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Two weeks ago in my post Pachter’s P-value Prize I offered {\bf \frac{\$100}{p}} for justifying a reasonable null model and a p-value (p) associated to the statement “”Strikingly, 95% of cases of accelerated evolution involve only one member of a gene pair, providing strong support for a specific model of evolution, and allowing us to distinguish ancestral and derived functions” in the paper

M. Kellis, B.W. Birren and E.S. Lander, Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisaeNature 2004 (hereafter referred to as the KBL paper).

Today I am happy to announce the winner of the prize. But first, I want to thank the many readers of the blog who offered comments (>135 in total) that are extraordinary in their breadth and depth, and that offer a glimpse of what scientific discourse can look like when not restricted to traditional publishing channels. You have provided a wonderful public example of what “peer review” should mean. Coincidentally, and to answer one of the questions posted, the blog surpassed one million views this past Saturday (the first post was on August 19th, 2013), a testament to the the fact that the collective peer reviewing taking place on these pages is not only of very high quality, but also having an impact.

I particularly want to thank the students who had the courage to engage in the conversation, and also faculty who published comments using their name. In that regard, I admire and commend Joshua Plotkin and Hunter Fraser for deciding to deanonymize themselves by agreeing to let me announce here that they were the authors of the critique sent to the authors in April 2004  initially posted as an anonymous comment on the blog.

The discussion on the blog was extensive, touching on many interesting issues and I only summarize a few of the threads of discussion here. I decided to touch on a number of key points made in order to provide context and justification for my post and selection of the prize winner.

The value of post-publication review

One of the comments  made in response to my post that I’d like to respond to first was by an author of KBL who dismissed the entire premise of the my challenge writing “We can keep debating this after 11 years, but I’m sure we all have much more pressing things to do (grants? papers? family time? attacking 11-year-old papers by former classmates? guitar practice?)”

This comment exemplifies the proclivity of some authors to view publication as the encasement of work in a casket, buried deeply so as to never be opened again lest the skeletons inside it escape. But is it really beneficial to science that much of the published literature has become, as Ferguson and Heene noted, a vast graveyard of undead theories? Surely the varied and interesting comments posted in response to my challenge (totaling >25,000 words and 50 pages in Arial 11 font), demonstrate the value of communal discussion of science after publication.

For the record, this past month I did submit a paper and also a grant, and I did spend lots of time with my family. I didn’t practice the guitar but I did play the piano. Yet in terms of research, for me the highlight of the month was reading and understanding the issues raised in the comments to my blog post. Did I have many other things to do? Sure. But what is more pressing than understanding if the research one does is to be meaningful?

The null model

A few years ago I introduced a new two-semester freshman math course at UC Berkeley for intended biology majors called Math 10- Methods of Mathematics: Calculus, Statistics and Combinatorics“. One of the key ideas we focus on in the first semester is that of a p-value. The idea of measuring significance of a biological result via a statistical computation involving probabilities is somewhat unnatural, and feedback from the students confirms what one might expect: that the topic of p-values is among the hardest in the course. Math for biologists turns out to be much harder than calculus. I believe that at Berkeley we are progressive in emphasizing the importance of statistics for biology majors at the outset of their education (to be clear, this is a recent development). The prevailing state is that of statistical illiteracy, and the result is that p-values are frequently misunderstood, abused, and violated in just about every possible way imaginable (see, e.g., here, here and here).

P-values require a null hypothesis and a test statistic, and of course one of the most common misconceptions about them is that when they are large they confirm that the null hypothesis is correct. No! And worse, a small p-value cannot be used to accept an alternative to the null, only to (confidently) reject the null itself. And rejection of the null comes with numerous subtle issues and caveats (see arguments against the p-value in the papers mentioned above). So what is the point?

I think the KBL paper makes for an interesting case study of when p-values can be useful. For starters, the construction of a null model is already a useful exercise, because it is a thought experiment designed to test ones understanding of the problem at hand. The senior author of the KBL paper argues that “we were interested in seeing whether, for genes where duplication frees up at least one copy to evolve rapidly, the evidence better fits one model (“Ohno”: only one copy will evolve quickly) or an alternative model (both genes will evolve quickly).” While I accept this statement at face value, it is important to acknowledge that if there is any science to data science, it is the idea that when examining data one must think beyond the specific hypotheses being tested and consider alternative explanations. This is the essence of what my colleague Ian Holmes is saying in his comment. In data analysis, thinking outside of the box (by using statistics) is not optional. If one is lazy and resorts to intuition then, as Páll Melted points out, one is liable to end up with fantasy.

