Stephen Senn

Sir Harold Jeffreys’ (tail area) one-liner: Sat night comedy [draft ii]

You might not have thought there could be new material for 2014, but there is, and if you look a bit more closely, you’ll see that it’s actually not Jay Leno who is standing up there at the mike ….

It’s Sir Harold Jeffreys himself! And his (very famous) joke, I admit, is funny. So, since it’s Saturday night, let’s listen in on Sir Harold’s howler* in criticizing the use of p-values.

“Did you hear the one about significance testers rejecting H0 because of outcomes H0 didn’t predict?

Well, what’s unusual, is that they do it when these unpredicted outcomes haven’t even occurred!”

Much laughter.

[The actual quote from Jeffreys: Using p-values implies that “An hypothesis that may be true is rejected because it has failed to predict observable results that have not occurred. This seems a remarkable procedure.” (Jeffreys 1939, 316)]

I say it’s funny, so to see why I’ll strive to give it a generous interpretation.

We can view p-values in terms of rejecting H0, as in the joke: There’s a test statistic D such that H0 is rejected if its observed value d0 reaches or exceeds a cut-off d* where Pr(D > d*; H0) is small, say .025.
Reject H0 if Pr(D > d0H0) < .025.
The report might be “reject Hat level .025″.
Example:  H0: The mean light deflection effect is 0. So if we observe a 1.96 standard deviation difference (in one-sided Normal testing) we’d reject H0 .

Now it’s true that if the observation were further into the rejection region, say 2, 3 or 4 standard deviations, it too would result in rejecting the null, and with an even smaller p-value. It’s also true that H0 “has not predicted” a 2, 3, 4, 5 etc. standard deviation difference in the sense that differences so large are “far from” or improbable under the null. But wait a minute. What if we’ve only observed a 1 standard deviation difference (p-value = .16)? It is unfair to count it against the null that 1.96, 2, 3, 4 etc. standard deviation differences would have diverged seriously from the null, when we’ve only observed the 1 standard deviation difference. Yet the p-value tells you to compute Pr(D > 1; H0), which includes these more extreme outcomes! This is “a remarkable procedure” indeed! [i]

So much for making out the howler. The only problem is that significance tests do not do this, that is, they do not reject with, say, D = 1 because larger D values might have occurred (but did not). D = 1 does not reach the cut-off, and does not lead to rejecting H0. Moreover, looking at the tail area makes it harder, not easier, to reject the null (although this isn’t the only function of the tail area): since it requires not merely that Pr(D = d0 ; H0 ) be small, but that Pr(D > d0 ; H0 ) be small. And this is well justified because when this probability is not small, you should not regard it as evidence of discrepancy from the null. Before getting to this …. Continue reading

Categories: Comedy, Fisher, Jeffreys, P-values, Statistics, Stephen Senn

Stephen Senn: Dawid’s Selection Paradox (guest post)

Stephen Senn
Competence Center for Methodology and Statistics (CCMS),
Luxembourg

You can protest, of course, that Dawid’s Selection Paradox is no such thing but then those who believe in the inexorable triumph of logic will deny that anything is a paradox. In a challenging paper published nearly 20 years ago (Dawid 1994), Philip Dawid drew attention to a ‘paradox’ of Bayesian inference. To describe it, I can do no better than to cite the abstract of the paper, which is available from Project Euclid, here: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?

When the inference to be made is selected after looking at the data, the classical statistical approach demands — as seems intuitively sensible — that allowance be made for the bias thus introduced. From a Bayesian viewpoint, however, no such adjustment is required, even when the Bayesian inference closely mimics the unadjusted classical one. In this paper we examine more closely this seeming inadequacy of the Bayesian approach. In particular, it is argued that conjugate priors for multivariate problems typically embody an unreasonable determinism property, at variance with the above intuition.

I consider this to be an important paper not only for Bayesians but also for frequentists, yet it has only been cited 14 times as of 15 November 2013 according to Google Scholar. In fact I wrote a paper about it in the American Statistician a few years back (Senn 2008) and have also referred to it in a previous blogpost (12 May 2012). That I think it is important and neglected is excuse enough to write about it again.

