Linear Regression Doesn’t Work if the Underlying Process is not Linear

Normally, I would have classified the basic premise of Craig Loehle's recent paper, as summarized at Climate Audit, as a blinding glimpse of the obvious.  Unfortunately, the climate science world is in desperate need of a few BGO's, so the paper is timely.  I believe his premise can be summarized as follows:

  1. Many historical temperature reconstructions, like Mann's hockey stick, use linear regressions to translate tree ring widths into past temperatures
  2. Linear regressions don't work when the underlying relationship, here between tree rings and temperature, is not linear.

The relationship between tree ring growth and temperature is almost certainly non-linear.  For example, tree ring growth does not go up forever, linearly, with temperature.  A tree that grows 3mm in a year at 80F and 4mm at 95F is almost certainly not going to grow 6mm at 125F. 

However, most any curve, over a sufficiently narrow range, can be treated as linear for the purposes of most analyses.  The question here is, given the relationship between tree ring growth and temperatures, do historical temperatures fall into such a linear region?  I think it is increasingly obvious the answer is "no," for several reasons:

  1. There is simply not very good, consistent data on the behavior of tree ring growths with temperature from folks like botanists rather than climate scientists.  There is absolutely no evidence whether we can treat ring widths as linear with temperatures over a normal range of summer temperatures.
  2. To some extent, folks like Mann (author of the hockey stick) are assuming their conclusion.  They are using tree ring analysis to try to prove the hypothesis that historic temperatures stayed in a narrow band (vs. current temperatures that are, they claim, shooting out of that band).  But to prove this, they must assume that temperatures historically remained in a narrow band that is the linear range of tree ring growth.  Essentially, they have to assume their conclusion to reach their conclusion.
  3. There is strong evidence that tree rings are not very good, linear measurements of temperature due to the divergence issue.  In short — Mann's hockey stick is only hockey stick shaped if one grafts the surface temperature record onto the tree ring history.  Using only tree ring data through the last few decades shows no hockey stick.  Tree rings are not following current rises in temperatures, and so it is likely they underestimate past rises in temperature.  Much more here.

  4. Loehle's pursues several hypotheticals, and demonstrates that a non-linear relationship of tree rings to temperature would explain the divergence problem and would make the hockey stick a completely incorrect reconstruction.

  • Jennifer

    Actually, if you read the abstract of the Mann et al. paper you’re referring to, you’ll see that they also did reconstructions which didn’t use tree ring data at all, and the result – that recent warming is anomalous – was robust. Obviously your points 2, 3 and 4 are spurious.

    As for point 1, there is actually a very large amount of data on tree rings from botanists. The response of trees to the various things that affect their growth have been studied for decades. Look up Hal Fritts – he wrote the classic text book on this.

  • Quibbler

    “Actually, if you read the abstract of the Mann et al. paper you’re referring to, you’ll see that they also did reconstructions which didn’t use tree ring data at all, and the result – that recent warming is anomalous – was robust.

    Heh. Read past the abstract, and then read this. You will find that he replaced tree rings with even less reliable data. For example, he used some sediment data which seems, on the face of it, to indicate a radical drop in temperature recently. The authors of that paper explained that the sediments had been badly disturbed recently by agriculture, so recent data could not be trusted as a temperature proxy. Mann ignored that and just flipped the data over! What would have indicated a radical drop in temperature is instead interpreted as a radical rise in temperature.

  • Andy

    “Actually, if you read the abstract of the Mann et al. paper you’re referring to, you’ll see that they also did reconstructions which didn’t use tree ring data at all, and the result – that recent warming is anomalous – was robust.”

    What utter claptrap, I suggest a visit to climateaudit is needed to see how “robust” Mann’s evidence is.
    The tree rings don’t work.
    The lake sediments (inverted) don’t work.
    The speleotherms don’t work.

    Oh but there’s a “consensus” so it must be true……and the rain in spain still falls in…Africa?

