Category Archives: statistical theory


My favorite Neyman passage: on confidence intervals

I’ve been doing a lot of reading on confidence interval theory. Some of the reading is more interesting than others. There is one passage from Neyman’s (1952) book “Lectures and Conferences on Mathematical Statistics and Probability” (available here) that stands above the rest in terms of clarity, style, and humor. I had not read this before the last draft of our confidence interval paper, but for those of you who have read it, you’ll recognize that this is the style I was going for. Maybe you have to be Jerzy Neyman to get away with it.

Neyman gets bonus points for the footnote suggesting the “eminent”, “elderly” boss is so obtuse (a reference to Fisher?) and that the young frequentists should be “remind[ed] of the glory” of being burned at the stake. This is just absolutely fantastic writing. I hope you enjoy it as much as I did.

[begin excerpt, p. 211-215]

[Neyman is discussing using “sampling experiments” (Monte Carlo experiments with tables of random numbers) in order to gain insight into confidence intervals. \(\theta\) is a true parameter of a probability distribution to be estimated.]

The sampling experiments are more easily performed than described in
detail. Therefore, let us make a start with \(\theta_1 = 1\), \(\theta_2 = 2\), \(\theta_3 = 3\) and \(\theta_4 = 4\). We imagine that, perhaps within a week, a practical statistician is faced four times with the problem of estimating \(\theta\), each time from twelve observations, and that the true values of \(\theta\) are as above [ie, \(\theta_1,\ldots,\theta_4\)] although the statistician does not know this. We imagine further that the statistician is an elderly gentleman, greatly attached to the arithmetic mean and that he wishes to use formulae (22). However, the statistician has a young assistant who may have read (and understood) modern literature and prefers formulae (21). Thus, for each of the four instances, we shall give two confidence intervals for \(\theta\), one computed by the elderly Boss, the other by his young Assistant.

[Formula 21 and 22 are simply different 95% confidence procedures. Formula 21 is has better frequentist properties; Formula 22 is inferior, but the Boss likes it because it is intuitive to him.]

Using the first column on the first page of Tippett’s tables of random
numbers and performing the indicated multiplications, we obtain the following
four sets of figures.

The last two lines give the assertions regarding the true value of \(\theta\) made by the Boss and by the Assistant, respectively. The purpose of the sampling experiment is to verify the theoretical result that the long run relative frequency of cases in which these assertions will be correct is, approximately, equal to \(\alpha = .95\).

You will notice that in three out of the four cases considered, both assertions (the Boss’ and the Assistant’s) regarding the true value of \(\theta\) are correct and that in the last case both assertions are wrong. In fact, in this last case the true \(\theta\) is 4 while the Boss asserts that it is between 2.026 and 3.993 and the Assistant asserts that it is between 2.996 and 3.846. Although the probability of success in estimating \(\theta\) has been fixed at \(\alpha = .95\), the failure on the fourth trial need not discourage us. In reality, a set of four trials is plainly too short to serve for an estimate of a long run relative frequency. Furthermore, a simple calculation shows that the probability of at least one failure in the course of four independent trials is equal to .1855. Therefore, a group of four consecutive samples like the above, with at least one wrong estimate of \(\theta\), may be expected one time in six or even somewhat oftener. The situation is, more or less, similar to betting on a particular side of a die and seeing it win. However, if you continue the sampling  experiment and count the cases in which the assertion regarding the true value of \(\theta\), made by either method, is correct, you will find that the relative frequency of such cases converges gradually to its theoretical value, \(\alpha= .95\).

Let us put this into more precise terms. Suppose you decide on a number \(N\) of samples which you will take and use for estimating the true value of \(\theta\). The true values of the parameter \(\theta\) may be the same in all \(N\) cases or they may vary from one case to another. This is absolutely immaterial as far as the relative frequency of successes in estimation is concerned. In each case the probability that your assertion will be correct is exactly equal to \(\alpha = .95\). Since the samples are taken in a manner insuring independence (this, of course, depends on the goodness of the table of random numbers used), the total number \(Z(N)\) of successes in estimating \(\theta\) is the familiar binomial variable with expectation equal to \(N\alpha\) and with variance equal to \(N\alpha(1 – \alpha)\). Thus, if \(N = 100\), \(\alpha = .95\), it is rather improbable that the relative frequency \(Z(N)/N\) of successes in estimating \(\alpha\) will differ from \(\alpha\) by more than

2\sqrt{\frac{\alpha(1-\alpha)}{N}} = .042


This is the exact meaning of the colloquial description that the long run relative frequency of successes in estimating \(\theta\) is equal to the preassigned \(\alpha\). Your knowledge of the theory of confidence intervals will not be influenced by the sampling experiment described, nor will the experiment prove anything. However, if you perform it, you will get an intuitive feeling of the machinery behind the method which is an excellent complement to the understanding of the theory. This is like learning to drive an automobile: gaining experience by actually driving a car compared with learning the theory by reading a book about driving.

