Archive for hypothesis testing

a Bayesian interpretation of FDRs?

Posted in Statistics with tags , , , , , , , , , , on April 12, 2018 by xi'an

This week, I happened to re-read John Storey’ 2003 “The positive discovery rate: a Bayesian interpretation and the q-value”, because I wanted to check a connection with our testing by mixture [still in limbo] paper. I however failed to find what I was looking for because I could not find any Bayesian flavour in the paper apart from an FRD expressed as a “posterior probability” of the null, in the sense that the setting was one of opposing two simple hypotheses. When there is an unknown parameter common to the multiple hypotheses being tested, a prior distribution on the parameter makes these multiple hypotheses connected. What makes the connection puzzling is the assumption that the observed statistics defining the significance region are independent (Theorem 1). And it seems to depend on the choice of the significance region, which should be induced by the Bayesian modelling, not the opposite. (This alternative explanation does not help either, maybe because it is on baseball… Or maybe because the sentence “If a player’s [posterior mean] is above .3, it’s more likely than not that their true average is as well” does not seem to appear naturally from a Bayesian formulation.) [Disclaimer: I am not hinting at anything wrong or objectionable in Storey’s paper, just being puzzled by the Bayesian tag!]

a null hypothesis with a 99% probability to be true…

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , on March 28, 2018 by xi'an

When checking the Python t distribution random generator, np.random.standard_t(), I came upon this manual page, which actually does not explain how the random generator works but spends instead the whole page to recall Gosset’s t test, illustrating its use on an energy intake of 11 women, but ending up misleading the readers by interpreting a .009 one-sided p-value as meaning “the null hypothesis [on the hypothesised mean] has a probability of about 99% of being true”! Actually, Python’s standard deviation estimator x.std() further returns by default a non-standard standard deviation, dividing by n rather than n-1…

admissible estimators that are not Bayes

Posted in Statistics with tags , , , , , , on December 30, 2017 by xi'an

A question that popped up on X validated made me search a little while for point estimators that are both admissible (under a certain loss function) and not generalised Bayes (under the same loss function), before asking Larry Brown, Jim Berger, or Ed George. The answer came through Larry’s book on exponential families, with the two examples attached. (Following our 1989 collaboration with Roger Farrell at Cornell U, I knew about the existence of testing procedures that were both admissible and not Bayes.) The most surprising feature is that the associated loss function is strictly convex as I would have thought that a less convex loss would have helped to find such counter-examples.

p-values and decision-making [reposted]

Posted in Books, Statistics, University life with tags , , , , , , , , , , on August 30, 2017 by xi'an

In a letter to Significance about a review of Robert Matthews’s book, Chancing it, Nicholas Longford recalls a few basic facts about p-values and decision-making earlier made by Dennis Lindley in Making Decisions. Here are some excerpts, worth repeating in the light of the 0.005 proposal:

“A statement of significance based on a p-value is a verdict that is oblivious to consequences. In my view, this disqualifies hypothesis testing, and p-values with it, from making rational decisions. Of course, the p-value could be supplemented by considerations of these consequences, although this is rarely done in a transparent manner. However, the two-step procedure of calculating the p-value and then incorporating the consequences is unlikely to match in its integrity the single-stage procedure in which we compare the expected losses associated with the two contemplated options.”

“At present, [Lindley’s] decision-theoretical approach is difficult to implement in practice. This is not because of any computational complexity or some problematic assumptions, but because of our collective reluctance to inquire about the consequences – about our clients’ priorities, remits and value judgements. Instead, we promote a culture of “objective” analysis, epitomised by the 5% threshold in significance testing. It corresponds to a particular balance of consequences, which may or may not mirror our clients’ perspective.”