Philip Dawid is not responsible for my interpretation of his paradox but the way that I understand it can be explained by considering what it means to have a prior distribution. First, as a reminder, if you are going to be 100% Bayesian, which is to say that all of what you will do by way of inference will be to turn a prior into a posterior distribution using the likelihood and the operation of Bayes theorem, then your prior distribution has to satisfy two conditions. First, it must be what you would use to bet now (that is to say at the moment it is established) and second no amount of subsequent data will change your prior qua prior. It will, of course, be updated by Bayes theorem to form a posterior distribution once further data are obtained but that is another matter. The relevant time here is your observation time not the time when the data were collected, so that data that were available in principle but only came to your attention after you established your prior distribution count as further data.

Now suppose that you are going to make an inference about a population mean, θ, using a random sample from the population and choose the standard conjugate prior distribution. Then in that case you will use a Normal distribution with known (to you) parameters μ and σ2. If σ2 is large compared to the random variation you might expect for the means in your sample, then the prior distribution is fairly uninformative and if it is small then fairly informative but being uninformative is not in itself a virtue. Being not informative enough runs the risk that your prior distribution is not one you might wish to use to bet now and being too informative that your prior distribution is one you might be tempted to change given further information. In either of these two cases your prior distribution will be wrong. Thus the task is to be neither too informative nor not informative enough. Continue reading

Highly probable vs highly probed: Bayesian/ error statistical differences

A reader asks: “Can you tell me about disagreements on numbers between a severity assessment within error statistics, and a Bayesian assessment of posterior probabilities?” Sure.

There are differences between Bayesian posterior probabilities and formal error statistical measures, as well as between the latter and a severity (SEV) assessment, which differs from the standard type 1 and 2 error probabilities, p-values, and confidence levels—despite the numerical relationships. Here are some random thoughts that will hopefully be relevant for both types of differences. (Please search this blog for specifics.)

1. The most noteworthy difference is that error statistical inference makes use of outcomes other than the one observed, even after the data are available: there’s no other way to ask things like, how often would you find 1 nominally statistically significant difference in a hunting expedition over k or more factors?  Or to distinguish optional stopping with sequential trials from fixed sample size experiments.  Here’s a quote I came across just yesterday:

“[S]topping ‘when the data looks good’ can be a serious error when combined with frequentist measures of evidence. For instance, if one used the stopping rule [above]…but analyzed the data as if a fixed sample had been taken, one could guarantee arbitrarily strong frequentist ‘significance’ against H0.” (Berger and Wolpert, 1988, 77).

The worry about being guaranteed to erroneously exclude the true parameter value here is an error statistical affliction that the Bayesian is spared (even though I don’t think they can be too happy about it, especially when HPD intervals are assured of excluding the true parameter value.) See this post for an amusing note; Mayo and Kruse (2001) below; and, if interested, search the (strong)  likelihood principle, and Birnbaum.

2. Highly probable vs. highly probed. SEV doesn’t obey the probability calculus: for any test T and outcome x, the severity for both H and ~H might be horribly low. Moreover, an error statistical analysis is not in the business of probabilifying hypotheses but evaluating and controlling the capabilities of methods to discern inferential flaws (problems with linking statistical and scientific claims, problems of interpreting statistical tests and estimates, and problems of underlying model assumptions). This is the basis for applying what may be called the Severity principle. Continue reading

Stephen Senn: Open Season (guest post)

Stephen Senn
Competence Center for Methodology and Statistics (CCMS),
Luxembourg

“Open Season”

The recent joint statement(1) by the Pharmaceutical Research and Manufacturers of America (PhRMA) and the European Federation of Pharmaceutical Industries and Associations(EFPIA) represents a further step in what has been a slow journey towards (one assumes) will be the achieved  goal of sharing clinical trial data. In my inaugural lecture of 1997 at University College London I called for all pharmaceutical companies to develop a policy for sharing trial results and I have repeated this in many places since(2-5). Thus I can hardly complain if what I have been calling for for over 15 years is now close to being achieved.

However, I have now recently been thinking about it again and it seems to me that there are some problems that need to be addressed. One is the issue of patient confidentiality. Ideally, covariate information should be exploitable as such often increases the precision of inferences and also the utility of decisions based upon them since they (potentially) increase the possibility of personalising medical interventions. However, providing patient-level data increases the risk of breaching confidentiality. This is a complicated and difficult issue about which, however, I have nothing useful to say. Instead I want to consider another matter. What will be the influence on the quality of the inferences we make of enabling many subsequent researchers to analyse the same data?

