Trenberth on Statistics

I don’t think that people entirely appreciate the absurdity of the views of Gavin and Rasmus that consideration of persistence in climate somehow "pitches statistics against physics". If climate scientists are seeking more familiar authority for just how preposterous this claim is, they need look no further than Trenberth [1984], previously discussed on this blog here, which categorically asserts that many statistics will be seriously biased by autocorrelation. The forms of persistence discussed by Trenberth are less severe than the ones considered by Cohn and Lins and Koutsoyannis, but the persistence issue then becomes a matter of degree (rather than of "physics").

Here is an extended excerpt from Trenberth:

Climate is usually regarded as dealing with the average behavior over a relatively long time of the climate system and is not concerned with the daily fluctuations called weather… The focus of many climate studies is the difference between any climatic states that can be distinguished from the climatic noise. This “climatic signal” may arise from influences truly external to the climate system or it may arise from slowly varying modes of the entire climate system…Two other aspects are also important in time series analysis of meteorological parameters…(2) persistence, which gives rise to a lack of independence in the observations. Leith (1973, 1975) discussed the problem of signal-noise ratio in the predictability of climate and showed that the magnitude of the noise was related to persistence in the atmosphere…

However this climatic noise and the persistence, along with the finite size of the samples must be taken into consideration when computing statistics of the circulation or the resulting statistics may be significantly biased

Trenberth pointed out that”many publications have failed to take note of the potential problems” and goes on to point out severe biases in the estimation of variances, covariances and autocorrelations. He concluded as follows:

This paper has pointed out the need to take persistence into account in estimating population statistics from a finite sample”.

It’s pretty hard to see a more on-point and more categorical refutation of realclimate’s view of the statistical issues involved with persistence. I’ll probably leave these matters alone for a while, but will conclude by pointing out the following from the Climate Analysis Group, University of Reading:

Climate by definition is the statistics of weather. It therefore makes a lot of sense that climate researchers know something about the important subject of statistics.

It’s too bad that so many climate scientists, who hold themselves out as authorities, actually have such sketchy knowledge of statistics and that, despite this, have generally failed to involve statistical professionals in their work.

By the way, the University of Reading group has some interesting-looking (I haven’t inspected the packages) information on specialized R packages here and here .

Reference: Trenberth, K. [1984], Some effects of finite sample size and persistence on meteorological statistics. Part I: Autocorrelation. Monthly Weather Review, 112, 2359-2368


38 Comments

  1. Dave Dardinger
    Posted Jan 5, 2006 at 11:54 AM | Permalink

    It’s too bad that so many climate scientists, who hold themselves out as authorities, actually have such sketchy knowledge of statistics

    I can somewhat feel sorry for the Hockey Team in this situation. I used to play quite a bit of chess and went to tournaments when I could. As someone went who just played chess I was quite good, but I was only average in small-medium tournaments and when I watched the masters play I could sometimes understand their moves, but could not predict them, at least not during crucial situations.

    Anyway as far as statistics goes, I figure the Hockey Team is well above average when it comes to knowing statistics. Among scientists they’re probably average. But when they try claiming to be experts they’ll get wiped off the board every time when facing statistics masters. Not for nothing is humility known as a virtue.

  2. Steve Bloom
    Posted Jan 5, 2006 at 2:38 PM | Permalink

    There is a much more recent article by Trenberth in which he discusses how it is reasonable to expect strengthened tropical storms due to increased SSTs even though such a hypothesis may fail statistical tests (although note that more recent studies by Emanuel and Webster et al did provide statistical proofs); see http://www.sciencemag.org/cgi/content/full/308/5729/1753. Key quote: “Because of the weakness associated with statistical tests, it is vital to also gain a physical understanding of the changes in hurricane activity and their origins.” Weakness? Hmm.

    Speaking of statistics, I have a completely non-partisan question: There are many, many university departments filled with PhD statisticians who presumably are the most qualified to comment on all of this stuff, and yet they seem to be heard from rarely if ever. Why?

  3. Posted Jan 5, 2006 at 3:07 PM | Permalink

    Dear Steve,
    the 1984 Trenberth paper has 34 citations,

    http://scholar.google.com/scholar?q=link:qrqNCU50vwsJ:scholar.google.com/

    and the first of them (from 1997) proposes a method to evaluate the statistical significance of a correlation when the data are serially correlated. Do you know this one, for example?