Among other things, the sampling experiment will attract attention to
the frequent difference in the precision of estimating \(\theta\) by means of the two alternative confidence intervals (21) and (22). You will notice, in fact, that the confidence intervals based on \(X\), the greatest observation in the sample, are frequently shorter than those based on the arithmetic mean \(\bar{X}\). If we continue to discuss the sampling experiment in terms of cooperation between the eminent elderly statistician and his young assistant, we shall have occasion to visualize quite amusing scenes of indignation on the one hand and of despair before the impenetrable wall of stiffness of mind and routine of thought on the other.[See footnote] For example, one can imagine the conversation between the two men in connection with the first and third samples reproduced above. You will notice that in both cases the confidence interval of the Assistant is not only shorter than that of the Boss but is completely included in it. Thus, as a result of observing the first sample, the Assistant asserts that

.956 \leq \theta \leq 1.227.

On the other hand, the assertion of the Boss is far more conservative and admits the possibility that \(\theta\) may be as small as .688 and as large as 1.355. And both assertions correspond to the same confidence coefficient, \(\alpha = .95\)! I can just see the face of my eminent colleague redden with indignation and hear the following colloquy.

Boss: “Now, how can this be true? I am to assert that \(\theta\) is between .688 and 1.355 and you tell me that the probability of my being correct is .95. At the same time, you assert that \(\theta\) is between .956 and 1.227 and claim the same probability of success in estimation. We both admit the possibility that \(\theta\) may be some number between .688 and .956 or between 1.227 and 1.355. Thus, the probability of \(\theta\) falling within these intervals is certainly greater than zero. In these circumstances, you have to be a nit-wit to believe that
P\{.688 \leq \theta \leq 1.355\} &=& P\{.688 \leq \theta < .956\} + P\{.956 \leq \theta \leq 1.227\}\\
&& + P\{1.227 \leq \theta \leq 1.355\}\\
&=& P\{.956 \leq \theta \leq 1.227\}.\mbox{”}

Assistant: “But, Sir, the theory of confidence intervals does not assert anything about the probability that the unknown parameter \(\theta\) will fall within any specified limits. What it does assert is that the probability of success in estimation using either of the two formulae (21) or (22) is equal to \(\alpha\).”

Boss: “Stuff and nonsense! I use one of the blessed pair of formulae and come up with the assertion that \(.688 \leq \theta \leq 1.355\). This assertion is a success only if \(\theta\) falls within the limits indicated. Hence, the probability of success is equal to the probability of \(\theta\) falling within these limits —.”

Assistant: “No, Sir, it is not. The probability you describe is the a posteriori probability regarding \(\theta\), while we are concerned with something else. Suppose that we continue with the sampling experiment until we have, say, \(N = 100\) samples. You will see, Sir, that the relative frequency of successful estimations using formulae (21) will be about the same as that using formulae (22) and that both will be approximately equal to .95.”

I do hope that the Assistant will not get fired. However, if he does, I would remind him of the glory of Giordano Bruno who was burned at the stake by the Holy Inquisition for believing in the Copernican theory of the solar system. Furthermore, I would advise him to have a talk with a physicist or a biologist or, maybe, with an engineer. They might fail to understand the theory but, if he performs for them the sampling experiment described above, they are likely to be convinced and give him a new job. In due course, the eminent statistical Boss will die or retire and then —.

[footnote] Sad as it is, your mind does become less flexible and less receptive to novel ideas as the years go by. The more mature members of the audience should not take offense. I, myself, am not young and have young assistants. Besides, unreasonable and stubborn individuals are found not only among the elderly but also frequently among young people.

[end excerpt]


Two things to stop saying about null hypotheses

There is a currently fashionable way of describing Bayes factors that resonates with experimental psychologists. I hear it often, particularly as a way to describe a particular use of Bayes factors. For example, one might say, “I needed to prove the null, so I used a Bayes factor,” or “Bayes factors are great because with them, you can prove the null.” I understand the motivation behind this sort of language but please: stop saying one can “prove the null” with Bayes factors.