One of the reasons that many researchers have called for all trials to be published is that trials that are missing tend to be different from those that are present. Thus there is a bias in summarising evidence from published trial only and it can be a difficult task with no guarantee of success to identify those that have not been published. This is a wider reflection of the problem of missing data within trials. Such data have long worried trialists and the Food and Drug Administration (FDA) itself has commissioned a report on the subject from leading experts(6). On the European side the Committee for Medicinal Products for Human Use (CHMP) has a guideline dealing with it(7).

However, the problem is really a particular example of data filtering and it also applies to statistical analysis. If the analyses that are present have been selected from a wider set, then there is a danger that they do not provide an honest reflection of the message that is in the data. This problem is known as that of multiplicity and there is a huge literature dealing with it, including regulatory guidance documents(8, 9).

Within drug regulation this is dealt with by having pre-specified analyses. The broad outlines of these are usually established in the trial protocol and the approach is then specified in some detail in the statistical analysis plan which is required to be finalised before un-blinding of the data. The strategies used to control for multiplicity will involve some combination of defining a significance testing route (an order in which test must be performed and associated decision rules) and reduction of the required level of significance to detect an event.

I am not a great fan of these manoeuvres, which can be extremely complex. One of my objections is that it is effectively assumed that the researchers who chose them are mandated to circumscribe the inferences that scientific posterity can make(10). I take the rather more liberal view that provided that everything that is tested is reported one can test as much as one likes. The problem comes if there is selective use of results and in particular selective reporting. Nevertheless, I would be the first to concede the value of pre-specification in clarifying the thinking of those about to embark on conducting a clinical trial and also in providing a ‘template of trust’ for the regulator when provided with analyses by the sponsor.

However, what should be our attitude to secondary analyses? From one point of view these should be welcome. There is always value in looking at data from different perspectives and indeed this can be one way of strengthening inferences in the way suggested nearly 50 years ago by Platt(11). There are two problems, however. First, not all perspectives are equally valuable. Some analyses in the future, no doubt, will be carried out by those with little expertise and in some cases, perhaps, by those with a particular viewpoint to justify. There is also the danger that some will carry out multiple analyses (of which, when one consider the possibility of changing endpoints, performing transformations, choosing covariates and modelling framework there are usually a great number) but then only present those that are ‘interesting’. It is precisely to avoid this danger that the ritual of pre-specified analysis is insisted upon by regulators. Must we also insist upon it for those seeking to reanalyse?

To do so would require such persons to do two things. First, they would have to register the analysis plan before being granted access to the data. Second, they would have to promise to make the analysis results available, otherwise we will have a problem of missing analyses to go with the problem of missing trials. I think that it is true to say that we are just beginning to feel our way with this. It may be that the chance has been lost and that the whole of clinical research will be ‘world wide webbed’: there will be a mass of information out there but we just don’t know what to believe. Whatever happens the era of privileged statistical analyses by the original data collectors is disappearing fast.

[Ed. note: Links to some earlier related posts by Prof. Senn are:  “Casting Stones” 3/7/13, “Also Smith & Jones” 2/23/13, and “Fooling the Patient: An Unethical Use of Placebo?” 8/2/12 .]

References

1. PhRMA, EFPIA. Principles for Responsible Clinical Trial Data Sharing. PhRMA; 2013 [cited 2013 31 August]; Available from: http://phrma.org/sites/default/files/pdf/PhRMAPrinciplesForResponsibleClinicalTrialDataSharing.pdf.

2. Senn SJ. Statistical quality in analysing clinical trials. Good Clinical Practice Journal. [Research Paper]. 2000;7(6):22-6.

3. Senn SJ. Authorship of drug industry trials. Pharm Stat. [Editorial]. 2002;1:5-7.

4. Senn SJ. Sharp tongues and bitter pills. Significance. [Review]. 2006 September 2006;3(3):123-5.

5. Senn SJ. Pharmaphobia: fear and loathing of pharmaceutical research. [pdf] 1997 [updated 31 August 2013; cited 2013 31 August ]; Updated version of paper originally published on PharmInfoNet].

6. Little RJ, D’Agostino R, Cohen ML, Dickersin K, Emerson SS, Farrar JT, et al. The prevention and treatment of missing data in clinical trials. N Engl J Med. 2012 Oct 4;367(14):1355-60.