    Dave D: Your description of their being average statisticians among the scientists and above the average among the climate scientists reminds me of the well-known joke. Do you know what happens when a particle physicist switches to climate science? The average intelligence of both groups – particle physicists and climate scientists – jumps. ;-) (I am confident that the climate scientists won’t be offended because they can’t calculate what the previous sentence means.)

    Is not this whole business about evaluating the climate and confirming various conjectures about statistics? I just don’t know how can one do it properly without statistics. If it were an exact science where quantities are measured with the accuracy of 10 digits and where the truth is known independently by “mathematicaly beauty” and where the experiments confirm exactly known calculational schemes that only depend on a few parameters, the situation would be different. (This was an attempt to justify that the people in gravity and high-energy theoretical physics need to know less statistics than those in the “dirty” sciences.) ;-)

    Back to the autocorrelation assumptions: it is a dogma that the world must behave as in the holy GCM scriptures that can’t be improved anymore. They, much like God, are perfect already and only sinners would question their omnipotency.

    Thanks, Gavin and Rasmus, for the holy word.

    I am completely baffled by Steve Bloom’s choice of the “key quote” in the new Trenberth article and its interpretation. Does Steve Bloom suggest that Trenberth believes that there is evidence that CO2 emissions increase the rate of tropical storms? That may be enough for Trenberth to sue Bloom if he got a bit annoyed. ;-) Those who have learned how to read will see something very different in Trenberth’s (new) article:

    “Despite this enhanced activity, there is no sound theoretical basis for drawing any conclusions about how anthropogenic climate change affects hurricane numbers or tracks, and thus how many hit land.”

    I am also pretty sure that Steve Bloom misunderstood the statements about the weakness of statistics.

    Amen,
    Luboà…⟍

    P.S. Statisticians either do not care or do not want to work on things they’re not paid for – but most of them probably don’t want to irritate their pro-environmental friends from other fields by identifying bugs in the way how statistics in climate science is done.

  4. Steve McIntyre
    Posted Jan 5, 2006 at 3:52 PM | Permalink

    Re #2: I don’t know. They probably never thought about it.

    One problem might be the barriers to entry on the data side – not many people are prepared to go through the quasi-litigation to get at the data. At this point, the excuses and prevarications amuse me so I don’t mind working at it, but I suspect that a young PhD wouldn’t have the same patience or, for that matter, would lack the experience in dealing with quasi-litigation. Of course, the data concealment should not happen, but that’s a different story.

    Another is that much of the empirical work is pretty low-brow in terms of mathematical statistics and is simply data handling. If you’re young and smart, you probably think that that’s beneath your dignity. Finding seams in large data sets is a bit of a knack, which you don’t really learn at school. I’ve got some pretty good skills at it, which I’ve picked up over the years largely though other activities. There’s nothing high-falutin about it, but you still have to do it and not everyone can. You also have to do it efficiently to make it worthwhile and I’m probably a little quicker at it than a young PhD. I didn’t have the same skills when I was in my 20s, although my math was vastly more athletic than it is now.

    There are some interesting theoretical statistical issues, but many of them have emerged only by data-driven analyses. I might actually take some credit for discerning some of these issues within the climate corpus and think that there are some topics which would be of interest to such students. I haven’t touched on a number of these topics at this blog, but I have a few.

    Given the attention that our work has obtained, I’d have thought that some other people would have gone after some of the other multiproxy studies, if only because such articles would attract attention. I’ve got some articles in the pipleine and TCO gets on my case for not finishing them off. Right now there’s a lot of low-hanging fruit and lots of opportunity for enterprising stats students. I’d be happy to give suggestions to any that show up.

  5. John S
    Posted Jan 5, 2006 at 4:08 PM | Permalink

    Re #2

    You can generally tell by the comments whether someone has studied statistics and I would suggest that there are a number of contributors to the discussions who have studied statistics (and some who quite obviously, and in some cases admittedly, have not). Could you imagine someone popping up and announcing – “I have a PhD in statistics and think you are all smoking weed”? That would make them sound like a bit of a tosser as far as I am concerned.

    And actual PhDs in statistics (as opposed to a PhD in another discipline who specialised in statistics) are much more likely to be theoretical than applied and associate with pure mathematicians rather than do applied work. I would suggest that the greatest contributions would be from PhDs in economics/econometrics because they are actually interested in applied issues and have the requisite statistical knowledge (e.g. Kaufmann and one of his co-authors, Stock). And you already have a number of them coming out of the woodwork.