The first author of KBL suggests that the “paper was quite explicit about the null model being tested.” But I was unsure of whether to assume that the one-gene-only-speeds-up model is the null based on”we sought to distinguish between the Ohno one-gene-only speeds-up (OS) model and the alternative both-genes speed-up (BS) model” or was the null the BS model because “the Ohno model is 10^87 times more likely, leading to significant rejection of the BS null”?  Or was the paper being explicit about not having a null model at all, because  “Two alternatives have been proposed for post-duplication”, or was it the opposite, i.e. two null models: “the OS and BS models are each claiming to be right 95% of the time”? I hope I can be forgiven for failing, despite trying very hard, to identify a null model in either the KBL paper, or the comments of the authors to my blog.

There is however a reasonable null model, and it is the “independence model”, which to be clear, is the model where each gene after duplication “accelerates” independently with some small probability (80/914). The suggestions that “the independence model is not biologically rooted” or that it “would predict that only 75% of genes would be preserved in at least one copy, and that 26% would be preserved in both copies” are of course absurd, as explained by Erik van Nimwegen who explains why point clearly and carefully. The fact that many entries reached the same conclusion about the suitable null model in this case is reassuring. I think it qualifies as a “reasonable model” (thereby passing the threshold for my prize).

The p-value

One of my favorite missives about p-values is by Andrew Gelman, who in “P-values and statistical practice” discusses the subtleties inherent in the use and abuse of p-values. But as my blog post illustrates, subtlety is one thing, and ignorance is an entirely different matter. Consider for example, the entry by Manolis Kellis who submitted that p = 10^{-87} thus claiming that I owe him 903,659,165 million billion trillion quadrillion quintillion sextillion dollars (even more than the debt of the United States of America). His entry will not win the prize, although the elementary statistics lesson that follows is arguably worth a few dollars (for him). First, while it is true that a p-value can be computed from the (log) likelihood ratio when the null hypothesis is a special case of the alternative hypothesis (using the chi^2 distribution), the ratio of two likelihoods is not a p-value! Probabilities of events are also not p-values! For example, the comment that “I calculated p-values for the exact count, but the integral/sum would have been slightly better” is a non-starter. Even though KBL was published in 2004, this is apparently the level of understanding of p-values of one of the authors, a senior computational biologist and professor of computer science at MIT in 2015. Wow.

So what is “the correct” p-value? It depends, of course, on the test statistic. Here is where I will confess that like many professors, I had an answer in mind before asking the question. I was thinking specifically of the setting that leads to 0.74 (or 0.72/0.73, depending on roundoff and approximation). Many entries came up with the same answer I had in mind and therefore when I saw them I was relieved: I owed $135, which is what I had budgeted for the exercise. I was wrong. The problem with the answer 0.74 is that it is the answer to the specific question: what is the probability of seeing 4 or less pairs accelerate out of 76 pairs in which at least one accelerated. A better test statistic was proposed by Pseudo in which he/she asked for the probability of seeing 5% or less of the pairs accelerate from among the pairs with at least one gene accelerating when examining data from the null model with 457 pairs. This is a subtle but important distinction, and provides a stronger result (albeit with a smaller p-value). The KBL result is not striking even forgoing the specific numbers of genes measured to have accelerated in at least one pair (of course just because p=0.64 also does not mean the independence model is correct). What it means is that the data as presented simply weren’t “striking”.

One caveat in the above analysis is that the arbitrary threshold used to declare “acceleration” is problematic. For example, one might imagine that other thresholds produce more convincing results, i.e. farther from the null, but of course even if that were true the use of an arbitrary cutoff was a poor approach to analysis of the data. Fortunately, regarding the specific question of its impact in terms of the analysis, we do not have to imagine. Thanks to the diligent work of Erik van Nimwegen, who went to the effort of downloading the data and reanalyzing it with different thresholds (from 0.4 to 1.6), we know that the null cannot be rejected even with other thresholds.

The award

There were many entries submitted and I read them all. My favorite was by Michael Eisen for his creative use of multiple testing correction, although I’m happier with the direction that yields $8.79. I will not be awarding him the prize though, because his submission fails the test of “reasonable”, although I will probably take him out to lunch sometime at Perdition Smokehouse.