7. Committee for Medicinal Products for Human Use (CHMP). Guideline on Missing Data in Confirmatory Clinical Trials London: European Medicine Agency; 2010. p. 1-12.

8. Committee for Proprietary Medicinal Products. Points to consider on multiplicity issues in clinical trials. London: European Medicines Evaluation Agency2002.

9. International Conference on Harmonisation. Statistical principles for clinical trials (ICH E9). Statistics in Medicine. 1999;18:1905-42.

10. Senn S, Bretz F. Power and sample size when multiple endpoints are considered. Pharm Stat. 2007 Jul-Sep;6(3):161-70.

11. Platt JR. Strong Inference: Certain systematic methods of scientific thinking may produce much more rapid progress than others. Science. 1964 Oct 16;146(3642):347-53.

Stephen Senn: Indefinite irrelevance

Stephen Senn
Competence Center for Methodology and Statistics (CCMS),
Luxembourg

At a workshop on randomisation I attended recently I was depressed to hear what I regard as hackneyed untruths treated as if they were important objections. One of these is that of indefinitely many confounders. The argument goes that although randomisation may make it probable that some confounders are reasonably balanced between the arms, since there are indefinitely many of these, the chance that at least some are badly confounded is so great as to make the procedure useless.

This argument is wrong for several related reasons. The first is to do with the fact that the total effect of these indefinitely many confounders is bounded. This means that the argument put forward is analogously false to one in which it were claimed that the infinite series ½, ¼,⅛ …. did not sum to a limit because there were infinitely many terms. The fact is that the outcome value one wishes to analyse poses a limit on the possible influence of the covariates. Suppose that we were able to measure a number of covariates on a set of patients prior to randomisation (in fact this is usually not possible but that does not matter here). Now construct principle components, C1, C2… .. based on these covariates. We suppose that each of these predict to a greater or lesser extent the outcome, Y  (say).  In a linear model we could put coefficients on these components, k1, k2… (say). However one is not free to postulate anything at all by way of values for these coefficients, since it has to be the case for any set of m such coefficients that where  V(  ) indicates variance of. Thus variation in outcome bounds variation in prediction. This total variation in outcome has to be shared between the predictors and the more predictors you postulate there are, the less on average the influence per predictor.

The second error is to ignore the fact that statistical inference does not proceed on the basis of signal alone but also on noise. It is the ratio of these that is important. If there are indefinitely many predictors then there is no reason to suppose that their influence on the variation between treatment groups will be bigger than their variation within groups and both of these are used to make the inference. Continue reading

Categories: RCTs, Statistics, Stephen Senn

Gelman sides w/ Neyman over Fisher in relation to a famous blow-up

blog-o-log

Andrew Gelman had said he would go back to explain why he sided with Neyman over Fisher in relation to a big, famous argument discussed on my Feb. 16, 2013 post: “Fisher and Neyman after anger management?”, and I just received an e-mail from Andrew saying that he has done so: “In which I side with Neyman over Fisher”. (I’m not sure what Senn’s reply might be.) Here it is:

“In which I side with Neyman over Fisher” Posted by  on 24 May 2013, 9:28 am

As a data analyst and a scientist, Fisher > Neyman, no question. But as a theorist, Fisher came up with ideas that worked just fine in his applications but can fall apart when people try to apply them too generally.

Here’s an example that recently came up.

Deborah Mayo pointed me to a comment by Stephen Senn on the so-called Fisher and Neyman null hypotheses. In an experiment with n participants (or, as we used to say, subjects or experimental units), the Fisher null hypothesis is that the treatment effect is exactly 0 for every one of the n units, while the Neyman null hypothesis is that the individual treatment effects can be negative or positive but have an average of zero.

Senn explains why Neyman’s hypothesis in general makes no sense—the short story is that Fisher’s hypothesis seems relevant in some problems (sometimes we really are studying effects that are zero or close enough for all practical purposes), whereas Neyman’s hypothesis just seems weird (it’s implausible that a bunch of nonzero effects would exactly cancel). And I remember a similar discussion as a student, many years ago, when Rubin talked about that silly Neyman null hypothesis. Continue reading

Categories: Fisher, Statistics, Stephen Senn | Tags: ,

Guest post: Bad Pharma? (S. Senn)

Professor Stephen Senn*