“The p-value and statistical significance are at best half-baked products in the process of making decisions, and a distraction at worst, because the ultimate conclusion of a statistical analysis should be a proposal for what to do next in our clients’ or our own research, business, production or some other agenda. Let’s reflect and admit how frequently we abuse hypothesis testing by adopting (sometimes by stealth) the null hypothesis when we fail to reject it, and therefore do so without any evidence to support it. How frequently we report, or are party to reporting, the results of hypothesis tests selectively. The problem is not with our failing to adhere to the convoluted strictures of a popular method, but with the method itself. In the 1950s, it was a great statistical invention, and its popularisation later on a great scientific success. Alas, decades later, it is rather out of date, like the steam engine. It is poorly suited to the demands of modern science, business, and society in general, in which the budget and pocketbook are important factors.”

estimation versus testing [again!]

Posted in Books, Statistics, University life with tags , , , , , , , , , , on March 30, 2017 by xi'an

The following text is a review I wrote of the paper “Parameter estimation and Bayes factors”, written by J. Rouder, J. Haff, and J. Vandekerckhove. (As the journal to which it is submitted gave me the option to sign my review.)

The opposition between estimation and testing as a matter of prior modelling rather than inferential goals is quite unusual in the Bayesian literature. In particular, if one follows Bayesian decision theory as in Berger (1985) there is no such opposition, but rather the use of different loss functions for different inference purposes, while the Bayesian model remains single and unitarian.

Following Jeffreys (1939), it sounds more congenial to the Bayesian spirit to return the posterior probability of an hypothesis H⁰ as an answer to the question whether this hypothesis holds or does not hold. This however proves impossible when the “null” hypothesis H⁰ has prior mass equal to zero (or is not measurable under the prior). In such a case the mathematical answer is a probability of zero, which may not satisfy the experimenter who asked the question. More fundamentally, the said prior proves inadequate to answer the question and hence to incorporate the information contained in this very question. This is how Jeffreys (1939) justifies the move from the original (and deficient) prior to one that puts some weight on the null (hypothesis) space. It is often argued that the move is unnatural and that the null space does not make sense, but this only applies when believing very strongly in the model itself. When considering the issue from a modelling perspective, accepting the null H⁰ means using a new model to represent the model and hence testing becomes a model choice problem, namely whether or not one should use a complex or simplified model to represent the generation of the data. This is somehow the “unification” advanced in the current paper, albeit it does appear originally in Jeffreys (1939) [and then numerous others] rather than the relatively recent Mitchell & Beauchamp (1988). Who may have launched the spike & slab denomination.

I have trouble with the analogy drawn in the paper between the spike & slab estimate and the Stein effect. While the posterior mean derived from the spike & slab posterior is indeed a quantity drawn towards zero by the Dirac mass at zero, it is rarely the point in using a spike & slab prior, since this point estimate does not lead to a conclusion about the hypothesis: for one thing it is never exactly zero (if zero corresponds to the null). For another thing, the construction of the spike & slab prior is both artificial and dependent on the weights given to the spike and to the slab, respectively, to borrow expressions from the paper. This approach thus leads to model averaging rather than hypothesis testing or model choice and therefore fails to answer the (possibly absurd) question as to which model to choose. Or refuse to choose. But there are cases when a decision must be made, like continuing a clinical trial or putting a new product on the market. Or not.

In conclusion, the paper surprisingly bypasses the decision-making aspect of testing and hence ends up with a inconclusive setting, staying midstream between Bayes factors and credible intervals. And failing to provide a tool for decision making. The paper also fails to acknowledge the strong dependence of the Bayes factor on the tail behaviour of the prior(s), which cannot be [completely] corrected by a finite sample, hence its relativity and the unreasonableness of a fixed scale like Jeffreys’ (1939).

John Kruschke on Bayesian assessment of null values

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , , on February 28, 2017 by xi'an

John Kruschke pointed out to me a blog entry he wrote last December as a follow-up to my own entry on an earlier paper of his. Induced by an X validated entry. Just in case this sounds a wee bit too convoluted for unraveling the threads (!), the central notion there is to replace a point null hypothesis testing [of bad reputation, for many good reasons] with a check whether or not the null value stands within the 95% HPD region [modulo a buffer zone], which offers the pluses of avoiding a Dirac mass at the null value and a long-term impact of the prior tails on the decision, as well as the possibility of a no-decision, with the minuses of replacing the null with a tolerance region around the null and calibrating both the rejection level and the buffer zone. The December blog entry exposes this principle with graphical illustrations familiar to readers of Doing Bayesian Data Analysis.

As I do not want to fall into an infinite regress of mirror discussions, I will not proceed further than referring to my earlier post, which covers my reservations about the proposal. But interested readers may want to check the latest paper by Kruschke and Liddel on that perspective. (With the conclusion that “Bayesian estimation does everything the New Statistics desires, better”.) Available on PsyArXiv, an avatar of arXiv for psychology papers.

a concise introduction to statistical inference [book review]

Posted in Statistics with tags , , , , , , , , , , on February 16, 2017 by xi'an

[Just to warn readers and avoid emails about Xi’an plagiarising Christian!, this book was sent to me by CRC Press for a review. To be published in CHANCE.]

This is an introduction to statistical inference. And with 180 pages, it indeed is concise! I could actually stop the review at this point as a concise review of a concise introduction to statistical inference, as I do not find much originality in this introduction, intended for “mathematically sophisticated first-time student of statistics”. Although sophistication is in the eye of the sophist, of course, as this book has margin symbols in the guise of integrals to warn of section using “differential or integral calculus” and a remark that the book is still accessible without calculus… (Integral calculus as in Riemann integrals, not Lebesgue integrals, mind you!) It even includes appendices with the Greek alphabet, summation notations, and exponential/logarithms.

“In statistics we often bypass the probability model altogether and simply specify the random variable directly. In fact, there is a result (that we won’t cover in detail) that tells us that, for any random variable, we can find an appropriate probability model.” (p.17)

Given its limited mathematical requirements, the book does not get very far in the probabilistic background of statistics methods, which makes the corresponding chapter not particularly helpful as opposed to a prerequisite on probability basics. Since not much can be proven without “all that complicated stuff about for any ε>0” (p.29). And makes defining correctly notions like the Central Limit Theorem impossible. For instance, Chebychev’s inequality comes within a list of admitted results. There is no major mistake in the chapter, even though mentioning that two correlated Normal variables are jointly Normal (p.27) is inexact.

“The power of a test is the probability that you do not reject a null that is in fact correct.” (p.120)

Most of the book follows the same pattern as other textbooks at that level, covering inference on a mean and a probability, confidence intervals, hypothesis testing, p-values, and linear regression. With some words of caution about the interpretation of p-values. (And the unfortunate inversion of the interpretation of power above.) Even mentioning the Cult [of Significance] I reviewed a while ago.

Given all that, the final chapter comes as a surprise, being about Bayesian inference! Which should make me rejoice, obviously, but I remain skeptical of introducing the concept to readers with so little mathematical background. And hence a very shaky understanding of a notion like conditional distributions. (Which reminds me of repeated occurrences on X validated when newcomers hope to bypass textbooks and courses to grasp the meaning of posteriors and such. Like when asking why Bayes Theorem does not apply for expectations.) I can feel the enthusiasm of the author for this perspective and it may diffuse to some readers, but apart from being aware of the approach, I wonder how much they carry away from this brief (decent) exposure. The chapter borrows from Lee (2012, 4th edition) and from Berger (1985) for the decision-theoretic part. The limitations of the exercise are shown for hypothesis testing (or comparison) by the need to restrict the parameter space to two possible values. And for decision making. Similarly, introducing improper priors and the likelihood principle [distinguished there from the law of likelihood] is likely to get over the head of most readers and clashes with the level of the previous chapters. (And I do not think this is the most efficient way to argue in favour of a Bayesian approach to the problem of statistical inference: I have now dropped all references to the likelihood principle from my lectures. Not because of the controversy, but simply because the students do not get it.) By the end of the chapter, it is unclear a neophyte would be able to spell out how one could specify a prior for one of the problems processed in the earlier chapters. The appendix on de Finetti’s formalism on personal probabilities is very much unlikely to help in this regard. While it sounds so far beyond the level of the remainder of the book.