I also often hear other people say “but the null is never true.” I’d like to explain why we should avoid saying both of these things.

Null hypotheses are tired of your jibber jabber

Why you shouldn’t say “prove the null”

Statistics is complicated. People often come up with colloquial ways of describing what a particular method is doing: for instance, one might say a significance tests give us “evidence against the null”; one might say that a “confidence interval tells us the 95% most plausible values”; or one might say that a Bayes factor helps us “prove the null.” Bayesians often are quick to correct misconceptions that people use to justify their use of classical or frequentist methods. It is just as important to correct misconceptions about Bayesian methods.

In order to understand why we shouldn’t say “prove the null”, consider the following situation: You have a friend who claims that they can affect the moon with their mind. You, of course, think this is preposterous. Your friend looks up at the moon and says “See, I’m using my abilities right now!” You check the time.

You then decide to head to the local lunar seismologist, who has good records of subtle moon tremors. You ask her whether about what happened at the time your friend was looking at the moon, and she reports back to you that lunar activity at that time was stronger than it typically is 95% of the time (thus passes the bar for “statistical significance”).

Does this mean that there is evidence for your friend’s assertion? The answer is “no.” Your friend made no statement about what one would expect from the seismic data. In fact, your friend’s statement is completely unfalsifiable (as is the case with the typical “alternative” in a significance test, (muneq0)).

But consider the following alternative statements your friend could have made: “I will destroy the moon with my mind”; “I will make very large tremors (with magnitude (Y))”; “I will make small tremors (with magnitude (X)).” How do we now regard your friend’s claims in light of the what happened?

  • “I will destroy the moon with my mind” is clearly inconsistent with the data. You (the null) are supported by an infinite amount, because you have completely falsified his statement that he would destroy the moon (the alternative).
  • “I will make very large tremors (with magnitude (Y))” is also inconsistent with the data, but if we allow a range of uncertainty around his claim, may not be completely falsified. Thus you (the null) are supported, but not by as much in the first situation.
  • “I will make small tremors (with magnitude (X))” may support you (the null) or your friend (the alternative), depending on how the magnitude predicted and observed.

Here we can see that the support for the null depends on the alternative at hand. This is, of course, as it must be. Scientific evidence is relative. We can never “prove the null”: we can only “find evidence for a specified null hypothesis against a reasonable, well-specified alternative”. That’s quite a mouthful, it’s true, but “prove the null” creates misunderstandings about Bayesian statistics, and makes it appear that it is doing something it cannot do.

In a Bayesian setup, the null and alternative are both models and the relative evidence between them will change based on how we specify them. If we specify them in a reasonable manner, such that the null and alternative correspond to relevant theoretical viewpoints or encode information about the question at hand, the relative statistical evidence will be informative for our research ends. If we don’t specify reasonable models, then the relative evidence between the models may be correct, but useless.

We never “prove the null” or “compute the probability of the null hypothesis”. We can only compare a null model to an alternative model, and determine the relative evidence.

[See also Gelman and Shalizi (2013) and Morey, Romeijn and Rouder (2013)]

Why you shouldn’t say “the null is never true”

A common retort to tests including a point null (often called a ‘null’ hypothesis) is that “the null is never true.” This backed up by four sorts of “evidence”:

  • A quote from an authority: “Tukey or Cohen said so!” (Tukey was smart, but this is not an argument.)
  • Common knowledge / “experience”: “We all know the null is impossible.” (This was Tukey’s “argument”)
  • Circular: “The area under a point in a density curve is 0.” (Of course if your model doesn’t have a point null, the point null will be impossible.)
  • All models are “false” (even if this were true — I think it is actually a category error — it would equally apply to all alternatives as well)

The most attractive seems to be the second, but it should be noted that people almost never use techniques that allow finding evidence for null hypotheses. Under these conditions, how is one determining that the null is never true? If a null were ever true, we would not be able to accumulate evidence for it, so the second argument definitely has a hint of circularity as well.

When someone says “The null hypothesis is impossible/implausible/irrelevant”, what they are saying in reality is “I don’t believe the null hypothesis can possibly be true.” This is a totally fine statement, as long as we recognize it for what it is: an a priori commitment. We should not pretend that it is anything else; I cannot see any way that one can find universal evidence for the statement “the null is impossible”.

If you find the null hypothesis implausible, that’s OK. Others might not find it implausible. It is ultimately up to substantive experts to decide what hypotheses they want to consider in their data analysis, and not up to methodologists or statisticians to decide to tell experts what to think.