    PS. And sometimes its just too amusing to watch blog scrags and making sensible comments would just spoil the fun.

  6. Frank H. Scammell
    Posted Jan 5, 2006 at 4:33 PM | Permalink

    Steve, The shaft of the “hockeystick” is broken. As far as I can tell that is entirely due to you (with some help from Ross). If it’s broken , then it is of no use to the “hockeyteam”, and should be discarded. They will “move on (moveon.org?)”. The blade of the “hockeystick” still remains, however. It’s clearly wrong (yes, there is some global warming – look at the satellite data), but it doesn’t warrant any great alarm. How do we know it is anthropogenic? What is 1-2ËÅ¡ C compared to the daily and annual variations? It still remains unclear whether the GISS raw data is available, and whether UHI effects can be extracted, but this is still a statistics issue. See Warwick Hughes on Siberia (and probably N. Canada).

  7. Armand MacMurray
    Posted Jan 5, 2006 at 5:22 PM | Permalink

    Re: #2
    I am also very surprised by this. It is common in many fields of biology that deal with quantitative real-world data (e.g. clinical trials, disease incidence correlations, etc) to include a biostatistician as part of the team in order to correctly design and calculate the statistical tests. This doesn’t seem to happen in climate science (at least in the proxy world). Like Steve B., I ask “why not?”. If only the hockey team had had a stats goalie!

  8. Ian Castles
    Posted Jan 5, 2006 at 6:29 PM | Permalink

    Re #2, #5 and #7. Statistics is a broad field with many sub-specialisations, as the President of the International Statistical Institute, Sir David Cox, F R S, explained in his opening address at the ISI session at Istanbul in 1998:

    “Our subject spans so many different activities of science, technology, public affairs, business and everyday life. Thus our programme [at ISI sessions] ranges from pure probability theory, … applications in astrophysics and astronomy and physics … through to the major issues of government and business statistics and public policy that stem from those that so interested our founders nearly 150 years ago.”

    In correspondence with the Chairman of the IPCC in 2002, Professor David Henderson (a former Head of the Economics and Statistics Division at OECD) and I (a former Head of Australia’s national statistical office and a former President of the International Association of Official Statistics, a section of the ISI) urged that national statistical offices, the ISI and the United Nations Statistical Commission be involved in the preparation of the Panel’s Fourth Assessment Report. The Australian Government’s Submission to the IPCC on the Scoping of the Fourth Assessment Report (AR4) in March 2003 recommended that the Panel “draw upon a broader range of … statistical expertise in the development and explanation of the economic aspects of the emissions scenarios content of the AR4″. The Submission also suggested that “the second scoping meeting of the AR4 should engage a broader range of experts and ensure representative participation is sought at a national level from the … statistical … profession, and include representation from key international bodies such as the United Nations Statistical Commission.”

    These suggestions were not taken up, for reasons that seem obvious. I agree that experts in other fields of statistics – econometricians, biostaticians, survey statisticians, etc – could and should be contributing to various branches of climate change science. I welcome the fact that some are now “coming out of the woodwork.”

  9. Steve Bloom
    Posted Jan 5, 2006 at 9:52 PM | Permalink

    Lubos, recall I said Trenberth’s statement referred to “strengthened tropical storms,” whereas you referred to the “rate of tropical storms” and quoted Trenberth’s assertion that there is no basis for thinking that GW is affecting “hurricane numbers or tracks.” Let’s see, is “strength” perhaps different than “rate”/”numbers” or “tracks”? Either you a) are truly not paying attention, b) think everyone else in the world is so stupid they will not notice internally inconsistent arguments from you, or c) are having fun wasting time and think so little of others that you happily waste their time too without asking if they want to do that.

    Thanks to all who answered constructively (a word choice that of course excludes Lubos) on the statistician question. I asked it after reading the comment about “statistics masters,” which caused me to wonder why I’ve never seen a single PhD statistician involved on any side of this debate. Too much number-crunching combined with a lack of anything of theoretical interest to a statistician seems to be a fair answer, though.