I can’t review every single entry or this post, which is already too long, would become unbearable, but I did think long and hard about the entry of K. It doesn’t directly answer the question of why the 95% number is striking, nor do I completely agree with some of the assumptions (e.g. if neither gene in a pair accelerates then the parent gene was not accelerated pre-WGD). But I’ll give the entry an honorable mention.

The prize will be awarded to Pseudo for defining a reasonable null model and test statistic, and producing the smallest p-value within that framework. With a p-value of 0.64 I will be writing a check in the amount of $156.25. Congratulations Pseudo!!

The biology

One of the most interesting results of the blog post was, in my opinion, the extensive discussion about the truth. Leaving aside the flawed analysis of KBL, what is a reasonable model for evolution post-WGD? I am happy to see the finer technical details continue to be debated, and the intensity of the conversation is awesome! Pavel Pevzner’s cynical belief that “science fiction” is not a literary genre but rather a description of what is published in the journal Science may be realistic, but I hope the comments on my blog can change his mind about what the future can look like.

In lieu of trying to summarize the scientific conversation (frankly, I don’t think I could do justice to some of the intricate and excellent arguments posted by some of the population geneticists) I’ll just leave readers to enjoy the comment threads on their own. Comments are still being posted, and I expect the blog post to be a “living” post-publication review for some time. May the skeletons keep finding a way out!

The importance of wrong

Earlier in this post I admitted to being wrong. I have been wrong many times. Even though I’ve admitted some of my mistakes on this blog and elsewhere in talks, I would like to joke that I’m not going to make it easy for you to find other flaws in my work. That would be a terrible mistake. Saying “I was wrong” is important for science and essential for scientists. Without it people lose trust in both.

I have been particularly concerned with a lack of “I was wrong” in genomics. Unfortunately, there is a culture that has developed among “leaders” in the field where the three words admitting error or wrongdoing are taboo. The recent paper of Lin et al. critiqued by Gilad-Mizrahi is a good example. Leaving aside the question of whether the result in the paper is correct (there are strong indications that it isn’t), Mizrahi-Gilad began their critique on twitter by noting that the authors had completely failed to account for, or even discuss, batch effect. Nobody, and I mean nobody who works on RNA-Seq would imagine for even a femtosecond that this is ok. It was a major oversight and mistake. The authors, any of them really, could have just come out and said “I was wrong”. Instead, the last author on the paper, Mike Snyder, told reporters that “All of the sequencing runs were conducted by the same person using the same reagents, lowering the risk of unintentional bias”. Seriously?

Examples abound. The “ENCODE 80% kerfuffle” involved claims that “80% of the genome is functional”. Any self-respecting geneticist recognizes such headline grabbing as rubbish. Ewan Birney, a distinguished scientist who has had a major impact on genomics having being instrumental in the ENSEMBL project and many other high-profile bioinformatics programs defended the claim on BBC:

“EB: Ah, so, I don’t — It’s interesting to reflect back on this. For me, the big important thing of ENCODE is that we found that a lot of the genome had some kind of biochemical activity. And we do describe that as “biochemical function”, but that word “function” in the phrase “biochemical function”is the thing which gets confusing. If we use the phrase “biochemical activity”, that’s precisely what we did, we find that the different parts of the genome, [??] 80% have some specific biochemical event we can attach to it. I was often asked whether that 80% goes to 100%, and that’s what I believe it will do. So, in other words, that number is much more about the coverage of what we’ve assayed over the entire genome. In the paper, we say quite clearly that the majority of the genome is not under negative selection, and we say that most of the elements are not under pan-mammalian selection. So that’s negative selection we can detect between lots of different mammals. [??] really interesting question about what is precisely going on in the human population, but that’s — you know, I’m much closer to the instincts of this kind of 10% to 20% sort of range about what is under, sort of what evolution cares about under selection.”

This response, and others by members of the ENCODE consortium upset many people who may struggle to tell apart white and gold from blue and black, but certainly know that white is not black and black is not white. Likewise, I suspect the response of KBL to my post disappointed many as well. For Fisher’s sake, why not just acknowledge what is obvious and true?

The personal critique of professional conduct

A conversation topic that emerged as a result of the blog (mostly on other forums) is the role of style in online discussion of science. Specifically, the question of whether personal attacks are legitimate has come up previously on my blog pages and also in conversations I’ve had with people. Here is my opinion on the matter:

Science is practiced by human beings. Just like with any other human activity, some of the humans who practice it are ethical while others are not. Some are kind and generous while others are… not. Occasionally scientists are criminal. Frequently they are honorable. Of particular importance is the fact that most scientists’ behavior is not at any of these extremes, but rather a convex combination of the mentioned attributes and many others.