  • Jennifer

    Quibbler, how much of the paper did you read? Did you get to the specific discussion of the Finnish sediments, and the graph showing reconstructions which excluded them?

    Andy: never trust blogs over peer-reviewed journal papers.

  • Andy

    Really Jennifer, would you care to point out what is wrong with the analysis?

    Would you care to tell us, show us why the inverted sediment record is an appropriate modification/ appropriate use?
    Would you care to tell us, show us how the analysis regarding speleotherms is incorrect?
    Would you care to show us why an assumed linear relationship is useful, when the relationship is clearly none linear?

    Good science is good science no matter where it is located.
    The problems of peer review and climate science are well documented and it is hardly the gold standard it should be.

    What is evident is that Mann’s attempts at reconstructing a temperature history first time were flawed and second time round are no better, similar standards are found in your inevitable and tedious attempts to decry valid criticisms on this blog.

  • cfdman


    Source of arguments don’t matter – argue the point. Anything else is a concession.

    Your starting to fade already….same as always.

  • joshv

    Andy, Andy, Andy – you should know better than to ask direct pointed questions of Jennifer. She will get insulted and start ignoring you – like a good scientist does when they encounter uncomfortable questions they can’t answer.

  • Andy

    Silly me!

  • Anne

    NASA’s Gavin Schmidt can’t even agree that temperatures have been lower since the 1998 peak.

    In his latest piece on Real Climate, he attacks a journalist for not checking her facts,
    and then mealy-mouths around that facts are not certain.

    When I questioned one of his assertions (where he used this temperature plot (I don’t know where its from, there are no references..typical)

    He gave these “legalese” defensive answers. Typical of his approach, he, the scientist, can say anything he wants, without supporting his positions, but laymen must check their stuff against his facts.

    See comment 71:

    # Anne Says:
    2 December 2008 at 10:59 AM

    “She then throws in a few completely untrue ‘facts’ (i.e. “in every year since 1998, world temperatures have been getting colder” (not) ”

    maybe she checked Wikipedia

    which confirms her statement

    [Response: Not really. There are two issues – first, that there is uncertainty in what the actual anomaly is in any one year (different products with different assumptions give different anomalies) – in the NCDC or GISTEMP analyses your interpretation of the statement (that she meant that 1998 was the record warmest year) is not true. However, that is not the interpretation I took from her statement (actually David Bellamy’s) at all – “In every year… world temperatures have been getting colder” implies to me that specifically each year since 1998 has been colder than the last, and this is not true no matter what product you look at. And finally, and this is probably the most important point – these individual year rankings at the level of less than 0.1 deg C are pretty much meaningless and anyone drawing conclusions from them is just fooling themselves. They are simply not a robust measure of global warming – long term trends are much more relevant. – gavin]

    I didnt bother to respond, since Gavin edits comments first.


  • “you’ll see that they also did reconstructions which didn’t use tree ring data at all, and the result – that recent warming is anomalous – was robust.”

    To argue that the study is “robust” is to really argue from an ideological position. It’s one of the worst studies ever done.

    Everyone should read the links posted but in summary:

    If a particular series happens to respond in the opposite way to what Mann wanted, i.e., shows recent cooling instead of warming, Mann flips the proxy upside down in order to still get the result he wants. He also uses proxies that have been stated by the providers of the proxies to be corrupted for recent decades. When a very small number of these types of proxies are removed, a completely different result is produced. There is nothing “robust” about a result like that.

    This is a bad study done by a lousy scientist. Cranks are everywhere but a real scientist who does work like this hurts the reputation of science itself.

    AGW believers who try to defend appallingly bad work, show that they are interested in defending a conclusion rather than looking at the science in an honest way. One bad study doesn’t prove or disprove AGW so let’s just admit it’s bad and move on to looking at other issues. There is more to AGW than just one (exceptionally) bad study.

