a topic for a future blog post. With which I mean to say I haven't completely figured out how it should be done. If you have, I'd appreciate a comment below.

21 comments:

  1. Here's a naive thought. Given what we have learned from the reproducibility project that original findings with p values betw .02 and .05 replicated extremely rarely, would it be crazy to move away from .05 to a value such as .01?

    I've begun to do this myself in my own work (e.g., I recently observed an anticipated effect with a p value of .04; my next step will be to try to replicate this effect, and make some procedural changes to hopefully make it larger, before thinking about trying to publish it).

    Of course, this naive proposal assumes no p-hacking (otherwise, there would be a rash of findings just below p = .01 that would not replicate).

    ReplyDelete
    Replies
    1. Hi Tony, thanks, that's a great comment! Yes, this is directly related to why none of the original studies with p-values between .2 and .5 replicated (so far) in the Reproducibility Project. Although, absolutely none replicating is still a little unexpected, and can't just be accounted for by p-value distributions, but that's another story. Replicating high p-values is an excellent way to guarantee the robustness of your work. There have been many suggestions to lower significance levels (e.g., to .005, by Johnson, 2013), but I think it's difficult to set a single standard for all research areas. Nevertheless, we should be slightly more crictical about conclusions about the alternative hypothesis after a single p = .04 (and we can be a little (but not too much!) less critical about a single lower p-value.

      Delete
  2. in addition to reporting simply a p value, also report a p* value which indicates the likelihood of observing a test statistic of the magnitude found under the alternative hypothesis (to be specified in advance). if the study is underpowered, power will be low and hence the likelihood of observing a t value (for example) with p=.049 is not very much larger than the likelihood of observing that same t/p value pair under H1. You may have found a significant effect, but with much uncertainty as to which distribution that value comes from). p/p* will be closer to zero in a study with higher power however. if the study is adequately powered for the specified effect, then p* will exceed p to a higher degree, and p/p* will become smaller.

    if, however, the pre-specified effect is very large, then a test statistic with p = .049 - as you point out - might still be relatively more likely under the null than under the alternative hypothesis with specified huge effect size. then p/p* will become be > 1 and the result - albeit associated with a small p value and huge power - still speaks in favor of the null rather than the alternative, but with much uncertainty. hence, the p value should be much smaller (and with that, the likelihood of the test statistic occurring under the alternative hypothesis becomes larger again), so that p/p* again decreases.

    ideally p/p* should be zero (no chance of a significant finding being a false positive, but every chance to find the true effect).

    i believe that this is, however, the same idea as that behind using the bayes factor instead of simple p values that are based on the assumption that the null is true. it is also represented in classical neyman pearson testing, where you are not to only look at the p value, but also specify a target effect size beforehand and after results are in, inspect the found effect size for consistency with the target effect size you planned with (not necessarily whether it is larger or smaller, but whether it is in the vicinity of the planned effect). only if the empirical effect is approximately the size of the effect size planned with, the p value is a strong indication of the alternative being true. if the found effect is smaller than the one planned or much larger, then alpha and beta errors are out of control and either not much can be said anymore about what the study tells you (e.g., p value is very small, but p* is much smaller than p, but on the left side of the noncentral distribution - with a much larger planned effect) or you are running the risk that the found statistic - even though its likelihood / p value under the null is quite small -, the likelihood under the alternative is also small (this time on the right side of the noncentral distribution). then the beta error is also large and power quite low, so that finding something significant is arguably a lucky coincidence much more than a stable phenomenon you would bet a lot of money to find again with a similar small sample.

    ReplyDelete
    Replies
    1. i also think that the same idea was at the heart of the p rep measure that was popularized and then quickly abandonned a few years ago.

      Delete
    2. Hi Johann, is there work on this? I don't think the p-rep value did this (it was directly related to the p-value), but just out of interest, I'd like to read up on p-values for alternative hypotheses (or p*) as you describe,

      Delete
    3. ok, yes, I see, p rep is simply a function of p, so had no additional information. i misremembered because i did remember that the motivation of p rep was to indicate a sense of how likely the same found effect would be replicated if it indeed was the true effect. but that does not take into account power and simply takes the empirical effect as a face value estimate for the true effect, so it is useless for the present purpose. yes.
      but the neyman pearson, i am sure operates along these lines - i never read an original, but i was taught inferential statistics in that way by willi hager. i believe he has only german literature on it, and it's mainly books. but heres two online articles: http://www.dgps.de/fachgruppen/methoden/mpr-online/issue9/art1/hager.pdf and http://www.dgps.de/fachgruppen/methoden/mpr-online/issue11/art2/article.html.