In science it is people who benefit, or are hurt, by the behavior of scientists. Preprints on the bioRxiv do not collect salaries, the people who write them do. Papers published in journals do not get awarded or rejected tenure, people do. Grants do not get jobs, people do. The behavior of people in science affects… people.

Some argue for a de facto ban on discussing the personal behavior of scientists. I agree that the personal life of scientists is off limits. But their professional life shouldn’t be. When Bernie Madoff fabricated gains of $65 billion it was certainly legitimate to criticize him personally. Imagine if that was taboo, and instead only the technical aspects of his Ponzi scheme were acceptable material for public debate. That would be a terrible idea for the finance industry, and so it should be for science. Science is not special among the professions, and frankly, the people who practice it hold no primacy over others.

I therefore believe it is not only acceptable but imperative to critique the professional behavior of persons who are scientists. I also think that doing so will help eliminate the problematic devil–saint dichotomy that persists with the current system. Having developed a culture in which personal criticism is outlawed in scientific conversations while only science is fair fodder for public discourse, we now have a situation where scientists are all presumed to be living Gods, or else serious criminals to be outlawed and banished from the scientific community. Acknowledging that there ought to be a grey zone, and developing a healthy culture where critique of all aspects of science and scientists is possible and encouraged would relieve a lot of pressure within the current system. It would also be more fair and just.

A final wish

I wish the authors of the KBL paper would publish the reviews of their paper on this blog.

In this blog post I offer a cash prize for computing a p-value [update June 9th: the winner has been announced!]. For details about the competition you can skip directly to the challenge. But context is important:

Background

I’ve recently been reading a bioRxiv posting by X. Lan and J. Pritchard, Long-term survival of duplicate genes despite absence of subfunctionalized expression (2015) that examines the question of whether gene expression data (from human and mouse tissues) supports a model of duplicate preservation by subfunctionalization.

The term subfunctionalization is a hypothesis for explaining the ubiquity of persistence of gene duplicates in extant genomes. The idea is that gene pairs arising from a duplication event evolve, via neutral mutation, different functions that are distinct from their common ancestral gene, yet together recapitulate the original function. It was introduced in 1999 an alternative to the older hypothesis of neofunctionalization, which posits that novel gene functions arise by virtue of “retention” of one copy of a gene after duplication, while the other copy morphs into a new gene with a new function. Neofunctionalization was first floated as an idea to explain gene duplicates in the context of evolutionary theory by Haldane and Fisher in the 1930s, and was popularized by Ohno in his book Evolution by Gene Duplication published in 1970. The cartoon below helps to understand the difference between the *functionalization hypotheses (adapted from wikipedia):

Neo- and subfunctionalization
Lan and Pritchard examine the credibility of the sub- and neofunctionalization hypotheses using modern high-throughput gene expression (RNA-Seq) data: in their own words “Based on theoretical models and previous literature, we expected that–aside from the youngest duplicates–most duplicate pairs would be functionally distinct, and that the primary mechanism for this would be through divergent expression profiles. In particular, the sub- and neofunctionalization models suggest that, for each duplicate gene, there should be at least one tissue where that gene is more highly expressed than its partner.”

What they found was that, in their words, that “surprisingly few duplicate pairs show any evidence of sub-/neofunctionalization of expression.” The went further, stating that “the prevailing model for the evolution of gene duplicates holds that, to survive, duplicates must achieve non-redundant functions, and that this usually occurs by partitioning the expression space. However, we report here that sub-/neofunctionalization of expression occurs extremely slowly, and generally does not happen until the duplicates are separated by genomic rearrangements. Thus, in most cases long-term survival must rely on other factors.” They propose instead that “following duplication the expression levels of a gene pair evolve so that their combined expression matches the optimal level. Subsequently, the relative expression levels of the two genes evolve as a random walk, but do so slowly (33) due to constraint on their combined expression. If expression happens to become asymmetric, this reduces functional constraint on the minor gene. Subsequent accumulation of missense mutations in the minor gene may provide weak selective pressure to eventually eliminate expression of this gene, or may free the minor gene to evolve new functions.”