Any automatic behavior — either automatically rejecting all null hypothesis, or automatically testing null hypotheses — is bad. Hypothesis testing and estimation should be considered and deliberate. Luckily, Bayesian statistics allows both to be done in a principled, coherent manner, so informed choices can be made by the analyst and not by the restrictions of the method.

Statistical alchemy and the “test for excess significance”

[This post is based largely on my 2013 article for Journal of Mathematical Psychology; see the other articles in that special issue as well for more critiques.]

When I tell people that my primary area of research is statistical methods, one of the reactions I often encounter from people untrained in statistics is that “you can prove anything with statistics.” Of course, this rankles, first because it isn’t true (unless you use a very strange definition of prove) and second because I’ve spent years learning the limitations of statistics, and there are many limitations. These limitations exist, however, in the context of enormous successes. In the sciences, the field of statistics rightly has a place of honor.

This success is evidenced by the great number of scientific arguments that are supported by statistical methods. Not all statistical arguments are created equal, of course. But the respect with which statistics is viewed has the unfortunate downside that a statistical argument can apparently turn a leaden hunch into a golden “truth”. This post is about such statistical alchemy.

The gold: Justified substantive claims

One of the goals we all have as scientists is to make claims backed by solid evidence. This is harder than it seems. Ideally we would prefer that evidence be ironclad and assumptions unnecessary. In real-life cases, however, the strength of evidence does not provide certainty, and assumptions are needed. The key to good argument, then, is that all assumptions are made explicit, the chain of reasoning is clear and logical, and the resulting evidence is strong enough to garner agreement.

Such cases we might call the “gold standard” for scientific arguments. We expect this sort of argument when someone makes a strong claim. This is the stuff that the scientific literature should be made of, for the most part. Among other things, the gold standard requires careful experimental design and execution, deliberate statistical analysis and avoidance of post hoc reasoning, and a willingness to explore the effects of unstated assumptions in one’s reasoning.

The lead: Hunches

Hunches are a necessary part of science. Science is driven by a creative force that cannot (at this point) be quantified, and a sneaking suspicion that something is true is often the grounds on which we design experiments. Hunches are some of the most useful things in science, just as lead is an exceptionally useful metal. Like lead, hunches are terribly common. We all have many hunches, and often we don’t know where they come from.

What makes a hunch is that it doesn’t have solid grounds to back it up. Hunches often turn to dust upon closer examination: they may contradict other knowledge, they may be based on untenable assumptions, or the evidence for them may turn out to be much weak when we examine it. If a hunch survives a solid test, it is no longer a hunch; but so long we do not test them — or cannot test them — they remain hunches.

The alchemy of statistics

One of the most dangerous, but unfortunately common, ways in which statistics is used is to magically turn hunches into “truth”. The mother of all statistical alchemy is the Fisherian (p) value, by which hunches based on “low” (p) values are turned into statements about the implausibility of the null hypothesis. Although it seems reasonable, when the hunch on which (p) values rest is examined by either frequentists or Bayesians, it is found wanting.

However, my main focus here is not specifically (p) values. I’d like to focus on one particularly recent special case of statistical alchemy among methodologists called the “test for excess significance”. Here’s the hunch: in any series of typically-powered experiments, we expect some to fail to be non-significant due to sampling error, even if a true effect exists. If we see a series of five experiments, and they are all significant, one thinks that either they are either very high powered, the authors got lucky, or there are some nonsignificant studies missing. For many sets of studies, the first seems implausible because the effect sizes are small; the last is important, because if it is true then the picture we get of the results is misleading.

Just to be clear, this hunch makes sense to me, and I think to most people. However, without a formal argument it remains a hunch. Ioannidis and Trikalinos (2007) suggested formalising it:

We test in a body of $n$ published studies whether the observed number of studies $O$ with ‘positive’ results at a specified $alpha$ level on a specific research question is different from the expected number of studies with ‘positive’ results $E$ in the absence of any bias. (Ioannidis and Trikalinos, 2007, p246)

Of “biases”, Ioannidis and and Tikalinos say that “biases…result in a relative excess of published statistically significant results as compared with what their true proportion should be in a body of evidence.” If there are too many significant studies, there must be too few nonsignificant ones, hence the idea of “relative” excess.