  • Paul Maynard

    Re Jennifer

    The so called peer review of the first Mann opus failed to spot that his conclusion contradicted the known historical record from the Vikings in Greenland, to the paintings of Brughel, cathedral building, River Thames freezing (not done that since 1821 just before the Victorians started driving round in their 4x4s) and the flat thermometer record from the Armagh observatory. All this could be dealt with if the amply funded Mann team could go and update the tree record of the past 30 years which would confirm that his findings are peer reviewed (by his mates) nonsense.



  • Andy

    Where has the usually loud and volumous Jenny gone…………

  • Gavin Schmidt is a smart guy but as always deceptive…

    “They are simply not a robust measure of global warming – long term trends are much more relevant.”

    Without specifying what would be considered a ‘long term trend’. Apparently it’s more than 10 years. When GISS states we’ll see record temperatures hit on a 2-3 years basis (visit their website for confirmation of this), or the MET Office predicts specific years:

    It seems to be OK. But when things don’t pan out, the ‘opposition’ is being unreasonable by pointing to nearly 10 years of contradicting data because it’s too ‘short’ and such individuals are apparently fooling themselves.

    Each side should be held to the same standards of criticism. It’s a strange kind of one sided argumentation that always sounds reasonable if you don’t stop to think about what is actually being said.

  • hunter

    The frustrating thing is that even as the weaknesses and deceptions of AGW become more obvious, the momentum behind political changes demanded by AGW promoters are becoming more difficult to avoid.
    The best outcome at this time will be for a Kyoto style construct- one that does nothing except keep bureaucrats well entertained at nice phot ops.
    When I think of how much progress could have been made in worldwide standards to burn coal more cleanly, in next generation nuclear power plants, and in domestic drilling, instead of chasing the idiocy of CO2, it is clear we have wasted a lot of time on Hansen & co.’s scams.

  • The worst thing about the mann paper was the fabrication of the data in 90 percent of the proxies. If the proxies didn’t fit well, the data was actually chopped off and new data was added. The non-linearity just has to do with the fact that your instrument (in this case trees) must be verified. Mann08 does it by correlation. Dr. Loehle’s point is that if we assume some trees only react to one temp and not moisture, that doesn’t mean that the same tree would grow even faster at a higher temp. He then pointed out a number of studies which back up his claim that this may be the case.

    Here is my post which led to the CA post.

  • Andy

    Ohhh jennifer answer the questions, we know you are reading this having moved onto the next thread, come on now don’t be shy!

  • The other coyote

    Can somebody answer for me whether moisture is more important to tree growth than temperature? Has Mann and his ilk ever considered whether moisture is more important to tree growth than temperature?

    I’m not botanist, but I am an enthusiastic gardener. Unfortunately for me, I live in north Texas. Between the wind, the cold, the sometimes dry climate, and the blistering Texas summer sun, it’s not the greatest place to grow a showplace garden.

    I live west of Dallas, in an area that is dry 4 years out of 5, bone dry two of those four years, and rainy as all get out one year out of five. It’s brutally hot 5 summers out of 5, it’s just that two of them are a withering heat, two are a dry heat, and one is a mosquito-infested sauna. I have noticed over the past 12 years that my little tree collection doesn’t grow at all (and in fact, goes sort of dormant) in the withering years, grows a little in the dry years, and grows like crazy in the sauna year. I should add that I don’t water my trees after the first six months, because if something around here doesn’t grow really deep roots (which they will do if they have to look for water), it will blow away.

  • Andy,

    Dendroclimatologists look specifically for trees hypothesized to be non-moisture limited. This is usually based on observations of the terrain the trees are located. Often but not always they will then use statistical correlation to temp to demonstrate whether they are correct and throw away the rest (very bad science).

    There is no demonstration or verification of linearity, no demonstration that trees are responding to temp and no data to show how much certain species react to temp change. So basically you can say moisture is known to have a huge effect on the result.

    Mann in particular ignores all of the rules for dendro sites and piles them all together. (even with the knowledge that some trees he uses are collected specifically for moisture records). Mann’s mode of operation over several papers is to use any old proxy and let the BS math sort it out.

  • andy

    Look below not above the post Jeff!