The Lan and Pritchard paper is the latest in a series of works that examine high-browed evolutionary theories with hard data, and that are finding reality to be far more complicated than the intuitively appealing, yet clearly inadequate, hypotheses of neo- and subfunctionalization. One of the excellent papers in the area is

Dean et al. Pervasive and Persistent Redundancy among Duplicated Genes in Yeast, PLoS Genetics, 2008.

where the authors argue that in yeast “duplicate genes do not often evolve to behave like singleton genes even after very long periods of time.” I mention this paper, from the Petrov lab, because its results are fundamentally at odds with what is arguably the first paper to provide genome-wide evidence for neofunctionalization (also in yeast):

M. Kellis, B.W. Birren and E.S. Lander, Proof and evolutionary analysis of ancient genome duplication in the yeast Saccharomyces cerevisae, Nature 2004.

At the time, the Kellis-Birren-Lander paper was hailed as containing “work that may lead to better understanding of genetic diseases” and in the press release Kellis stated that “understanding the dynamics of genome duplication has implications in understanding disease. In certain types of cancer, for instance, cells have twice as many chromosomes as they should, and there are many other diseases linked to gene dosage and misregulation.” He added that “these processes are not much different from what happened in yeast.” and the author of the press releases added that “whole genome duplication may have allowed other organisms besides yeast to achieve evolutionary innovations in one giant leap instead of baby steps. It may account for up to 80 percent (seen this number before?) of flowering plant species and could explain why fish are the most diverse of all vertebrates.”

This all brings me to:

The challenge

In the abstract of their paper, Kellis, Birren and Lander wrote that:

Strikingly, 95% of cases of accelerated evolution involve only one member of a gene pair, providing strong support for a specific model of evolution, and allowing us to distinguish ancestral and derived functions.” [boldface by authors]
In the main text of the paper, the authors expanded on this claim, writing:

Strikingly, in nearly every case (95%), accelerated evolution was confined to only one of the two paralogues. This strongly supports the model in which one of the paralogues retained an ancestral function while the other, relieved of this selective constraint, was free to evolve more rapidly”.

The word “strikingly” suggests a result that is surprising in its statistical significance with respect to some null model the authors have in mind. The data is as follows:

The authors identified 457 duplicated gene pairs that arose by whole genome duplication (for a total of 914 genes) in yeast. Of the 457 pairs 76 showed accelerated (protein) evolution in S. cerevisiae. The term “accelerated” was defined to relate to amino acid substitution rates in S. cerevisiae, which were required to be 50% faster than those in another yeast species, K. waltii. Of the 76 genes, only four pairs were accelerated in both paralogs. Therefore 72 gene pairs showed acceleration in only one paralog (72/76 = 95%).

So, is it indeed “striking” that “in nearly every case (95%), accelerated evolution was confined to only one of the two praralogues”? Well, the authors don’t provide a pvalue in their paper, nor do they propose a null model with respect to which the question makes sense. So I am offering a prize to help crowdsource what should have been an exercise undertaken by the authors, or if not a requirement demanded by the referees. To incentivize people in the right direction,

I will award {\bf \frac{\$100}{p}}

to the person who can best justify a reasonable null model, together with a p-value (p) for the phrase “Strikingly, 95% of cases of accelerated evolution involve only one member of a gene pair” in the abstract of the Kellis-Birren-Lander paper. Notice the smaller the (justifiable) p-value someone can come up with, the larger the prize will be.

Bonus: explain in your own words how you think the paper was accepted to Nature without the authors having to justify their use of the word “strikingly” for a main result of the paper, and in a timeframe consisting of submission on December 17th 2003 (just three days before Hanukkah and one week before Christmas) and acceptance January 19th 2004 (Martin Luther King Jr. day).

Rules

To be eligible for the prize entries must be submitted as comments on this blog post by 11:59pm EST on Sunday May 31st June 7th, 2015 and they must be submitted with a valid e-mail address. I will keep the name (and e-mail address) of the winner anonymous if they wish (this can be ensured by using a pseudonym when submitting the entry as a comment). The prize, if awarded, will go to the person submitting the most complete, best explained solution that has a pvalue calculation that is correct according to the model proposed. Preference will be given to submission from students, especially undergraduates, but individuals in any stage of their career, and from anywhere in the world, are encouraged to submit solutions. I reserve the right to interpret the phrase “reasonable null model” in a way that is consistent with its use in the scientific community and I reserve the right to not award the prize if no good/correct solutions are offered. Participants do not have to answer the bonus question to win.

 

 

 

 

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