      Delete
  3. This idea is a dead-end. The power calculations depend on the effect size which is unknown. One approach is to use the post-hoc estimate of the effect size. In general this estimator over-estimates the magnitude of the effect size. So we need some kind of correction. In bayesian stats this is handled by a prior which locates most probability mass around zero. (Another approach is to use hierarchical prior.) Another option is to derive the effect size (distribution) from the literature. Or as an third option you derive the effect size distribution based on some domain/topic-specific theoretic considerations. In any case all of this has been attempted and is routinely done in bayesian literature, when researchers try to justify their priors. The question then is, shouldn't we use the bayesian approach right on, instead of attempting to patch p-values so that they emulate bayesian inference?

    ReplyDelete
    Replies
    1. Hi Matus, is there any work on a direct comparison between post-hoc estimates of effect sizes vs. priors? Which are easier to get right, and have less bias? That would be an interesting question. I'm very pragmatic when it comes to statistics - perhaps people should use Bayesian inference (if they can formulate a sensible alternative hypothesis, which is not always the case), but they don't. Many excellent researchers are doing their best to convince people to use Bayesian inferences. I'm more interested in how to ammend the worst problems in the use of p-values. The idea to bring p-values and Bayesian inference closer together is not a dead-end: it's a very practical and logical solution for some of the problems in the way people use statistics. The question is whether people would be more likely to adjust their significance level as a function of sample size, than that they will switch to Bayesian inferences. I think that's likely. Even if they don't, people need to understand and evaluate p-values in the literature, so this blog post seemed wortwhile.

      Delete
    2. It's true that this is a roundabout way of approximating Bayesian model comparison through p-value. However, I think this post does a good job of illustrating the relationship between p-value and Bayes factors for those who are more comfortable with the frequentist framework. It shows how easy it is to set a prior on an effect size by making the judgment as to whether you're powered at 50% or 99%.

      This post also does a good job of pointing out that p=.04 at large N is really not very informative at all! Whether you assume low power or high power, you're never much more than 2 : 1 odds one way or the other. Felix Schonbrodt makes a similar point with his Shiny app here: http://www.nicebread.de/interactive-exploration-of-a-priors-impact/

      Always glad to read your stuff, Daniel.

      Delete
    3. Hi Daniel,

      I'm not sure how one would test the performance of the various methods. Quality of the post-hoc estimate depends on what kind of estimator you use. I didn't exress myself precisely. When I wrote "post-hoc estimate of the effect size" I meant the sample average which is biased. (This follows from basic sampling theory.) Then you can consider the performance of the various frequentist estimators and the various priors. Ultimately, some priors/estimator will perform better under certain conditions. Unfortunately, when working with real data it is difficult to say precisely which conditions are given and which prior/estimator should be used. In this respect all data analysis is subjective. Bayesians would probably argue that bayesian analysis manages to make at least this subjective element explicit.

      As to what method are psych researchers mostly likely to accept and report in their paper this is easy to say. Three conditions need to be fullfilled: 1) the researchers need to be told by editors to use the method. 2) the method is implemented in SPSS 3) the method does not require any thinking - it should have the usual SPSS workflow - load data, select a method from the menu, click through the method's wizard to obtain the output and copy some values from the output into the manuscript. It should be clear that this has nothing to do with statistics or data analysis and hence I tend to ignore researcher's taste - at least when we talk about psychologists who lack the math & comp sci background that is required to understand and appreciate the methods they use.

      This kind of work - balancing between the mathematical requirements and the intuitive appeal and between the frequentist and bayesian approach, has a long history in psych research but in certain sense it always leads to more trouble. If youre familiar with gigerenzer's 89 paper on ego, super-ego and id in statistics - your attempt looks to me like another chapter in this psychoanalytic saga.

      Delete
    4. sorry, that's this one:

      Gigerenzer, G. (1993). The superego, the ego, and the id in statistical reasoning. A handbook for data analysis in the behavioral sciences: Methodological issues, 311-339.

      Delete
    5. Hi, thanks for the reference - I'll read it. As I said - my main focus is making things a little better. Although I agree with most what you said, but I'm a very pragmatic guy. See this post on my old blog for an criticism for statisticians who are not pragmatic: https://sites.google.com/site/lakens2/blog/thelimitedpracticalsignificanceofstatisticians

      Delete
    6. Matus, I've read it, but I don't see the relevance to my blog post. My blog post is work in progress to fix some of the issues described in the Gigerenzer paper (and in many papers before and after). If you can be more specific about what you mean, let me know, but I think you're wrong.

      Delete
    7. Gigerenzer argues that bayesian stats is more intuitive and therefore appealing while frequentist stats represents the objective scientific standard. From the conflict of these two arises the hybrid approach. Your attempt is a hybrid approach - it attempts to combine the intuitive appeal of bayes with objectivity of frequentists.
      Why I think this won't work? I have no proof. But as Gigerenzer's work illustrates it failed to work many times before.

      I'm positive towards work that attempts to bridge bayes and frequentis stats e.g. berger's likelihood principle or jaynes/gelman's objective bayes. But such work needs to build from first principles. Lot of work however offers only ad-hoc patches to some accute problems. This is the hybrid approach. These hacks are only local and lead to inconsistencies and work for only for some data/exp designs. I perceive your proposal as such hack. I prefer to stick with orthodox p-values or with bayes, or some intermediate approach that builds from a sound fundament.

      Delete
    8. Regarding the pragmatism: I agree that this is partly a problem of stats people - some of whom have rare experience with actual psychological data. But partly this is because the problem with psychology lies with inapproriate use of hypothesis testing and poor experiment design. If your experiment and data are rubish no amount of statistics will save them. Then it is of no surprise that statisticians dont want to help you with your data.

      As an example consider the latest example of psychologists "chasing noise" on the Gelman channel: http://tinyurl.com/lca27s2

      Delete
    9. Hi Matus, your example nicely illustrates how unimportant statistics is, in the life of a researcher (and rightly so). Researchers should first spend (years of) their time understanding the literature, and make theoretically guided predicitions. Then, they need to learn how to create questionnaires, program experiments, learn to collect physiological data, and other skills necessary to collect data. Only then do they need to have sufficient understanding of statistics to draw some reasonable inference, while understanding that at the very best their single study is imput for a future meta-analysis.

      Given this situation, researchers are mainly interesting in statistics as an applied science. I really don't care about 'first principles' - I want a decent hammer to hit a nail. It should have a good handle and get the job done. Sometimes if I don't have time, I'll use another tool to hammer in a nail. At the end of the day, as long as my paintings don't fall down, I'm happy.

      If you want to be a theoretical statistician, then you don't need to worry about researchers. If you want your work to be applied, you need to understand how researchers work. I read some of your blogs, and I will never understand them in my life. I could, if I had the time, but I don't have the time, and the returns are too slim to warrant the time investment. My solutions take 3 minutes to understand, and make the tool people are already using better. If you think you can do better, give it a try, and let history decide who had a bigger influence.

      Delete
    10. To be clear, I hope you will have a huge influence on how people do statistics. And I don't think the 1000 hits on my blog will make such of a difference. I'm just trying to explain what I think is important to consider if you want to teach people better statistics.

      Delete
    11. The "first principles" is the stuff that allows us to predict whether the painting will still hang tomorrow. That's why I consider them essential.

      The material on my blog is very raw in terms of time invested into revision (and a bit too python-heavy), so I don't blame you if it didn't catch you interest :) Though it often features re-analysis of real data-set so I would consider it applied. The blog is useful for me in putting down the ideas that I may wish to organize, simplify and publish at some later point. That's also my motivation for engaging in discussions - it helps to formulate ideas and see what my position is and how consistent it is. I don't think I will convince anyone, nor that many people read this.

      Delete
  4. Hi,
    Thanx fort this great post. Could you provide any R script to reproduce those figures?

    ReplyDelete
    Replies
    1. This was not created in R - you can do the calculations yourself here: http://rpsychologist.com/d3/pdist/

      Delete
  5. The blog contain valuable information thanks for sharing it
    Selenium Training in Chennai

    ReplyDelete