Suppose there is a true effect size that is being pursued by study $i (i = 1,ldots,n)$ and its size is $theta_i$…[T]he expected probability that a specific single study $i$ will find a ‘positive’ result equals $1 – beta_i$, its power at the specified $alpha$ level. (Ioannidis and Trikalinos, 2007, p246)

So far so good; this is all true. Ioannidis and Trikalinos continue:

Assuming no bias, $E$ equals the sum of the expected probabilities across all studies on the same question: [ E = sum_{i=1}^n (1 – beta_i). ] (Ioannidis and Trikalinos, 2007, p246)

Here we have bit of a mystery. That (E) equals the sum of the expected probabilities is merely asserted. There is no explanation of what assumptions were necessary to derive that fact. Moreover, it is demonstrably false. Suppose I run experiments until I obtain (k) nonsignificant studies ((k>0)). The expected number of significant studies in a set of (n) is exactly (n-k). Depending on the stopping rule for the studies, which is unknown (and unknowable or even meaningless, in most cases), (E) can be chosen to be 0 (stop after (n) nonsignificant studies), (n) (stop after (n) significant studies), or any number in between!

Ioannidis and Trikalinos go on to say that “[t]he expected number (E) is compared against the observed number (O) of ‘positive’ studies” and if there are an “excess” then bias is claimed, by standard significance test logic. Here, things go off the rails again. First, as we have seen, (E) could be anything. Second, a significance test is performed by computing the probability of observing an outcome as extreme or more extreme than the one observed, given no “bias”. What is more extreme? Suppose we observe 4 significant results in 5 studies. It seems clear that 5/5 is more extreme. Is 6/6 possible? No mention is made of the assumed sampling process, so how are we to know what the more extreme samples would be? And if a sampling assumption were made explicit, how could we know whether that was a reasonable assumption for the studies at hand? The (p) value is simply incalculable from the information available.

Suppose I find a “significant” result; what do I infer? Ioannidis and Trikalinos claim that they “have introduced an exploratory test for examining whether there is an excess of significant findings in a body of evidence” (p 251). This is a very strange assertion. When we do a statistical test, we are not asking a question about the data itself; rather, we are inferring something about a population. The “body of evidence” is the sample; we infer from the sample to the population. But what is the population? Or, put in frequentist terms, what is the sampling process from which the studies in question arise? Given that this question is central to the statistical inference, one would think it would be addressed, but it is not. Dealing with this question would require a clear definition of a “set” of studies, and how this set is sampled.

Are these studies one sample of hypothetical sets of studies from all scientific fields? Or perhaps they are a sample of studies within a specific field; say, psychology? Or from a subfield, like social psychology? Or maybe from a specific lab? There’s no way to uniquely answer this question, and so it isn’t clear what can be inferred. Am I inferring bias in all of science, in the field, the subfield, or the lab? And if any of these are true, why do they discuss bias in the sample instead? They have confused the properties of the population and sample in a basic way.

But even though these critical details are missing — details that are necessary to the argument — the authors go on to apply this to several meta-analyses, inferring bias in several. Other authors have applied the method to claim “evidence” of bias in other sets of studies.

…and the alchemy is complete

We see that Ioannidis and Trikalinos have unstated assumptions of enormous import, they have failed to clearly define any sort of sampling model, and they have not made clear the link between the act of inference (“we found a ‘significant’ result”) and what is to be inferred (“Evidence for bias exists in these studies.”). And this is all before even addressing the problematic nature of (p) values themselves, which cannot be used as a measure of evidence. The test for “excess significance” is neither a valid frequentist procedure (due to the lack of a clearly defined sampling process) nor a valid Bayesian procedure.

But through the alchemy of statistics, the Ioannidis and Trikalinos’ test for “excess significance” has given us the appearance of a justified conclusion. Bodies of studies are called into doubt, and the users of the approach continue to get papers published using the approach despite its utter lack of justification. We would not accept such shoddy modeling and reasoning for studying other aspects of human behavior. As Val Johnson put it in his comment on the procedure, “[We] simply cannot quite determine the level of absurdity that [we are] expected to ignore.” Why is this acceptable for deploying against groups of studies in the scientific literature?

The reason is simple: we all have the hunch. It seems right. Ioannidis and Trikalinos have given us a way to transmute our hunch that something is amiss into the gold of a publishable, evidence-backed conclusion. But it is an illusion; the argument simply falls apart under scrutiny.

This is bad science, and it should not be tolerated. Methodologists have the same responsibility as everyone else to justify their conclusions. The peer review system has failed to prevent the leaden hunch passing for gold, which is acutely ironic given how methodologists use the test to accuse others of bad science.

Further reading: