Archive for confidence intervals

exoplanets at 99.999…%

Posted in Books, pictures, Statistics, University life with tags , , , , , on January 22, 2016 by xi'an

The latest Significance has a short article providing some coverage of the growing trend in the discovery of exoplanets, including new techniques used to detect those expoplanets from their impact on the associated stars. This [presumably] comes from the recent book Cosmos: The Infographics Book of Space [a side comment: new books seem to provide material for many articles in Significance these days!] and the above graph is also from the book, not the ultimate infographic representation in my opinion given that a simple superposition of lines could do as well. Or better.

¨A common approach to ruling out these sorts of false positives involves running sophisticated numerical algorithms, called Monte Carlo simulations, to explore a wide range of blend scenarios (…) A new planet discovery needs to have a confidence of (…) a one in a million chance that the result is in error.”

The above sentence is obviously of interest, first because the detection of false positives by Monte Carlo hints at a rough version of ABC to assess the likelihood of the observed phenomenon under the null [no detail provided] and second because the probability statement in the end is quite unclear as of its foundations… Reminding me of the Higgs boson controversy. The very last sentence of the article is however brilliant, albeit maybe unintentionaly so:

“To date, 1900 confirmed discoveries have been made. We have certainly come a long way from 1989.”

Yes, 89 down, strictly speaking!

evolution of correlations [award paper]

Posted in Books, pictures, Statistics, University life with tags , , , , , on September 15, 2015 by xi'an

“Many researchers might have observed that the magnitude of a correlation is pretty unstable in small samples.”

On the statsblog aggregator, I spotted an entry that eventually led me to this post about the best paper award for the evolution of correlation, a paper published in the Journal of Research in Personality. A journal not particularly well-known for its statistical methodology input. The main message of the paper is that, while the empirical correlation is highly varying for small n’s, an interval (or corridor of stability!) can be constructed so that a Z-transform of the correlation does not vary away from the true value by more than a chosen quantity like 0.1. And the point of stability is then defined as the sample size after which the trajectory of the estimate does not leave the corridor… Both corridor and point depending on the true and unknown value of the correlation parameter by the by. Which implies resorting to bootstrap to assess the distribution of this point of stability. And deduce quantiles that can be used for… For what exactly?! Setting the necessary sample size? But this requires a preliminary run to assess the possible value of the true correlation ρ. The paper concludes that “for typical research scenarios reasonable trade-offs between accuracy and confidence start to be achieved when n approaches 250”. This figure was achieved by a bootstrap study on a bivariate Gaussian population with 10⁶ datapoints, yes indeed 10⁶!, and bootstrap samples of maximal size 10³. All in all, while I am at a loss as to why the Journal of Research in Personality would promote the estimation of a correlation coefficient with 250 datapoints, there is nothing fundamentally wrong with the paper (!), except for this recommendation of the 250 datapoint, as the figure stems from a specific setting with particular calibrations and cannot be expected to apply in every and all cases.

bespprActually, the graph in the paper was the thing that first attracted my attention because it looks very much like the bootstrap examples I show my third year students to demonstrate the appeal of bootstrap. Which is not particularly useful in the current case. A quick simulation on 100 samples of size 300 showed [above] that Monte Carlo simulations produce a tighter confidence band than the one created by bootstrap, in the Gaussian case. Continue reading →

Bureau international des poids et mesures [bayésiennes?]

Posted in pictures, Statistics, Travel with tags , , , , , , , , , , , , , on June 19, 2015 by xi'an

The workshop at the BIPM on measurement uncertainty was certainly most exciting, first by its location in the Parc de Saint Cloud in classical buildings overlooking the Seine river in a most bucolic manner…and second by its mostly Bayesian flavour. The recommendations that the workshop addressed are about revisions in the current GUM, which stands for the Guide to the Expression of Uncertainty in Measurement. The discussion centred on using a more Bayesian approach than in the earlier version, with the organisers of the workshop and leaders of the revision apparently most in favour of that move. “Knowledge-based pdfs” came into the discussion as an attractive notion since it rings a Bayesian bell, especially when associated with probability as a degree of belief and incorporating the notion of an a priori probability distribution. And propagation of errors. Or even more when mentioning the removal of frequentist validations. What I gathered from the talks is the perspective drifting away from central limit approximations to more realistic representations, calling for Monte Carlo computations. There is also a lot I did not get about conventions, codes and standards. Including a short debate about the different meanings on Monte Carlo, from simulation technique to calculation method (as for confidence intervals). And another discussion about replacing the old formula for estimating sd from the Normal to the Student’s t case. A change that remains highly debatable since the Student’s t assumption is as shaky as the Normal one. What became clear [to me] during the meeting is that a rather heated debate is currently taking place about the need for a revision, with some members of the six (?) organisations involved arguing against Bayesian or linearisation tools.

This became even clearer during our frequentist versus Bayesian session with a first talk so outrageously anti-Bayesian it was hilarious! Among other things, the notion that “fixing” the data was against the principles of physics (the speaker was a physicist), that the only randomness in a Bayesian coin tossing was coming from the prior, that the likelihood function was a subjective construct, that the definition of the posterior density was a generalisation of Bayes’ theorem [generalisation found in… Bayes’ 1763 paper then!], that objective Bayes methods were inconsistent [because Jeffreys’ prior produces an inadmissible estimator of μ²!], that the move to Bayesian principles in GUM would cost the New Zealand economy 5 billion dollars [hopefully a frequentist estimate!], &tc., &tc. The second pro-frequentist speaker was by comparison much much more reasonable, although he insisted on showing Bayesian credible intervals do not achieve a nominal frequentist coverage, using a sort of fiducial argument distinguishing x=X+ε from X=x+ε that I missed… A lack of achievement that is fine by my standards. Indeed, a frequentist confidence interval provides a coverage guarantee either for a fixed parameter (in which case the Bayesian approach achieves better coverage by constant updating) or a varying parameter (in which case the frequency of proper inclusion is of no real interest!). The first Bayesian speaker was Tony O’Hagan, who summarily shred the first talk to shreds. And also criticised GUM2 for using reference priors and maxent priors. I am afraid my talk was a bit too exploratory for the audience (since I got absolutely no question!) In retrospect, I should have given an into to reference priors.

An interesting specificity of a workshop on metrology and measurement is that they are hard stickers to schedule, starting and finishing right on time. When a talk finished early, we waited until the intended time to the next talk. Not even allowing for extra discussion. When the only overtime and Belgian speaker ran close to 10 minutes late, I was afraid he would (deservedly) get lynched! He escaped unscathed, but may (and should) not get invited again..!

eliminating an important obstacle to creative thinking: statistics…

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , , , , on March 12, 2015 by xi'an

“We hope and anticipate that banning the NHSTP will have the effect of increasing the quality of submitted manuscripts by liberating authors from the stultified structure of NHSTP thinking thereby eliminating an important obstacle to creative thinking.”

About a month ago, David Trafimow and Michael Marks, the current editors of the journal Basic and Applied Social Psychology published an editorial banning all null hypothesis significance testing procedures (acronym-ed into the ugly NHSTP which sounds like a particularly nasty venereal disease!) from papers published by the journal. My first reaction was “Great! This will bring more substance to the papers by preventing significance fishing and undisclosed multiple testing! Power to the statisticians!” However, after reading the said editorial, I realised it was inspired by a nihilistic anti-statistical stance, backed by an apparent lack of understanding of the nature of statistical inference, rather than a call for saner and safer statistical practice. The editors most clearly state that inferential statistical procedures are no longer needed to publish in the journal, only “strong descriptive statistics”. Maybe to keep in tune with the “Basic” in the name of the journal!

“In the NHSTP, the problem is in traversing the distance from the probability of the finding, given the null hypothesis, to the probability of the null hypothesis, given the finding. Regarding confidence intervals, the problem is that, for example, a 95% confidence interval does not indicate that the parameter of interest has a 95% probability of being within the interval.”

The above quote could be a motivation for a Bayesian approach to the testing problem, a revolutionary stance for journal editors!, but it only illustrate that the editors wish for a procedure that would eliminate the uncertainty inherent to statistical inference, i.e., to decision making under… erm, uncertainty: “The state of the art remains uncertain.” To fail to separate significance from certainty is fairly appalling from an epistemological perspective and should be a case for impeachment, were any such thing to exist for a journal board. This means the editors cannot distinguish data from parameter and model from reality! Even more fundamentally, to bar statistical procedures from being used in a scientific study is nothing short of reactionary. While encouraging the inclusion of data is a step forward, restricting the validation or in-validation of hypotheses to gazing at descriptive statistics is many steps backward and does completely jeopardize the academic reputation of the journal, which editorial may end up being the last quoted paper. Is deconstruction now reaching psychology journals?! To quote from a critic of this approach, “Thus, the general weaknesses of the deconstructive enterprise become self-justifying. With such an approach I am indeed not sympathetic.” (Searle, 1983).

“The usual problem with Bayesian procedures is that they depend on some sort of Laplacian assumption to generate numbers where none exist (…) With respect to Bayesian procedures, we reserve the right to make case-by-case judgments, and thus Bayesian procedures are neither required nor banned from BASP.”

The section of Bayesian approaches is trying to be sympathetic to the Bayesian paradigm but again reflects upon the poor understanding of the authors. By “Laplacian assumption”, they mean Laplace´s Principle of Indifference, i.e., the use of uniform priors, which is not seriously considered as a sound principle since the mid-1930’s. Except maybe in recent papers of Trafimow. I also love the notion of “generat[ing] numbers when none exist”, as if the prior distribution had to be grounded in some physical reality! Although it is meaningless, it has some poetic value… (Plus, bringing Popper and Fisher to the rescue sounds like shooting Bayes himself in the foot.)  At least, the fact that the editors will consider Bayesian papers in a case-by-case basis indicate they may engage in a subjective Bayesian analysis of each paper rather than using an automated p-value against the 100% rejection bound!

[Note: this entry was suggested by Alexandra Schmidt, current ISBA President, towards an incoming column on this decision of Basic and Applied Social Psychology for the ISBA Bulletin.]