  10. Doug L
    Posted Jan 5, 2006 at 10:09 PM | Permalink

    Re: Hurricanes

    Perhaps there is insufficient data on hurricane strength, for anyone to say anything definitive about it. Hurricanes of cat 5 are only that way for a short period. They often don’t get the chance to hit land at that strength. These would be missed in the data from the old days.

  11. Martin Ringo
    Posted Jan 5, 2006 at 11:20 PM | Permalink

    Re: #2 and climate and econometrics

    The following papers will give an idea of the practice of econometrics to the trend in temperature issue. If you want to understand the statistics, you may have to do a little work. Vogelsang was very helpful and sent me his Gauss code for the statistics he developed (elsewhere) and used. (I haven’t tried either Stern or Kaufman.)

    Fomby, T and T. Vogelsang, “Tests of Common Deterministic Trend Slopes Applied to Quarterly Global Temperature Data”

    http://faculty.smu.edu/tfomby/Warming.pdf

    Fomby, T and T. Vogelsang, “Vogelsang The Application of Size Robust Trend Statistics to Global Warming Temperature Series”

    http://www.people.cornell.edu/pages/tjv2/globwarm.pdf

    Stern, D, and R. Kaufmann, “Is There a Global Warming Signal in Hemispheric Temperature Series?”

    http://www.bu.edu/cees/research/workingp/pdfs/9903.pdf

    I think these papers are simply working papers, and I do not suggest them for data or conclusion, merely for the examples of econometric technique applied to non-economic time series.

    Bon appetite

  12. Steve McIntyre
    Posted Jan 5, 2006 at 11:59 PM | Permalink

    #11. Martin, you concluding phrase reminds of a puzzle that I’m trying to figure out (and I hope that you pardon an indulgent grandfather.) My wife and I were playing with our 5 year old grand-daughter, who had taken into her head that she would plan the menu of a restaurant – which she named Ekovia, for some reason known only to her. After a while, she announced that her next project was a restaurant for dogs, for which she also had a name – Bon Appetit, a pun which is far out of her reach. I presume that there must be some cartoon for kids, where the writers wittily endowed the dogs with a such a restaurant, but for now it remains a mystery. But I can no longer hear the phrase without thinking about her doggie restaurant and the clever bilingual pun.

  13. ET SidViscous
    Posted Jan 6, 2006 at 12:28 AM | Permalink

    Okay Steve Give it up. I’m far from Bilingual, but I spend enough time around Quebecois that I should be able to get the Pun, but I’m at a complete loss. Maybe the dog bit is a bit of obfuscation.

    I’ve also noticed that Quebecois (some) are fascinated with Puns, but don’t seem able to do them well.

    On a side note, how much you want to bet there is a restaurant in a Western German city near the Belgian border (short little tank ride) known as Bonn Appetit.

    Oh yeah, this dog restaurant a nice Plaice?

  14. ET SidViscous
    Posted Jan 6, 2006 at 12:40 AM | Permalink

    Damn I get it now.

    My pronunciation was off. Sorry for the distraction.

  15. James Lane
    Posted Jan 6, 2006 at 2:30 AM | Permalink

    Ian Castles, you’re being dissed over at John Quiggan’s website, ypu might like to pay a visit.

  16. Steve Bloom
    Posted Jan 6, 2006 at 2:34 AM | Permalink

    Re #10: You really need to read the Emanuel and Webster papers to see the complete analysis, but the point you make was taken into account. I understand the journals will have a fair amount more on the subject this year.

  17. Peter Hearnden
    Posted Jan 6, 2006 at 5:48 AM | Permalink

    Re #6

    What is 1-2ËÅ¡ C compared to the daily and annual variations?

    Enough to be the difference between LIA conditions in Europe and the balmy conditons we have these day. A lot!

    You don’t seem to understand the difference between weather and climate.

  18. Ian Castles
    Posted Jan 6, 2006 at 7:41 AM | Permalink

    Re #15, thanks James. I’ve posted a comment on John Quiggin’s website.

  19. Posted Jan 6, 2006 at 7:57 AM | Permalink

    Dear #9 Steve Bloom,

    Trenberth and I do not talk just about “rates” but about the “hurricane numbers” which includes the strength. There is no evidence of a relation between the hurricane numbers and CO2 emissions, and this includes the average strength, both I and Trenberth say.

    Don’t be silly. Of course that if you could raise the intensity of hurricanes, you could also raise their rate because a higher fraction of tropical storms would be promoted to hurricanes.

    Best
    Lubos

  20. John Hekman
    Posted Jan 6, 2006 at 2:39 PM | Permalink

    Lubos
    Re: #19. Very polite correction of Steve Bloom.

    Re: #3. Your joke about the particle physicist changing to climate science is a little condescending. It also doesn’t work unless you set it up that the physicist is below average in his field. This is actually a version of a very old joke that is to be used with reference to a specific individual. I heard one academic say about another academic, “And then he moved from Yale to Harvard, and the average IQ at both institutions rose as a result.” See, now we have only insulted Harvard. That won’t offend anyone, right?

  21. Larry Hulden
    Posted Jan 6, 2006 at 3:28 PM | Permalink

    RE #3 ??????
    “the 1984 Trenberth paper has 34 citations,
    ……………………..
    and the first of them (from 1997) proposes”
    Unfortunately my English is quite limited and I hope somebody could explain to me this statement.

  22. Steve McIntyre
    Posted Jan 6, 2006 at 4:31 PM | Permalink

    #21. You’re thinking about references listed in the paper; the google link lists papers that have cited Trenbrth [1984], the first one being just the first in the google list, not the first one ever.

  23. IL
    Posted Jan 6, 2006 at 5:28 PM | Permalink

    #20 I guess the really stinging implication of the joke is that ONLY a very below average physicist would move to climate science.

  24. Posted Jan 6, 2006 at 5:53 PM | Permalink

    Dear John #20,

    I think that these jokes are just what they are – innocent jokes – and of course that you can’t insult anyone at Harvard because we will always humbly treat such comments as jokes, especially if the other school you refer to is the school known as “Harvard for dummies”. :-)

    There is one more climate-science-specific thing about the joke. Note that when the particle physicist moves to climate science, it must have been – as all intelligent readers understood – that she was below the average of particle physics, but she will still be above the average of climate science. So both averages will increase.

    But what’s funny is that the typical global warming advocates will have problems with this “double-growth” because they tend to think that the overall average IQ should still be unchanged much like the global average temperature – so how it can change just by moving one person around? It’s just like if you reclassify a weather station on the equator from the Northern Hemisphere to the Southern Hemisphere, and as a result, both averages for both hemispheres will go up. How it can ever be?

    Well, the paragraph before the previous one makes it clear that the average may change nevertheless – in a doubly positive direction. It changes because the weights are different.

    It indeed matters how we exactly define the average global temperatures. Different definitions will lead to different averages, different fluctuations, different autocorrelation rates, and other characteristics will differ, too. Defining and describing an average global temperature may be a very subtle thing, and it is very questionable whether it is a useful concept at all.

    All the best
    Lubos

  25. Frank H. Scammell
    Posted Jan 6, 2006 at 7:25 PM | Permalink

    Peter,- Re #17. Perhaps I do. If you are willing to concede the existence of the LIA, how about the MWP? If both, how can the last 2K years be viewed as stable (please define stable)? The intent of the 1-2ËÅ¡C comment is to point out that this signal is being extracted from a very large signal (the daily and annual variations – non anthropogenic except for UHI) and naturally one should expect a fair amount of noise. (It should be noted that these small numbers result from the differencing of large, nearly equal numbers and are an additional source of ropundoff/truncation errors as mentioned previously). To my knowledge, there is is no argument that physically links the magnitude of the rising CO² signal (essentially linear except for an annual (sine?) variation) to the “noisy” tempurature residual. There is a lot of handwaving about the climate sensitivity to CO² increases, but if it is significant, it should follow the annual variations in CO².It doesn’t. Furthermore, let me reassure you that, if you build a GCM that only handles positive feedback, then you will see the effects of positive feedback in the output. How do your models explain the transitions from ice ages to warm and back? For that matter, how could we have transitioned from the MWP to the LIA?

  26. Ron
    Posted Jan 6, 2006 at 10:41 PM | Permalink

    Well, Frank, obviously the sun must have cooled (or its effect otherwise blunted) to transition from the MWP to the LIA. Then CO2 forcing warmed us to the present. Sun effects only cause cooling, never warming. Only GHGs cause warming, and anthropogenic GHGs at that, in the current era.

  27. Frank H. Scammell
    Posted Jan 7, 2006 at 11:05 AM | Permalink

    Ron,- “Sun effects only cause cooling, never warming.” A strange “diode” effect that I am not familar with. Could you explain? Sure, I understand the “or its effect otherwise blunted” argument. The Earth’s absorption, reflectance, scattering, etc. can all change over time, but these are all passive effects. They are modifiers of the intercepted energy from the sun. The models, I believe, all view the Earth as a blackbox – with no internal energy source. Thus the blackbox radiates (or reradiates) to space only energy it has originally received from the sun. Thus, if the sun can only cool, then, in the long run, the earth can only cool.
    “Only GHGs cause warming, and anthropogenic GHGs at that, in the current era.” A fine authoritative statement. But, no further explanation? references?
    OK, let’s get rid of the absorbers (a bit of a problem, since we are dependent on plant life for our food and O2, and our CO2 feeds plants – very complex) by sequestering or whatever. Note that we didn’t get rid of the energy. Our blackbox only allows energy disposal by radiation to space. Have we fixed the problem? Doesn’t seem so to me. What do you suggest?

  28. Dave Dardinger
    Posted Jan 7, 2006 at 12:47 PM | Permalink

    Er, Frank,

    That was sarcasm by Ron, not a real assertion.

  29. Ron
    Posted Jan 7, 2006 at 1:50 PM | Permalink

    Sorry, Frank, I forgot my [sarcasm][/sarcasm] tags.
    I was practicing my “moron-speak” in case I ever decide on a career in journalism or cli. . . . no I won’t say it.

  30. Frank H. Scammell
    Posted Jan 7, 2006 at 2:41 PM | Permalink

    Whew! Thanks Dave and Ron for the explanation. Sorry I missed the sarcasm, so much I read on other sites seems to me to be “moron-speak”. I thought I was beginning to loose it. A career in journalism or cli. sounds like it would be “interesting”.

  31. JerryB
    Posted Jan 7, 2006 at 5:40 PM | Permalink

    Re #6

    “It still remains unclear whether the GISS raw data is available, and whether UHI effects can be extracted, but this is still a statistics issue.”

    Frank,

    GISS raw surface station temperature data are available. Are there some other GISS data, or perhaps some other group’s data, that you have in mind?

  32. Posted Jan 8, 2006 at 8:02 PM | Permalink

    If anyone is interested how our friend who is a senior climate modeller tried to calculate an extremely simple Brownian model of the climate using Monte Carlo methods and overshot the correct result by 60 percent :-), read

    http://motls.blogspot.com/2006/01/connolley-in-monte-carlo.html

  33. Paul
    Posted Jan 8, 2006 at 8:34 PM | Permalink

    Frank,

    I’m sure you’re familiar with the Junkscience.com global temperature. Check there for how they put theirs together. You might find the beginning threads of what you’re looking for, if you haven’t already.

  34. ET SidViscous
    Posted Jan 8, 2006 at 8:40 PM | Permalink

    :)

    Paul Luboà…⟠has a link to the junkscience Kyoto countdown on his site. I think he’s aware.
    ;)

  35. Posted Jan 8, 2006 at 10:39 PM | Permalink

    Dear ET,

    when Luboà…⟠is aware, does it mean that Frank, is aware, too? :-)

    Thanks, Luboà…⟠non-Frank

  36. ET SidViscous
    Posted Jan 8, 2006 at 11:10 PM | Permalink

    DOH!

    I done dere screwed up dere.

    Sorry all

  37. ET SidViscous
    Posted Jan 8, 2006 at 11:13 PM | Permalink

    And apologizes to Luboà…⟠and Paul

  38. Paul
    Posted Jan 9, 2006 at 7:41 AM | Permalink

    On the problems of trying to model and forecast endogenous systems, there is a wonderful analagous example in this week’s economist which refers to a paper in Nature(page 72 for those interested).

    It concerns the ecology of coral reefs in the Bahamas. A ban on fishing in the 1980s fueled conservationists fears that risnig populations of predatory fish would lead to a drop in herbiverous fish that keep the reefs clear of seaweed (sounds like a predicted response to a forcing in a closed system to me).

    In the end the system responded in such a way that the seaweed did not take over the reefs. Why? Because the feedback was not as simple as assumed. In short, although the numbers of predator fish increased, there was also an increase in the number of larger (and less edible) herbivourous fish that eat more.

    Result, more fish less seaweed, more attractive reefs, conservations off to their next alarmist issue.

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